text:
## Thursday, October 12, 2006

### Arché Logic Group

As I've mentioned earlier, this semester Arché has started a new logic group (boringly called 'Arché Logic Group', but unofficially named 'The Deviants'). There's now an Arché Twiki page for the seminar, although still quite provisional. The first part of the semester we're working through Graham Priest's "classic" An Introduction to Non-Classical Logic , preparing to go through a forthcoming publication by the same author. The usual (Scandinavian) suspects have been spending late office hours going through all of the exercises. True, there is quite a bit of tedious and repetitious work, but we have encountered a great deal of interesting material in some of the harder parts of the book.

However, this is only the warm-up: In early November we fly in the big guns. Then Priest himself will join in the fun when we proceed to uncharted terrain. By then there will hopefully be some comments on the first book on the Twiki page. I'll come back with more later.

Meanwhile: Today's session reminded me of a theorem that a friend of mine persistently has brought to my attention in another context. This is Glivenko's theorem (Glivenko 1929), which Priest has relegated to a footnote on p. 103.

Theorem. (a) If Gamma |- A in classical logic then ¬¬Gamma |- ¬¬A in intuitionistic logic, assuming that CL negations and implications are replaced by their IL counterparts. (b) If ¬Gamma,Sigma |- ¬A in CL then ¬Gamma,¬¬Sigma |- ¬A in IL.

Priest suggests that this theorem says that although IL is a sublogic of CL, CL is "in a sense" contained within IL. Of course, it is this "in a sense" which makes the theorem philosophically loaded - how are we to understand the resurfacing of classical truths within the intuitionistic system? Has it any bearing on the philosophical projects associated with the logics? Etc. I will not get into that here. Rather, I just want to point to a corollary of Glivenko's result that I was unaware of. Priest directs our attention to what he calls the "unobvious" fact that if |-A in classical logic, and A contains no logical notation except negation and conjunction, then |-A in intuitionistic logic as well (with intuitionistic conj. and neg.). Somewhat surprising perhaps, since intuitionistic and classical negation usually are thought to behave quite differently.

I checked out Priest's reference, which is to the standard Introduction to Metamathematics (1952) by Kleene (see pp. 492-493). The proof is something like this: Consider A as a conj. of n formulae (n is 1 or greater), where each conjunct is itself not a conjunction. Then, all conjuncts are either propositional variables or negated. Furthermore, if A is provable in CL, then all of its conjuncts are provable; but no propositional variables are provable. So, the conjuncts are all negated, and by Glivenko's theorem (b), they are provable in IL.

Any comments on this - philosophical or logical? (Paul Simon, if this isn't an invitation, nothing is.)

Hopefully, the ALG will prosper in the next few years, warranting the 'logic' part of the Arché research description. I'm already conspiring to make it into a proof-theory seminar next semester - one interesting suggestion was to read the manuscript for Greg Restall's forthcoming Proof-theory and Philosophy.

Update: Thanks to Aidan for this link.

Aidan said...

Stephen Read's course on algebraic logic two years ago helped me a lot in seeing the 'sense' in which CL is contained in IL. It's really worth talking to him about it.

Philosophically, CL is going to be acceptable to the intuitionist so long as we have some guarentee that none of the statements in question are undecidable. So it would be interesting to be clearer on the relationship between the fragment involved in the proof of the corollary of Glivenko's result and decidability. If restricting our attention to this fragment guarentees decidability, then this fact may be "unobvious", but it wouldn't be surprising. (For some discussion of this partial acceptance of CL, see here.

Aidan said...

Ps. The maths project spent a huge amount of time on the status of HOL, the modality conference has spent countless hours on modal logic, and the vagueness project gave a vast amount of time to exploring non-classical logics. If you want a proper challenge, try justifying the 'mind' in Arche's description....

Ole Thomassen Hjortland said...

Aidan:

Thanks for the link. I've put it as an update on the post.

Some questions:

"Philosophically, CL is going to be acceptable to the intuitionist so long as we have some guarentee that none of the statements in question are undecidable."

I assume that the 'statements in question' refer to the conj.-neg. formulae in the corollary? Logically, of course, the corollary itself ensures the decidability of these formulae, since provability in CL gives provability in IL. But are you perhaps alluding to a non-formal concept of decidability?

"So it would be interesting to be clearer on the relationship between the fragment involved in the proof of the corollary of Glivenko's result and decidability."

Or, are you rather refering to the decidability of the statements in the post, i.e., the proofs? I agree that it is interesting whether the proof is itself intuitionistically acceptable. I haven't looked into that in detail, but as far as I can see, there is only intuitionistically valid reasoning in the proofs (that is, both in Glivenko's theorem and in the corollary). But I'm prepared to be corrected.

Ole Thomassen Hjortland said...

I completely agree about the 'mind' part. Actually, I think they've decided to skip 'mind' in the description when the AHRC funding runs out. 'Logic', however, will still be in there.

Btw, do you know an elegant way of including logical symbols in html?

Paul Simon Svanberg said...

Sorry to have waited so long to comment on a very interesting post. I cannot access internet from home, so there goes.

As I'm working on an article (which you, Ole, have seen the first seed of) detailing the relationship IL-CL, I have too many thoughts on this subject. So I'll stick to a simple observation.

The reason why Glivenko's theorem works, algebraically, is that double pseudo-complement is a closure.

The "unobvious" fact you mention is interesting, but it is a corollary of the fact that all you need of intuitionistic logic to reconstruct classical logic is conjunction and intuitionistic negation.

So we could say that classical logic is the conjunction-negation-universal quantifier fragment of intuitinoistic logic: A rather symmetric sublogic of intuitionistic logic.

In any case, we should dispense entirely with the idea that intuitionistic logic is a sublogic of classical logic. It is simply false. More on this next week...

Aidan said...

Yeah, sorry Ole, I wasn't being clear. When talking of decidability I meant the generalised notion that features heavily in Dummett's writings on anti-realism (and of course others like Wright and Tennant). By the 'statements in question' I meant some class of statements in dispute between the realist and the anti-realist (so mathematical statements, or statements about the past, etc).

When we have some domain of discourse where we are sure there aren't any potentially verification-transcendent (this is basically what I meant by undecidable) statements, the intuitionist can use classical logic and semantics. So this really was an answer to your request for philosophical comments rather than logic ones (sorry for being quite so inexplicit).

Sorry, no idea about how to put logical symbols in.

Pål said...

As a comment on Dummett on undecidability. Dummett has identified three kinds of undecidability that figures in the revisionary arguments: subjunctive conditionals, past tense and quantification of infinite totalities. Without undecidability, Dummett says, the debate between the intuitionist and the classicisit is of no practical consequences, since both their meaning-theories will correspond to the linguistic behaviour of the masters of the discourse in question. So, undecidability is usely taken as a premise in a revisionary argument, even though, e.g. Cogburn takes the possibility of undecidability to be sufficient for revisionary purposes.

Ole Thomassen Hjortland said...

Paul Simon:
"In any case, we should dispense entirely with the idea that intuitionistic logic is a sublogic of classical logic. It is simply false."

I take it that you don't propose that we revise the current def. of sublogic at large. That, of course, will have dramatic consequences for other systems, e.g., the order of modal logics. However, I wonder if you can make a general definition of when L is a sublogic L' s.t. CL is a sublogic of IL, but which avoids undesirable results for other systems. The problem, as I see it, is that such a definition must allow for the corresponding L' to be inside the scope of some L'-constant (e.g., IL negation) when it is *not* the case that the original L theorem is inside the scope of the corresponding L-constant.

Aidan & Pål:

Ad undecidability. Don't know about you, but I find the idea that the *actual* absence of undecidable statements warrants CL for the intuitionist a bit unsettling. I'm more comfortable with what seems to be Cogburn's position, that possibility of undecidability is sufficient for revisionary purposes. There could be, say, mathematical discourses where we don't know of any undecidable statements, but if the intuitionist allows himself CL resources when exploring the field, he might end up proving something he lacks philosophical warrant for.

Paul Simon Svanberg said...

I haven't really thought about how to formulate a general definiton of sublogic which puts CL inside IL. But hopefully such one already exists.

The two ways to answering this question proceeds along the stony road of syntax or the serene path of semantics.

On the one hand, given a logic L as a pair (S , T), where S is its signature and T is its set of provable formulae, we can think of a sublogic L' of L (i.e. L'{L )as a pair (S', T'), such that S'[S and T'[T, where '[' denotes inclusion. However, this approach leaves much to be desired. For example, when are two signatures equivalent? (E.g. when they produce the same set T?) Furthermore, the exclusive focus on syntax is perhaps not very illuminating (endless inductions, term-rewriting etc), seeing that there is always one more x \in T to prove...

On the other hand, one could say that the logic L whose models subsume those of L' also contains L'. I personally think this is good way to think of the ordering of logics. A sublogic L'{L is then the restricting of attention to only certain operations and elements of L. I don't think this will upset anything in e.g. the hierarchy of modal logics, as I think that ordering arose from investigating modal logics from the point of view of universal algebra. (I think Blackburn et. al. writes this in the first chapter of their "modal logic", but my memory has failed me before.)

So, if this semantical approach to sublogics is taken, CL turns out to be a sublogic of IL. The corresponding syntactical fact can of course also be proved. (See e.g. section 2.3 in Troelstra: Basic Proof Theory for a number of embeddings of CL into IL.) But since we almost never consider the syntax of CL as derived from the syntax of IL, the syntactic result is perhaps less convincing. And also, I think, more difficult to get a clear view of.

Still, I'm torn on this. I am hesitant to claim, for instance, that the higher up in the hierarchy L' { ... { L { ... one gets, the logics get more fundamental. In many instances, such orderings will not reflect anything but a notational convenience. On the other hand, I enjoy the idea of sublogics since it seems that given an ordering L' { ... { L, we can isolate a family, perhaps an entire species of logics.

Somebody has probably written a book about this stuff. Anybody know of any literature on this topic?

Ole Thomassen Hjortland said...

I assume we agree that there are at least to usual ways of ordering logics: one proof-theoretical (syntactical) and one semantical. The one which most frequently is used to define sublogicality is the proof-theoretic one: L' is a sublogic of L iff for every A s.t. |-A in L', |-A in L as well; and (if strict) there is an A in L such that |-A but not |-A in L'. This, for instance, is the definition on which the ordinary ordering of the modal systems from K and upwards are ordered (where K is, in a sense, the smallest logic). However, I perfectly agree that by looking at it semantically, the ordering is turned up-side-down: say that L' is a sublogic of L iff every L'-model is is an L-model; and (if strict) there is an L-model which is not an L'-model. In other words, the ordering would now have the modal system K as the "largest logic", model-theoretically containing the other systems.

This duality, however, is not what my challenge consists in. Needless to say, I grant that on the semantic conception CL is a sublogic of IL (I guess this was your point as well). But, claiming that CL is a sublogic of IL proof-theoretically as well (say, because of some mapping like that provided by Glivenko) seems like a more interesting claim. And - it is for this claim that I want a new def. of sublogicality, i.e. a revised proof-theoretical definition.

Why is this a challenge? Because the mappings you refer to (Troelstra 2.3) all seem to use the so-called negative fragment of CL. I see no reason why this particular relation between CL and IL should give rise to a general definition of sublogicality.

Philosophically, it seems that it is this fact, that the negation is essential, which load the dice.

Paul Simon Svanberg said...
This comment has been removed by a blog administrator.
Paul Simon Svanberg said...
This comment has been removed by a blog administrator.
Paul Simon Svanberg said...

I made a mess, alas. Busy weekend, also alas.

Long comment due on sunday.

Paul Simon Svanberg said...

Your challenge points directly to issues I have been obsessing over for a while. I will try to answer it, and make some additional comments in an attempt to present a perspective on the more general questions at stake.

"I assume we agree that there are at least to usual ways of ordering logics: one proof-theoretical (syntactical) and one semantical. The one which most frequently is used to define sublogicality is the proof-theoretic one: L' is a sublogic of L iff for every A s.t.
|-A in L', |-A in L as well... However, I perfectly agree that by looking at it semantically, the ordering is turned up-side-down"

The contravarians between syntax and semantics seems to me to be misleading in this case. That is, not in the isolated case of CL and IL, but in the question of how to order logics w.r.t. some sort of inclusion.

Now, why is this contravarians misleading? If the models M(L') of a logic L' is contained in the models M(L) of a logic L, then this fact should find a syntactic expression as well, if for no other reason than completeness, whenever that obtains.

I suggest we don't think of the ordering of logics as dependent on whether we induce an order from syntax or semantics. A logic should be considered as a pair, syntax and semantics. In doing this we should not be forced to accept dialogues like

a: "L' is a sublogic of L"
b: "... ah, I see, you mean from the syntactic perspective..."

because it seems that a and b have no clear notion of a logic. To put it bluntly, either IL is a sublogic of CL or CL is a sublogic of IL. It should depend at from which angle one is looking at the matter. Dialetism is not an option.

Our claim is semantical sublogicality obtains iff syntactic sublogicality obtains.

Why bother with completeness in the first place? If I remember correctly, Priest's argument that IL is a sublogic of CL is based on semantical (Kripke-style) considerations. This is then a common way to approach the issue of sublogicality, and, depending on the choice of models, one which may yield different results. As an aside, one can note that Kripke-semantics for IL produces wrong metalogical results. This may indicate that something is amiss with Kripke-semantics as a semantical framework.

But in what sense are we justified in holding that IL cannot be a sublogic of CL? in order to answer this we should consider the question what does it mean when a logic proves less formulae?. The sole reason for claiming that IL is a sublogic of CL is that there is some formula which CL proves but not IL, e.g. A v ~A. When CL proves "A v ~A" it claims that this can never fail to hold. I.e. it disregards all instances, all possible situations, where it might fail. This is to say that CL consider a limited range of possible situations. On the other hand, IL does not claim that "A v ~A" holds all the time, but maybe sometimes. One can think of this as the logics ability to make disctinctions. This points to the duality between syntax and semantics you refer to, but we interpret it differently. If L proves some formula A which L' does not, other things being equal, then there is a syntactic distinction which L is not aware of, which L' respects. Think of e.g. ~~A = A in Cl and the de Morgan dualities and so on. They are distinct to our eyes, notation-wise and so forth, but they are equivalent are far as provability goes. This is a part of the syntactic expression of the fact that the less formulae some logic proves, the more models does its semantics subsume.

Keeping this in mind, we can proceed to justifying our above claim. Though, perhaps this is only partially an answer to your challenge, which was a revised proof-theoretical definition of sublogicality, since semantics somehow jumped on the train too. But I think everything will be covered in the end.

"Why is this a challenge? Because the mappings you refer to (Troelstra 2.3) all seem to use the so-called negative fragment of CL. I see no reason why this particular relation between CL and IL should give rise to a general definition of sublogicality."

CL appear as a restriction on the negative fragment of IL. This relation is particular to CL and IL -- it is just the way things are between those logics. What makes the relation between them general and interesting, is the fact that the embedding of CL in IL is faithful. All of CL can be reconstructed from IL. We are less concerned with the particulars of the embedding, in the case of IL-CL, that negation is essential, so long as the embedding exists and is faithfull.

So, what is the syntactic counterpart of model-inclusion? Embeddability! And it must be faithful too.

SUBLOGIC:
A logic L' is a subglogic of L iff there exists a map m() s.t.

L' |- A iff L |- m(A)

I mean, this is bound to have been covered in the literature on abstract model-theory and institutions in one way or another, so I have no delusions of originality here, but it is in my mind the best way to think of the ordering of logics. This way, a sublogic is actually a living fragment of some other logic, not just a subset of a set of wffs.

For example, we get that CL is a sublogic of IL, that both CL and IL are sublogics of S4. Also, we have that CL and IL are sublogics of linear logic (LL). Also, LL and S4 share the same modalities, but with different handling of contexts, or "resources", so I don't really know what happens there, but S4 is probably a sublogic of LL. etc..

To be continued...

Ole Thomassen Hjortland said...

Paul Simon:

"If the models M(L') of a logic L' is contained in the models M(L) of a logic L, then this fact should find a syntactic expression as well, if for no other reason than completeness, whenever that obtains."

Precisely how do you figure that the model-theoretic inclusion will manifest itself? Take for instance the modal systems K and T: M(T) is a (proper) subset of M(K). So, if a sentence A has a T-model (is T-satisfiable), then it has a K-model (is K-satisfiable), but not vice versa. The only immediate syntactic upshot of this is that if the formula A is T-consistent, then it is K-consistent. This follows by soundness.

"As an aside, one can note that Kripke-semantics for IL produces wrong metalogical results. This may indicate that something is amiss with Kripke-semantics as a semantical framework."

What do you mean by "wrong" metalogical results. Granted, they are different from algebraic semantics, but does that make them wrong?

"If L proves some formula A which L' does not, other things being equal, then there is a syntactic distinction which L is not aware of, which L' respects. Think of e.g. ~~A = A in Cl and the de Morgan dualities and so on. They are distinct to our eyes, notation-wise and so forth, but they are equivalent are far as provability goes. This is a part of the syntactic expression of the fact that the less formulae some logic proves, the more models does its semantics subsume."

I more or less agree with this part.

"So, what is the syntactic counterpart of model-inclusion? Embeddability! And it must be faithful too.

SUBLOGIC:
A logic L' is a subglogic of L iff there exists a map m() s.t.

L' |- A iff L |- m(A)"

But I thought that this was satisfied both for CL = L', IL = L and for CL = L, IL = L'. In other words, that there is a faithful embedding both ways. Maybe I'm getting something wrong here, but if this is the case then it seems that it contradicts your statement earlier in the comments:

"To put it bluntly, either IL is a sublogic of CL or CL is a sublogic of IL. It should depend at from which angle one is looking at the matter. Dialetism is not an option."

I'm prepared to be corrected on this.

Paul Simon Svanberg said...

Last things first.

"To put it bluntly, either IL is a sublogic of CL or CL is a sublogic of IL. It should depend at from which angle one is looking at the matter. Dialetism is not an option."

"To put it bluntly, either IL is a sublogic of CL or CL is a sublogic of IL. It should not depend at from which angle one is looking at the matter. Dialetism is not an option."

So, on to the meat.

""SUBLOGIC:
A logic L' is a subglogic of L iff there exists a map m() s.t.

L' |- A iff L |- m(A)"

But I thought that this was satisfied both for CL = L', IL = L and for CL = L, IL = L'. In other words, that there is a faithful embedding both ways"

There is only a faithful embedding of CL into IL. One can, however, embed IL into CL by some map m, but we cannot recover the distinctions collapsed in CL by this m. I think I gave a counterexample to show we lose faithfulness in my thesis, but I don't remember exactly how it got off the ground. The point of embeddings is to preserve the operational meaning of the logical constants. If the embedding is faithful, then we have identified a syntactic fragment L' of a logic L, s.t. all the logical operations (the rules!) are intact. This means that they are not weakened (in their "sense", so to speak). When we embed IL into CL, we are in effect weakening all the intuitionistic operations. Since CL is not fine grained enough, i.e. proves to many equivalences, we cannot reverse this weakening. This can also be thought of in terms of the non-invertibility of Weakening in sequent systems, though this is only a metaphor.

""As an aside, one can note that Kripke-semantics for IL produces wrong metalogical results. This may indicate that something is amiss with Kripke-semantics as a semantical framework."

What do you mean by "wrong" metalogical results. Granted, they are different from algebraic semantics, but does that make them wrong?"

My claim that Kripke-semantics produces wrong metalogical results is not based on the fact that algebraic semantics gives a different metalogical result. However, the upshot of the differences between Kripke-semantics and algebraic semantics, is that I think algebraic semantics produces the right metalogical results. I see it this way. A logic is a language used to describe some structure. The more distinctions the language is able to make, the more differences the language is able to discern -- then, if such a language, a logic L, admits a faithful embedding of another logic L', then I would think that L' is a sublogic of L, since L is able to describe all that L' describes, but at the same time make more distinctions. It appears we agree more or less that on the syntactic level, IL makes more distinctions than CL in that IL does not collapse e.g. ~~A and A. Also, since CL is faithfully embeddable in IL, but not the other way around, I think the coast is clear to say that CL is a sublogic of IL. It is from this line of thought I would argue that Kripke-semantics produces wrong metalogical results.

"Precisely how do you figure that the model-theoretic inclusion will manifest itself? (...)"

I'm not sure. I haven't really thought about it. Maybe my understanding of sublogicality (a very strict one indeed) is applicable and limited only to the class of logics closely related to IL: CL, S4, LL. However, I do not believe this to be the case.

"The only immediate syntactic upshot of this is that if the formula A is T-consistent, then it is K-consistent. This follows by soundness."

I think I can account for the relationship between CL, IL, S4 and LL. I don't know about S5, nor the smaller modal logics like K, T, K4 etc. My feeble attempt at defining sublogicality is probably too strong, and thus also likely to give us wrong metalogical results every now and then. But I think that behind the the hierarchy of modal logics, the intelligent way of defining sublogicality is probably already given. And I think semantics is indispensible in dealing with this question.

Paul Simon Svanberg said...

"My feeble attempt at defining sublogicality is probably too strong, and thus also likely to give us wrong metalogical results every now and then."

I withdraw this comment.

There is a certain amount of relativity in the talk about sublogics, e.g. as evidenced by the "less" tautologies, "more" models duality. It seems that we have some liberty in chosing along which dimensions we would like to measure the unit of sublogicality.

Should we start with the axiomatization, the semantics or perhaps with something else?

I think the right place to start when dealing with the notion of sublogicality, is logical strentgth.

This is what it boils down to: If some logic L' is expressible by another L, then we should say L' is a sublogic of L. Thus we measure sublogicality according to what structures can be described/named by the logics, which seems to me to be fair to the nature of Logic.

Embeddings is the natural way of establishing the syntactic result about logical strength. "Model theoretic inclusion" is the natural way of establishing the semantics result about logical strength. Both of these have non-trivial "methods" of verification, except for in the simplest cases.

One could argue that the notion of logical strength is at heart a semantical one. This is unproblematic to me, but if one so desires, one can counter this by pointing out that it is not a semantical notion exclusively. The whole point of ordering logics according to their expressive power, is to capture both their language and their models. Otherwise the relation between language and models, syntax and semantics, would appear fragmentary and to a certain degree arbitrary. If one by preference would like to work exclusively in the one realm at the expense of the other, one should nonetheless use notions which preserve the "good" qualities which lie inherent in Logic, and respects the nexus between syntax and semantics.

Ole Thomassen Hjortland said...

I must admit that I'm still puzzled. I take it that you want to have a proof-theoretic and a model-theoretic notion of sublogicality, such that these coincide (I assume, only when there is completeness and soundness).

So, according to the two points you have been discussing, such a relation would mean something along the following lines.

Let L, L' be two logics, and M(L), M(L') be their models:

M(L') is a subset of M(L) iff
there is a faithful embedding of L' into M. Is this what you suggest?

A further question: What about the relation between classical propositional logic and classical first-order logic? What is the sublogicality relation between these on your account?

henri galinon said...

For logical symbols in HTML, add this keyboard to your toolbar. The following is a tentative application.

Humberstone writes (The Connectives, p.275):
"One reaction to this [ie : the 'unobvious" corollary of Glivenko you mentionned in your post] has been to suggest that far from being a subsytem of CL, IL is actually an extension, every classical tautology being rewritten in terms of the functionally complete connectives ¬ and ∧ (thus giving a formula by classical lights equivalent to the original, with intuitionistic → and ∨ regarded as additional connectives, like the ⇑ of modal logic [...]. However the temptation is best resisted since the result in question does not extend to the consequence relation concerned (e.g. it is not an IL truth that : ¬¬p l- p ) " (Humbestone's emphasis)

Note also that Glivenko's theorem does not hold for predicate IL and CL logics :
It is classically true that : l- ∀ (Fx ∨ ¬ Fx)
but not intuitionistically true that l- ¬¬ ∀ (Fx ∨ ¬ Fx)

Finally, you have probably noticed that in the definition of the UNILOG 2007 (here) contest, reference is made to a result of Wojcicki and more generally to different notions of translation of a logic in another logic. I don't know what Wojcicki's result states, and I'd be interested in earing from someone having any lights on this.

Best

Ole Thomassen Hjortland said...

Thanks Henri,

This was very helpful. It is quite true that Glivenko's theorem is restricted, but although the provability relation (e.g., in your example ¬¬p |- p) is not within the scope of the corollary in question, extending Glivenko's theorem to a full translation (i.e., the Gödel-Gentzen translation), the above example becomes provable as ¬¬¬¬p |- ¬¬p since all atomic formulae are doubly negated.

Furthermore, the translation also works for the first-order case by devising a translation s.t. exists x (Ax) := ¬ forall ¬Ax. I'll have to see what more Humberstone has to say about this; it's definitely an interesting passage.

I have noticed the references on the UNILOG page, but I've had a hard time finding some of them. I would especially like to look at the Prawitz & Malmnäs paper from 1968, but so far I haven't been able to get the anthology it's printed in.

Best,


text:
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# Percent of a Number

## Use multiplication and proportions to find the percent of a number.

0%
Progress
Practice Percent of a Number
Progress
0%
Percent of a Number

Credit: WDPG Share Erik Thureson/Fox Sports
Source: https://www.flickr.com/photos/wdpg_share/3177157643/

Lee loves basketball. He hopes to be just as good as his favorite player, Kobe Bryant. On average, Kobe makes 83% of all his free throws. Lee practices shooting 50 free throws every day. If Lee wants to shoot free throws as well as Kobe, how many does he need to make?

In this concept, you will learn how to find the percent of a number.

### Guidance

A percent is a ratio that describes a quantity out of 100. A percent is also a part of a whole. Let’s take a look at finding the percent of a number is the part of a number that is equivalent to the percent. You can find the percent of a number two ways. One method is to use a proportion, an equation that shows equivalent ratios.

Let’s, find 10% of 25 using a proportion.

First, write the percent as a fraction. 10% is also 10 out of 100.

10%=10100

Next, write a proportion to find the equivalent fraction of 10% of 25. The second fraction is the unknown quantity, represented by x\begin{align*}x\end{align*}, over the whole, 25.

10100=x25

Then, cross multiply to find the cross products. To cross multiply, multiply the numerator of one fraction with the denominator of the other fraction. The products should be equal.

10(25)=100x

Finally, simplify the equation to solve for x\begin{align*}x\end{align*}. Multiply 10 by 25 and divide both sides by 100.

10(25)2501002.5===100x100x100x

10% of 25 is 2.5.

Let’s review the steps of using a proportion.

Another method is to use key words and multiplication. Let’s look again at the problem.

What 10% of 25?

First, look for key words that suggest an operation. The word “of” tells you to multiply 25 by 10%.

Next, convert 10% to a decimal.

10%=0.1

Then, write an equation to find the unknown number, x\begin{align*}x\end{align*}.

x=25×0.1

Finally, solve for x\begin{align*}x\end{align*} by multiplying 25 by 0.1.

25×0.1=2.5

The answer is the same. 10% of 25 is 2.5.

### Guided Practice

What is 15% of 200?

The keyword “of” tells you to multiply 200 by 15%. First, convert 15% to a decimal.

15%=0.15

Next, write an equation to find the unknown number, x\begin{align*}x\end{align*}.

x=200×0.15

Finally, solve for x\begin{align*}x\end{align*} by multiplying 200 by 0.15.

200×0.15x==30.0030

15% of 200 is 30.

### Examples

Find the percent of each number.

#### Example 1

What is 10% of 54?

First, convert 10% to a decimal.

10%=0.1

Next, write an equation to find the unknown number, x\begin{align*}x\end{align*}.

x=54×0.1

Then, solve for x\begin{align*}x\end{align*} by multiplying 54 by 0.1

10% of 54 is 5.4.

54×0.1x==5.45.4

#### Example 2

What is 25% of 80?

First, convert 25% to a decimal.

25%=0.25

Next, write an equation to find the unknown number, x\begin{align*}x\end{align*}.

x=80×0.25

Then, solve for x\begin{align*}x\end{align*} by multiplying 80 by 0.25.

80×0.25x==20.0020

25% of 80 is 20.

#### Example 3

What is 5% of 78?

First, convert 5% to a decimal.

5%=0.05

Next, multiply 78 by 0.05.

78×0.05x==3.903.9

5% of 78 is 3.9.

Credit: Keonl Cabral
Source: https://www.flickr.com/photos/keoni101/5106592086/

Remember Lee practicing his free throw?

Lee wants to make just as many free throws as Kobe. To find out how many he needs to make, Lee must multiply 50 by 83%.

First, convert 83% to a decimal.

83%=0.83

Next, write an equation to find the unknown number, x\begin{align*}x\end{align*}.

x=50×0.83

Then, solve for x\begin{align*}x\end{align*} by multiplying 50 by 0.83.

50×0.83x==41.5041.5

83% of 50 is 41.5. Lee needs to make at least 41 out of 50 free throws to be just as good as Kobe.

### Explore More

Find the percent of each number.

1. What is 2% of 10?
2. What is 5% of 50?
3. What is 10% of 30?
4. What is 25% of 18?
5. What is 20% of 36?
6. What is 11% of 40?
7. What is 8% of 80?
8. What is 15% of 45?
9. What is 20% of 100?
10. What is 25% of 250?
11. What is 4% of 60?
12. What is 5% of 85?
13. What is 2% of 18?
14. What is 15.5% of 90?
15. What is 20.5% of 70?

### Vocabulary Language: English

Percent

Percent

Percent means out of 100. It is a quantity written with a % sign.
Proportion

Proportion

A proportion is an equation that shows two equivalent ratios.

format: plain


text:
Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Random peptide mixtures entrapped within a copper-cuprite matrix: new antimicrobial agent against methicillin-resistant Staphylococcus aureus

## Abstract

The emergence of global antibiotic resistance necessitates the urgent need to develop new and effective antimicrobial agents. Combination of two antimicrobial agents can potentially improve antimicrobial potency and mitigate the development of resistance. Therefore, we have utilized metal molecular doping methodology whereby antimicrobial random peptides mixture (RPMs) are entrapped in a bactericidal copper metal matrix. The copper/RPM composite exhibits greater antimicrobial activity toward methicillin-resistant Staphylococcus aureus (MRSA) than either copper or RPMs alone. Our findings indicate that this bactericidal antimicrobial biomaterial could be utilized to efficiently eradicate antibiotic-resistant pathogenic bacteria for health, agricultural and environmental applications.

## Introduction

Bacterial resistance to antibiotics is a global concern and there is a continuing need to the develop new antimicrobial agents. Resistant bacteria in both hospitals and the environment cause prolonged hospitalization and the death of tens of thousands of people annually, worldwide. For example, methicillin-resistant Staphylococcus aureus (MRSA) claims the lives of approximately 19,000 people per annum in the United States alone1. Current solutions to this global crisis includes the use of new classes of antimicrobial agents and traditional bactericides1. The combination of two antimicrobial agents that possess different modes of action has been suggested as a more effective solution to eradicate resistant bacteria than using just one2,3.

Low concentrations of metals such as copper and silver exhibit bactericidal properties and are thought to work either extracellularly by binding and inactivating membrane proteins or intracellularly after transport into the cell4,5. In fact, copper has been used in ancient Egypt as early as 2000 B.C. to sterilize water and wounds6. Although silver has greater bactericidal activity, copper is less expensive and therefore utilized extensively in both the health and agricultural sectors7,8. For example, it has been used as an additive in water treatment facilities, as a disinfectant in the food industry, for hospital sterilization, to incorporate antifouling properties in paint and for wound healing6,9. Copper is also commonly used on a large scale in the agricultural sector for crop protection6,10. The microbial toxicity of copper is mediated through multifactorial pathways, one of which exploits its ability to act as a catalyst to generate reactive oxygen species (ROS), which cause damage to vital cell constituents such as proteins, lipids and DNA5,7,11. Copper ions can also deactivate intracellular and membrane proteins by direct interaction or via competition for the binding sites of essential metals. In both cases, inhibition or deactivation of the protein is the result of conformational changes in the protein structure6,7. Unfortunately, copper-resistant bacterial strains have emerged and are responsible for significant reductions in crop yields9,12, therefore, there is an urgent need for novel solutions to combat these resistant pathogens5. This has spurred study of its antimicrobial or anti-biofilm activity together with combinations of amino acids13, organic acids14, hepcidin15 or copper binding peptides16,17. Here, we present a new combination of antimicrobial agents: a copper matrix doped with antimicrobial peptides.

Antimicrobial random peptide mixtures (RPMs)18 were originally inspired by natural antimicrobial peptides (AMPs), which are produced by eukaryotes as part of their innate immune response to bacterial infection19,20. AMPs are typically cationic and act primarily via electrostatic interactions with the anionic bacterial membrane, followed by insertion whereby the hydrophobic residues cause membrane disruption and bacterial cell death. Despite the broad structural diversity of natural AMPs, we identified several common features, which we incorporated into the RPM design, to enable targeting of bacterial cell membranes. RPMs are synthesized via solid-phase peptide synthesis (SPPS) by mixing an equimolar ratio of one hydrophobic and one cationic amino acid to generate a mixture of peptides with the desired chain length and defined stereochemistry but with random sequences. Homochiral RPMs have been shown to disrupt synthetic lipid bilayers via a pore-like mode of action18 and were found to possess potent antimicrobial and anti-biofilm activity21,22. Their synthesis is both efficient and cost-effective and generates a cocktail of peptides, which may confer a broader spectrum of activity and potentially a lower probability of developing bacterial resistance23.

We decided to evaluate a combination treatment of antimicrobial agents, which is a strategy that has previously been successful against MRSA3,24,25. It has been demonstrated that the incorporation and entrapment of biologically active molecules in metals can result in synergistic activity26. Composites made by entrapping the antimicrobial agent chlorohexidine and silver (CH@Ag) have shown synergistic antimicrobial activity27,28. When the silver was replaced with copper, the composite had even greater antimicrobial activity8. Strong antibacterial activity was even observed by the incorporation of anti-inflammatory agents within silver29. Based on previous work that demonstrated the entrapment of proteins in silver30 and Nafion™ in copper31, we have developed a protocol to entrap antimicrobial RPMs in copper (Fig. 1). We selected a leucine/lysine (LK) RPM for entrapment within copper since this mixture possesses potent antibacterial activity21,22. The resulting composite proved to be an efficient growth inhibitor of MRSA.

## Materials and Methods

### Chemicals and bacterium

Fmoc-L-Leu-OH, Fmoc-L-Lys(Boc)-OH, 2-(1H-benzotriazole-1-yl)-1,1,3,3-tetramethyluronium hexafluorophosphate (HBTU) and rink amide resin (0.53 mmol/g) were purchased from Chem-Impex (USA). Sodium chloride, N,N-dimethylformamide (DMF), diethyl ether, trifluoroacetic acid (TFA), and N,N-diisopropylethylamine (DIEA) were purchased from Biolab, Israel. Yeast and tryptone were obtained from BD (Franklin Lakes, NJ). Copper sulfate, zinc (granular) and all other chemicals were purchased from Sigma Aldrich (Israel). The methicillin-resistant Staphylococcus aureus 1206 strain32 was obtained from Prof. B. Weisblum (UW-Madison, Wisconsin, USA).

### Peptide synthesis

RPMs were synthesized via microwave-assisted SPPS on Rink amide resin (substitution: 0.53 mmol/g, 25 μmol) in filter tubes (Silicol) as described previously21. Briefly, coupling reactions were conducted with binary combinations of Fmoc-L-Leu-OH and Fmoc-L-Lys (Boc)-OH. A freshly prepared stock solution containing the protected amino acids in a 1:1 molar ratio was used for each reaction. Prior to each coupling step, an aliquot containing 4 equivalents (100 μmol) of the amino acid mixture was activated with 4 equivalents of HBTU and 8 equivalents of DIEA in DMF and added to the resin. The reaction mixture was then heated to 70 °C in a MARS VI multimode microwave reactor (CEM, USA) (2 minutes ramp to 70 °C, 4 minutes hold at 70 °C) with stirring. For Fmoc deprotection we added 20% piperidine in DMF and heated the reaction solution to 80 °C (2 minutes ramp to 80 °C, 3 minutes hold at 80 °C) with stirring. After each coupling/deprotection cycle, the resin was washed 3 times with DMF. At the end of the synthesis, the peptide mixtures were cleaved from the resin by adding a solution containing 95% trifluoroacetic acid (TFA), 2.5% doubly-distilled water (DDW) and 2.5% triisopropylsilane and stirred for 3 hours. The mixture was then filtered, and the peptides precipitated by the addition of cold diethyl ether to the TFA solution and centrifuged. The supernatant was then removed and the peptide pellet dried under a stream of nitrogen, dissolved in acetonitrile/DDW and frozen with liquid nitrogen and lyophilized. The synthesis was validated by MALDI-TOF mass spectrometry. The resultant leucine and lysine peptide mixtures are referred to as the LK 10-mer and LK 20-mer RPMs, based on their chain length.

### Composite preparation

CuSO4 (0.200 g, 1.25 × 10−3 mol) was dissolved in 2.0 mL of DDW. Zinc powder (0.081 g, 1.25 × 10−3 mol) was added and then after 30 s of stirring, 1.0 mL of a peptide solution (0.003 g, LK 10-mer or 20-mer) was added at a ~50:1 ratio of copper to amino acid. The solution was mixed for 24 h at room temperature (~23 °C). The resulting precipitate was filtered through a sintered glass funnel (pore size of 10–16 μm), washed with 30 mL of DDW and dried overnight under vacuum. Non-doped copper composite was prepared by the same procedure without adding any peptides.

### Measurement of copper ion concentration

The copper ion concentration was measured as follows: 10 mg of the composites were dispersed in 50 mL Luria broth (LB) growth media at 37 °C and shaken at 180 rounds per minute (RPM) for 8 hours. A 5.0 mL sample was then removed from the suspension, filtered through a 0.45 µm filter and digested with 3.0 mL of 65% HNO3 and 2.0 mL of 30% H2O2. The samples were dissolved for analysis and measured by inductively coupled plasma mass spectrometry (ICP-MS)33,34.

### Growth inhibition assays

LB inoculated with MRSA was incubated overnight at 37 °C with shaking at 180 RPM. A bacterial suspension with optical density (O.D.) of 0.1 at 600 nm was then prepared and diluted 1,000-fold in LB (to approximately 105 CFU/mL). The tested composites (10 mg, 200 ppm) were added to sterile 100 mL Erlenmeyer flasks and 50 mL of the MRSA bacterial suspension was added. The flasks were incubated at 37 °C with shaking at 180 RPM for 24 hours. Bacterial growth was monitored by measuring the O.D.600 nm of 1.0 mL samples which were removed at different time points. Flasks without composites or with LK RPMs (4.0 μg/mL) were used as negative and positive controls, respectively. The percentage of growth inhibition was calculated at each time point (Ti) according to Eq. 1:

$$\% \,Growth\,Inhibition\,({T}_{i})=100-\frac{O.D.\,treatment\,at\,{T}_{i}}{O.\,D.\,negative\,comtrol\,at\,{T}_{i}}$$
(1)

### Mixing LK RPMs and copper ions assay

MRSA cells were grown as described to prepare a bacterial suspension with O.D600 nm of 0.1. Aliquots (1 mL) of the suspension were mixed with an equal volume of LB containing copper sulfate and various concentrations of LK 20-mer or 10-mer peptides to give final concentrations of 400 ppm copper sulfate with 25 or 12 μg/mL of LK 20-mer, or with 200 or 100 μg/mL of LK 10-mer. After vortex mixing, a 1 mL sample was taken for O.D. measurement and the remaining 1 mL sample was incubated for 6 hours at 37 °C, 100 RPM. After this time, the O.D. was remeasured and the percentage growth was calculated according to Eq. 2:

$${\rm{ \% }}\,Growth\,({T}_{i})=\frac{O.D.\,{T}_{6htreatment}-O.D.\,{T}_{0htreatment}\,}{O.D.\,{T}_{6hcontrol}-O.D.\,{T}_{0hcontrol}}\times 100$$
(2)

### Assessment of peptide release

Two methods were used to examine the release of peptides from the composites. (I) Composites (10 mg) were suspended in 50 mL DDW and incubated for 6.5 hours with shaking at 37 °C, 180 RPM. The suspension was then filtered through a sintered glass funnel (size pores of 10–16 μm). The filtered composite was then dried under vacuum and the organic element content was analyzed as described below. A lack of change in the weights of the elements indicated that the peptides were not released from the composite. (II) The filtered solution was lyophilized and then suspended with 80 μL DDW. The protein content of a 50 μL aliquot was analyzed by the addition of 450 μL of 4-fold diluted Bradford reagent. After 5 minutes of incubation in the dark, the peptide content was measured by reading the O.D. at 595 nm against a standard curve prepared with serial dilutions of free LK RPMs in DDW. The linear range was 6–100 μg/mL.

### Instrumentation

RPMs were characterized on a MALDI-TOF MS Microflex LRF (Bruker Daltonik GmbH, Germany). UV-Vis absorbance spectroscopy was carried out with Genesys UV-vis spectrophotometer (Thermo Spectronic). Elemental analysis (nitrogen, carbon, hydrogen, and sulfur) of at least 4 different batches was carried out using a Thermo Elemental Analyzer 1120. Thermogravimetric analysis (TGA) from 50 °C to 800 °C was conducted with a Mettler-Toledo TGA/SDTA 851e, at a heating rate of 10 °C per minute under N2. Density measurements were carried out with a Micromeritics AccuPyc 1340 instrument using helium as the displacing gas. The copper ion concentration was measured by an axial inductively coupled plasma optical emission spectrometer (ICP-OES) model ‘ARCOS’ from Spectro GMBH (Germany). Scanning electron microscope (SEM) and energy-dispersive X-ray spectroscopy (EDAX) analysis was carried out on a Sirion (FEI) high resolution (HR) SEM instrument. X-ray powder diffraction (XRD) measurements were performed on a D8 Advance diffractometer (Bruker AXS, Karlsruhe, Germany) with secondary Graphite monochromator, 2° Sollers slits and a 0.2 mm receiving slit. The powder samples were placed on low background quartz sample holders. X-ray diffraction (XRD) patterns from 5° to 85° 2θ were recorded at room temperature using CuKα radiation (λ = 0.15418 nm) under the following measurement conditions: tube voltage of 40 kV, tube current of 40 mA, step scan mode with a step size of 0.02° 2θ and a counting time of 1 s per step. The Scherrer equation was used to obtain the crystalline size from the experimental XRD data. The instrumental broadening was determined using LaB6 powder (NIST SRM 660).

## Results

LK 10-mer and 20-mer RPMs were synthesized via 9-fluorenylmethyloxycarbonyl (Fmoc) SPPS21. Characterization was validated by mass spectrometry (Fig. SI1). Copper was doped with the RPMs by a modified version of the heterogeneous doping methodology, which is based on the reduction of the copper cation with metallic zinc (see Fig. 1 and Material and methods section)31. A copper ([Cu]) control was prepared by the same procedure, but in the absence of the RPMs.

The new materials, LK20-mer@[Cu] and LK10-mer@[Cu], were characterized by several chemical and physical measurements. The densities of the LK20-mer@[Cu] and LK10-mer@[Cu] were lower (7.2 and 7.3 g/cm3 respectively) than that of the non-doped copper composite (8.0 g/cm3) as the peptides interfere with the growth of the copper crystals during reduction. Energy dispersive X-ray analysis (Fig. 2A) showed that the composites contained only copper and organic material and no detectable traces of zinc. The use of a lower copper to zinc ratio (1:1.2) reveals residual traces of zinc (Fig. SI2), indicating that the 1:1 ratio is optimal for a contamination-free reduction. The structure and morphology of the composites were characterized by SEM (Fig. 2B), and showed nanocrystals of copper tightly aggregated to form micron-sized particles form larger clusters. Interestingly, as observed by the X1000 magnification (Fig. 2B, upper row), the peptide entrapment appears to reduce the particle size with the longer chain random peptides resulting a smaller particle. This reduction in size might be attributed to inhibition of nanocrystal growth caused by the adsorptive interactions of the LK RPM with copper.

Elemental analysis detected the presence of organic elements in the LK@[Cu] composites as opposed to the undoped copper, which confirmed entrapment of the RPMs in the copper matrix (Fig. 3A). Higher percentages of nitrogen and carbon (0.21% and 1.16%) were detected in the LK20-mer@[Cu] composites than in the LK10-mer@[Cu] composites (0.07% and 0.66%). Thermogravimetric analysis was performed to quantify the amount of entrapped organic material (Figs 3B and SI4), which measured a 2.98% weight loss for the LK10-mer@[Cu] and 6.51% for the LK20-mer@[Cu]. XRD measurements revealed that the composites contained cuprite (Cu2O) as well as metallic copper (Fig. SI3). The copper ([Cu]), LK10-mer@[Cu], and LK20-mer@[Cu] contained 25.8%, 33.1% and 44.1% cuprite, respectively (Fig. SI3). This result indicates that the random peptides entrapped in a copper matrix that contains both copper and cuprite, therefore the matrix termed as [Cu] along the research.

The biological activity of each composite was then evaluated. Figure 4 shows the effect of the composites on the growth of MRSA bacterial cells. While copper ([Cu]) had only a minor effect on MRSA growth, the entrapped peptide composites at the same concentration strongly inhibited bacterial growth. This inhibition was maintained over the course of the culture, with a narrower gap after 24 hours. As presented in Fig. 4, the free LK 20-mer random peptide that was used as a control (at 4 μg/mL, in accordance with the estimated maximal amount of peptide in 200 ppm composite) did not inhibit bacterial growth.

Since the combination of both components display an enhanced antimicrobial effect, it was important to evaluate the release of copper ions from the composites. Copper ions might be released from the composites in a liquid medium according to the oligodynamic effect35. After 8 h incubation in growth medium, no significant differences in amount of released copper ions were detected by ICP-MS from the LK@[Cu] composites compared to the non-doped composite [Cu] (Table SI1). There was no detectable release of peptides from the composites as determined by a Bradford assay, even after extended incubation time. To verify this finding, we also performed elemental analysis of composites before and after incubation in water. The results did not show composites weight decrease, supporting the observation that there was no significant release of RPMs during incubation of the composites (Table SI2).

The antimicrobial activity of the LK@[Cu] composites was then compared with a simple mixture of the two active agents. As presented in Fig. 5, copper ions have a weak effect on the growth of MRSA (~15% lower than the control). The free LK20-mer peptide had a concentration-dependent inhibitory effect (grey bars), which was enhanced in the presence of copper ions (black bars). In contrast, the addition of copper ions to the LK10-mer did not enhance its antimicrobial activity. These findings support the notion that physical mixing of LK and copper ions is not enough to achieve the enhanced antimicrobial effect that the composites possess.

## Discussion

The LK@[Cu] composites were generated by the entrapment of random peptide mixtures in a copper-cuprite matrix represent a new material with unique properties and characteristics, rather than being simply the sum of the component parts separately. They were prepared according to the heterogeneous reduction method31 with zinc as the reducing agent, whereby copper agglomerates to microcrystals. RPMs interact with the copper and are entrapped within the aggregates; the zinc is then washed away as confirmed by EDAX analysis. Evidence for the formation of a new composite material is given by the reduction in particle density and size after entrapment of the peptides. This is consistent with the trend of decreasing densities of other composites (e.g. Nafion@Cu and Thionin@Cu)31 and may be attributable to perturbations in a typical crystal structure. The absence of detectable peptide release from the composites also suggests the formation of strong interactions between the LK RPMs and copper, in addition to their physical entrapment. This is in accord with reports that large molecules such as Nafion31 and the enzyme acid phosphatase30 are not released from similar composites, although it is possible for small molecules8,27,29. According to the results of the TGA and elemental analysis, the LK 20-mer RPM was entrapped more efficiently than the LK 10-mer RPM. Since copper competes with water for interaction with the peptides, we hypothesize that a longer peptide chain length, with its larger surface area, has a higher propensity to interact with the copper.

In addition to their physical properties, the composites also possessed stronger antimicrobial activity against MRSA than would be expected from the sum of its constituents. This activity is due to the unique combined effect of the two components and was not observed when the copper ions and LK 10-mer or 20-mer RPMs were tested alone or combined via physical mixing. These results indicate that the entrapment formed a new material with its own antimicrobial activity that both the copper and cuprite have a role in addition to the LK RPMs.

Interestingly, while our previous work indicated that higher concentrations of the LK 10-mer RPM was needed for growth inhibition of bacteria compared with the LK 20-mer RPM (Fig. 5) and as described at21, the results presented here showed that both the LK10-mer@[Cu] and LK20-mer@[Cu] composites possess similar activity. The reason for this discrepancy is not entirely clear although it probably relates to the structure of the composites. Since no peptide release from the composites was detected, we propose that the antimicrobial activity may derive from the entrapped state. In other words, the exposed 10-mer and 20-mer peptide chains within the copper matrix can interact with and disrupt bacterial cell membranes. Therefore, the LK@[Cu] composites possess an “antimicrobial surface”. In solution, copper ions are slowly released, which alters the metal matrix, thus leading to greater exposure of the bacteria to the internal RPM. According to our findings we proposed that the bacteria interacts with the composite surface then the cationic RPM attracts and disrupts the cell membrane permitting the entry of copper ions into the bacterial cells.

## Conclusion

In summary, this study describes the preparation, characterization and antimicrobial activity of LK@[Cu] composites, which are composed of copper and RPMs. These composites represent a new class of material with improved antimicrobial activity against the “superbug” MRSA. Activity towards other multidrug resistant pathogens and copper-resistant bacterial strains will require further investigation. The methodology and findings described here may be readily adapted to produce a wide variety of composites, to enable discovery of novel bactericidal agents that target resistant strains of bacteria.

## References

1. 1.

Blaskovich, M. A. T., Butler, M. S., Cooper, M. A., Martens, E. & Demain, A. L. The antibiotic resistance crisis, with a focus on the United States 70, 520–526, https://doi.org/10.1042/EBC20160077 (2017).

2. 2.

Fischbach, M. A. Combination therapies for combating antimicrobial resistance, Curr. Opin. Microbiol. 14 (n.d.) 519–523, https://doi.org/10.1016/j.mib.2011.08.003.

3. 3.

De Evgrafov, M. R., Gumpert, H., Munck, C., Thomsen, T. T. & Sommer, M. O. A. Collateral resistance and sensitivity modulate evolution of high level resistance to drug combination treatment in Staphylococcus aureus. Mol. Biol. Evol. 32, 1175–1185, https://doi.org/10.1093/molbev/msv006 (2015).

4. 4.

Olivares, M. & Uauy, R. Copper as an essential nutrient. Am. J. Clin. Nutr. 63 (1996).

5. 5.

Ladomersky, E. & Petris, M. J. Copper tolerance and virulence in bacteria. Metallomics. 7, 957–964, https://doi.org/10.1039/c4mt00327f (2015).

6. 6.

Borkow, G. & Gabbay, J. Copper, an ancient remedy returning to fight microbial, fungal and viral infections. Curr. Chem. Biol. 3, 272–278, https://doi.org/10.2174/187231309789054887 (2009).

7. 7.

O’Gorman, J., Humphreys, H., O’Gorman, H. & Humphreys, J. Application of copper to prevent and control infection. Where are we now?, J. Hosp. Infect. 81, 217–223, https://doi.org/10.1016/j.jhin.2012.05.009 (2012).

8. 8.

Ben-Knaz Wakshlak, R., Pedahzur, R., Menagen, B. & Avnir, D. An antibacterial copper composite more bioactive than metallic silver. J. Mater. Chem. B. 4, 4322–4329, https://doi.org/10.1039/C6TB00719H (2016).

9. 9.

Keller, A. A. et al. NanoImpact comparative environmental fate and toxicity of copper nanomaterials, 7 28–40, https://doi.org/10.1016/j.impact.2017.05.003 (2017).

10. 10.

Ingle, A. P., Duran, N. & Rai, M. Bioactivity, mechanism of action, and cytotoxicity of copper-based nanoparticles: A review. Appl. Microbiol. Biotechnol. 98, 1001–1009, https://doi.org/10.1007/s00253-013-5422-8 (2014).

11. 11.

Lemire, J. A., Harrison, J. J. & Turner, R. J. Antimicrobial activity of metals: mechanisms, molecular targets and applications. Nat. Rev. Microbiol. 11, 371–384, https://doi.org/10.1038/nrmicro3028 (2013).

12. 12.

Scheck, H. J., Student, F. G. & Pscheidt, J. W. Effect of copper bactericides on copper-resistant and -sensitive strains of Pseudomonas syringae pv. syringae. Plant Dis. 82, 397–406 (1998).

13. 13.

Li, J. & Wang, N. Foliar application of biofilm formation – Inhibiting compounds enhances control of citrus canker caused by Xanthomonas citri subsp. citri. Phytopathology. 104, 134–142 (2014).

14. 14.

Zhitnitsky, D., Rose, J. & Lewinson, O. The highly synergistic, broad spectrum, antibacterial activity of organic acids and transition metals. Sci. Rep. 7, 1–13, https://doi.org/10.1038/srep44554 (2017).

15. 15.

Maisetta, G. et al. Antimicrobial activity of human hepcidin 20 and 25 against clinically relevant bacterial strains: Effect of copper and acidic pH. Peptides. 31, 1995–2002, https://doi.org/10.1016/j.peptides.2010.08.007 (2010).

16. 16.

Libardo, M. D., Cervantes, J. L., Salazar, J. C. & Angeles-Boza, A. M. Improved bioactivity of antimicrobial peptides by addition of amino-terminal copper and nickel (ATCUN) binding motifs. ChemMedChem. 9, 1892–1901, https://doi.org/10.1002/cmdc.201402033 (2014).

17. 17.

Conklin, S. E. et al. Specific histidine residues confer histatin peptides with copper-dependent activity against Candida albicans. Biochemistry. 56, 4244–4255, https://doi.org/10.1021/acs.biochem.7b00348 (2017).

18. 18.

Hayouka, Z. et al. Binary encoding of random peptide sequences for selective and differential antimicrobial mechanisms, Angew. Chemie. 129, 8211–8215, https://doi.org/10.1002/ange.201702313 (2017).

19. 19.

Brogden, K. A. Antimicrobial peptides: pore formers or metabolic inhibitors in bacteria? Nat. Rev. Microbiol. 3, 238–50, https://doi.org/10.1038/nrmicro1098 (2005).

20. 20.

Wimley, W. C. & Hristova, K. Antimicrobial peptides: Successes, challenges and unanswered questions. J. Membr. Biol. 239, 27–34, https://doi.org/10.1007/s00232-011-9343-0 (2011).

21. 21.

Hayouka, Z. et al. Interplay among subunit identity, subunit proportion, chain length, and stereochemistry in the activity profile of sequence-random peptide mixtures. J. Am. Chem. Soc. 135, 11748–11751, https://doi.org/10.1021/ja406231b (2013).

22. 22.

Stern, T., Zelinger, E. & Hayouka, Z. Random peptide mixtures inhibit and eradicate methicillin-resistant Staphylococcus aureus biofilms. Chem. Commun. 52, 7102–7105, https://doi.org/10.1039/C6CC01438K (2016).

23. 23.

Zanchi, C., Johnston, P. R. & Rolff, J. Evolution of defence cocktails: Antimicrobial peptide combinations reduce mortality and persistent infection. Mol. Ecol. 26, 5334–5343, https://doi.org/10.1111/mec.14267 (2017).

24. 24.

Mun, S. et al. Phytomedicine synergistic antibacterial effect of curcumin against methicillin-resistant Staphylococcus aureus. Phytomedicine. 20, 714–718, https://doi.org/10.1016/j.phymed.2013.02.006 (2013).

25. 25.

Feng, Q. L. et al. A mechanistic study of the antibacterial effect of silver ions on Escherichia coli and Staphylococcus aureus. J. Biomed. Mater. Res. 52 662–668, 10.1002/1097-4636(20001215)52:4<662:AID-JBM10>3.0.CO;2-3 (2000).

26. 26.

Avnir, D. Molecularly doped metals. Acc. Chem. Res. 47, 579–592, https://doi.org/10.1021/ar4001982 (2014).

27. 27.

Ben-Knaz, R., Pedahzur, R. & Avnir, D. A concept in bactericidal materials: The entrapment of chlorhexidine within silver. Adv. Funct. Mater. 20, 2324–2329, https://doi.org/10.1002/adfm.200902393 (2010).

28. 28.

Ben-Knaz, R., Pedahzur, R. & Avnir, D. Bioactive doped metals: high synergism in the bactericidal activity of chlorhexidine@silver towards wound pathogenic bacteria. RSC Adv. 3, 8009, https://doi.org/10.1039/c3ra41196f (2013).

29. 29.

Menagen, B., Pedahzur, R. & Avnir, D. Sustained release from a metal - Analgesics entrapped within biocidal silver. Sci. Rep. 7, 1–11, https://doi.org/10.1038/s41598-017-03195-w (2017).

30. 30.

Ben-Knaz, R. & Avnir, D. Bioactive enzyme-metal composites: The entrapment of acid phosphatase within gold and silver. Biomaterials. 30, 1263–1267, https://doi.org/10.1016/j.biomaterials.2008.11.026 (2009).

31. 31.

Yosef, I. & Avnir, D. Metal-organic composites: The heterogeneous organic doping of the coin metals-copper, silver, and gold. Chem. Mater. 18, 5890–5896, https://doi.org/10.1021/cm0615368 (2006).

32. 32.

Weisblum, B. & Demohn, V. Erythromycin-inducible resistance in Staphylococcus aureus: survey of antibiotic classes involved. J. Bacteriol. 98, 447–452 (1969).

33. 33.

Sureshkumar, M., Siswanto, D. Y. & Lee, C. K. Magnetic antimicrobial nanocomposite based on bacterial cellulose and silver nanoparticles. J. Mater. Chem. 20, 6948–6955, https://doi.org/10.1039/c0jm00565g (2010).

34. 34.

Antonoglou, O. et al. Elucidation of one step synthesis of PEGylated CuFe bimetallic nanoparticles. Antimicrobial activity of CuFe@PEG vs Cu@PEG. J. Inorg. Biochem. 177, 159–170, https://doi.org/10.1016/j.jinorgbio.2017.09.014 (2017).

35. 35.

Mckhann, C. F., Carlson, H. J. & Douglas, H. Oligodynamic action of metallic elements and of metal alloys on certain bacteria and viruses. Pediatrics. 2, 272–289 (1948).

Download references

## Acknowledgements

The authors would like to thank the interdepartmental core facility of The Robert H. Smith Faculty of Agriculture, Food and Environment and the Center for Nanoscience and Nanotechnology at The Hebrew University of Jerusalem, for technical support and access to instrumentation. In addition, TSB would like to acknowledge the Center for Nanoscience and Nanotechnology of The Hebrew University of Jerusalem for her PhD fellowship. We also thank Dr John A. Karas for his help in reviewing and editing this manuscript.

## Author information

Authors

### Contributions

T.S.B. and B.M. performed the study. D.A. and Z.H. designed and planned the study and wrote the manuscript.

### Corresponding author

Correspondence to Zvi Hayouka.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

## Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

## About this article

### Cite this article

Bauer, T.S., Menagen, B., Avnir, D. et al. Random peptide mixtures entrapped within a copper-cuprite matrix: new antimicrobial agent against methicillin-resistant Staphylococcus aureus. Sci Rep 9, 11215 (2019). https://doi.org/10.1038/s41598-019-47315-0

Download citation

• Received:

• Accepted:

• Published:

## Further reading

• ### Inactivation of RNA and DNA viruses in water by copper and silver ions and their synergistic effect

• Mona Y.M. Soliman
• , Gertjan Medema
• , Boris Estrada Bonilla
• , Stan J.J. Brouns
•  & Doris van Halem

Water Research X (2020)

• ### Antimicrobial Activity of Nitrogen-Containing 5-α-Androstane Derivatives: In Silico and Experimental Studies

• Lela Amiranashvili
• , Nanuli Nadaraia
• , Maia Merlani
• , Charalampos Kamoutsis
• , Anthi Petrou
• , Athina Geronikaki
• , Pavel Pogodin
• , Dmitry Druzhilovskiy
• , Vladimir Poroikov
• , Ana Ciric
• , Jasmina Glamočlija
•  & Marina Sokovic

Antibiotics (2020)

## Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

## Search

### Quick links

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing


text:
Chapter 4 - Polynomials - 4.7 Polynomials in Several Variables - 4.7 Exercise Set - Page 284: 58

$0.04-r^2t^2$

Work Step by Step

Using $(a+b)(a-b)=a^2-b^2$ or the product of the sum and difference of like terms, the given expression, $(rt+0.2)(-rt+0.2) ,$ simplifies to \begin{array}{l}\require{cancel} (0.2+rt)(0.2-rt) \\\\= (0.2)^2-(rt)^2 \\\\= 0.04-r^2t^2 .\end{array}

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.


text:
# How to calculate the length orthogonal projection of a point onto a line

Let's say you have point A, B, C, these are represented by vectors:

($$x_a$$, $$y_a$$), ($$x_c$$, $$y_c$$), ($$x_c$$, $$y_c$$)

Imagine you draw a line across B and C, how do I find the length of the orthogonal projection of A to the line represented by B,C

I know how to calculate the orthogonal projection of 2 vectors (Which I learned in undergrad linear algebra). But I don't think I learned how to project a vector onto a line that is formed by 2 vectors

The line whose length you have labeled as "$$?$$" is the vector $$\overrightarrow{AB}-\text{proj}_{\overrightarrow{BC}}(\overrightarrow{AB})$$. So if you can calculate the projection of one vector onto another, then you can calculate the length of that line.

• Is vector AB = A - B? God I am so stupid. Sorry this seems really obvious now Feb 5 '19 at 2:31
• No, $\overrightarrow{AB}= B-A$.
– kccu
Feb 5 '19 at 2:32
• I have marked it as the correct answer. I don't know how to face my linear algebra teacher. Thank you very much for helping me Feb 5 '19 at 2:40
• Actually does the direction of the vector even matter? like vector AB vs vector BA. The length of projection should be same regardless of which direction no? Feb 5 '19 at 2:41
• @samol - don't be too hard on yourself. Feb 5 '19 at 3:21


text:
This is a testing site only. See the live Public Lab site here »

Nitrogen nitrogen

Nitrogen or #nitrate contamination of drinking and surface water poses serious health risks, and can result from agricultural runoff in rivers. It is often associated with algal blooms as well. Also see the topic #phosphate. ## Nitrogen: Nitrates, Nitrite, Ammonia, & Ammonium (moved from the [Common Water Contaminants page](/common-water-contaminants)) Nitrates, Nitrite, Ammonia, & Ammonium are all "fixed" forms of nitrogen available to living organisms, and represent different stages of nitrogen in the [nitrogen cycle.](https://en.wikipedia.org/wiki/Nitrogen_cycle) Nitrogen is a major limiting nutrient in plant growth-- when nitrates occur in large quantities in water from fertilizers, manure, or sewage runoff, they can cause algal blooms that create dead zones. Nitrates have also been linked to increased risks of [cancer, and complications with a number of diseases, including asthma](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1310926/). The EPA limits drinking water concentrations of Nitrates to 10mg/L or lower, however, health threats can occur even at those levels. Ammonia (NH3) and ammonium ion (NH4+) are related chemically by an acid / base reaction: NH3 + H+ <=> NH4+ The equilibrium between ammonia and ammonium is pH dependent, with the two species being at the same concentration only under fairly alkaline conditions, pH = 9.25. At neutral pH, the concentration of free NH3 is less than 1% that of NH4+. Since this is the case, the concentration of ammonia is usually not significant and can be determined from the concentration of ammonium and the pH. **** ## Questions Questions can be either frequently asked questions, or "next step" challenges we're looking to solve. [questions:nitrogen] ## Activities [activities:nitrogen] ...

Author Comment Last activity Moderation
GLwater "The Great Lakes basin has a few nonprofits as well as water treatment utilities working with gardening groups and some county extension Master Gard..." | Read more » over 2 years ago
joyofsoy "Shared this on the Public Lab Twitter and received this response: Measuring conductivity might be a simple proxy...— Jérôme Pellet (@jeromepellet)..." | Read more » over 2 years ago
Ag8n " My mistake, it chemiluminesces in the red, not the blue. Sorry about that. " | Read more » about 3 years ago
Ag8n " I've used ion chromatography(IC) to analyze for a number of ions, including nitrate and nitrite. The new instruments dedicated to IC are much bet..." | Read more » about 3 years ago
Ag8n "Warren got most of the problems with kjeldahl. Here are a few more. Kjeldahl testing takes some specialty glassware. It's not terribly hard to l..." | Read more » about 3 years ago
guolivar " All the way in New Zealand we also use Gradko for all our passive tubes needs. " | Read more » about 3 years ago
Bronwen "I did a little browsing and so far tthe Ormantine link @warren shared is the only US supplier that comes up quickly (will keep looking), and I have..." | Read more » about 3 years ago
warren " https://ormantineusa.com/diffusion-tubes/ is in the US! Louise Francis mentioned the lab they work with gets them from Gradko -- https://www.grad..." | Read more » about 3 years ago
stevie "Yes! here's the report back on the event https://publiclab.org/notes/Delaney/05-04-2018/water-quality-testing-at-turkey-creek " | Read more » over 3 years ago
belkinsa "Is the report ready? " | Read more » over 3 years ago
WaterGeek "Great work and super useful for citizen's looking to track the quality and state of open water. We've been looking into something like this for dri..." | Read more » over 4 years ago
MadTinker "I would add the following: alkalinity, hardness, and perhaps ORP " | Read more » over 4 years ago
jpkearns "Our systems are simpler, lower in cost, and made from common local materials. Please get in touch if you have questions - www.aqsolutions.org " | Read more » over 6 years ago
emontoya57 "Very inspiring nice work!!! " | Read more » over 6 years ago
eustatic "very cool. good question regaring the water pressure control, I've often wondered that myself. " | Read more » over 6 years ago
liz "FYI Public Lab, additional informal conversation on this is also happening on the WeChat app -- let us know if you want to join there " | Read more » over 6 years ago
chemcarr "This is very interesting. I am also working towards this kind of analysis, and especially open source accessible, whether fabrication or software/k..." | Read more » about 7 years ago
Frikkie "Just noticed video1's name show nitrate. Its meant to be nitrite, sorry about that. " | Read more » about 7 years ago
Phil "This is wicked Stuff! Thanks for sharing! " | Read more » about 7 years ago
Frikkie "For anyone who wants to try out the excel tool, here is the video’s for the recordings I made for the colorimetric nitrite analysis, the EC and fla..." | Read more » about 7 years ago
cfastie "Guillaume, Moving the entrance slit farther from the grating might help. I found that the grating does not have to be angled so sharply when the gr..." | Read more » over 7 years ago
Guillaume123 "To Chris It will be relatively easy for me to take out the slit adjusting mechanism and move it further from the diffraction grating adjustment by..." | Read more » over 7 years ago
cfastie "Guillaume, That was a great idea to allow adjustments to the angle of the slit and grating. You should be able to find positions that produce very..." | Read more » over 7 years ago
Guillaume123 "Thanks idahoboy1941! I have looked at the Google entries. Many of them require payment, but one by WG Fastie is immediately interesting. I can eas..." | Read more » over 7 years ago


text:
# Tag Info

110

In short, it is more than a belief: there is strong evidence that humans are not good entropy sources. There is a test for this Man vs. Machine. Or, why Man is not a Particularly Good Source of Entropy. Try to win! So we don't rely on whether generating a random number from the mind or random keyboard typings and mouse movements that seem like a monkey ...

38

What are the criteria that make an RNG cryptographically secure? In short, a DRBG [deterministic random bit generator] is formally considered computationally secure if a computationally-limited attacker has no advantage in distinguishing it from a truly random source. What does this mean? Given a DRBG F and a truly random oracle G, let A be a probabilistic ...

35

The ISO/IEC 9899:1990 edition of the C standard contains: EXAMPLE     The following functions define a portable implementation of rand and srand. static unsigned long int next = 1; int rand(void) // RAND_MAX assumed to be 32767 { next = next * 1103515245 + 12345; return (unsigned int)(next/65536) % 32768; } void srand(unsigned int ...

32

A colleague of mine told me about a website that, given a sufficient quantity of output from an PRNG, had been able to deduce which application the PRNG was from. As you correctly identified this would present an immediate and probably devastating attack to any cryptographic PRNG as it indeed would allow you to easily distinguish a random string from a PRNG ...

31

For me, the fraud-related applications of Benford's Law come to mind. When people make up data they tend to create overly uniform data, even when it's not appropriate. There's a definite psychology going on that may cause people to be less random than they are intending to be (Wikipedia links to a paper claiming humans are in fact bad at this). Or perhaps ...

28

One tool that tries to do this is untwister. It's almost certainly not the tool you were thinking of, though, as it cannot determine if the output came from OpenSSL specifically. It can determine Glibc's rand(), Mersenne Twister (MT19937), PHP's MT-variant (php_mt_rand), Ruby's MT-variant DEFAULT::rand(), and Java's Random() class, though, and can recover ...

22

A one-time pad requires a true random sequence that is as long as the material you want to encrypt. If you have a pseudo-random sequence, then you don't have a one-time pad: you have a stream cipher. If you have a stream of data that is only “nearly random”, then you don't have a one-time pad, you have a broken stream cipher. Concretely, if the nearly-...

21

I have the first 40 numbers of the sequence. Is there a way to recover the seed or find the next 460 numbers in the sequence? The first thing to know is that Python's random module uses Mersenne Twister as the PRNG. That is not a cryptographically secure RNG, in fact it is easy to recover the state as long as you have enough samples. 40 numbers of the ...

21

The other answers provide very good lists of reasons not to use Twitter as an entropy source. What follows is the flip side of your question:- Why would you want to? Tweets are typically read on tablets, PCs and phones. All of those have access to hardware entropy sources that can produce oodles of truly random bits for seeding anything. The zeitgeist is ...

20

What you are suggesting is not a good idea for a general purpose random number generator. It could be meaningful for very specific use cases if you need a random number generator whose output can be verified independently by a third party. Even in those cases there are other sources of entropy which are potentially more suitable. The oldest mention of this ...

18

Why would a dice rolled be "more random" than simply coming up with a sequence in your head, and then changing some of them? Humans have too many biases regarding what a random sequence is. If you ask humans to generate a random sequence, they will probably pay attention not to use the same character in a row, i.e., aa or bb, as they think that ab ...

17

The answer is given by Henrick is good, but I try to give a explanation with more details in security area. When you think about PRF (Pseudo Random Function), you will think that there are three elements with PRF, which are $K, X$, and $Y$. $K$ is the keyspace, $X$ the message or input space and $Y$ the output space. PRF is a function, when you give this ...

17

I tried to use mostly non-jargon to make it more understandable than the current top answer. What are the criteria that make an RNG cryptographically secure? From en.wikipedia.org/wiki/CSPRNG: Given all outputs so far, there must not be any algorithm that predicts future outputs with anything better than guessing. If you can say "there is a 50.0001% ...

15

Python uses a Mersenne twister PRNG, and though it is not secure it does have a large state. You have here 40 numbers, the first one gives you 1 bit and each subsequent number has an extra bit for a total of 800 bits. This is significantly smaller then the internal state of the MT-19937. This page explains how to find the internal state of python's PRNG: ...

15

How are you going to decide which tweet to use? Randomly? This quickly leads to a chicken / egg problem. What if the chosen tweet is one word? That would not add a lot of entropy. What if twitter is unavailable? Are you just stopping your service that relies on the entropy or are you going to continue regardless? How are you going to keep the chosen tweet ...

15

Randomness is a measurable, statistical property of a set of values. It doesn't mean the same as "hard for a human to guess." Your sample string is hard for a human to guess, but it isn't very random. There is a tool called "ent" for most Unix systems that can quantify the randomness, by some measures, of a file. Available here: https://...

14

Notations: $v=u\bmod m$ means $m$ divides $u-v$ and $0\le v<m$, including if $u<0$. All variables are non-negative integers (except for the above). $a=214013$, $b=2531011$ are the LCG parameters. $X_n$ is the 31-bit state with $X_{n+1}=(a\cdot X_n+b)\bmod 2^{31}$. $R_n=\lfloor X_n/2^{16}\rfloor$ is the 15-bit output. $S_n=X_n-2^{16}\cdot R_n$ is the ...

14

No, that would not be a true RNG, because these physics engines would just repeat the exact same calculation and thus repeat the whole sequence of random numbers - like a PRNG. The starting conditions are the seed of this PRNG. Dice are truly random in the real world. Well, are they? If we ignore quantum effects, we could measure all relevant values of the ...

14

On modern CPUs, a fast Cryptographically Secure Pseudo-Random Number Generator runs sizably faster than one cycle per byte. We are talking >40Gbit/s. See numbers there. Top contenders are AES-CTR assisted by special instructions, and ARX ciphers like ChaCha. When using dedicated hardware, the true limit is moving around the generated random bits. We can ...

14

People are not that bad, but we're slow. See How were one-time pads and keys historically generated? In summary, MB's of 100% secure key material were generated for one time pads by people simply key smashing on type writers. Sufficient to win three world wars. It's just that a human's entropy rate is a little lower than a laser phase based TRNG. ...

13

The problem with questions that ask for “the fastest” is, that such questions always raise the counter-question: compared to what exactly? Also, your question doesn’t specify if you mean cryptographically secure physical random number generators, or any physical random number generator. Anyway… 400 Mbps doesn’t really come anywhere near the word “fastest”. ...

13

I once played this online game, it was an old-school MUD. You log in, chat, kill some goblins. It had a casino. You go into the casino and you bet X gold, and there was a 40% chance you win double your bet. Obviously in the long run, the casino will always win, right? But here's the thing. I knew the game was written in C++, and I knew the rand() ...

13

There is a black-box separation between one-way functions and collision resistant hash functions. This was proven at Eurocrypt 1998 by Dan Simon, in the paper entitled Finding collisions on a one-way street: Can secure hash functions be based on general assumptions?. Of course, this doesn't mean that it's not possible using non-black-box reductions, but no ...

12

The seed of a pseudorandom number generator — whether cryptographically secure of not — is the initial input that defines the pseudorandom sequence of outputs generated from it. It's not really a term that's specific to cryptography, except insofar as there's a considerable amount of overlap between pseudorandom number generation and cryptography, which ...

11

I am the designer of the random number generator that is behind the Intel RdRand instruction. How feasible is it that the chip's manufacturer can predict the output of this PRNG when it passed tests from the people applying the use of this RdRand instruction in kernels? It isn't. We cannot. It passes the tests because it is a cryptographically ...

11

...wouldn't key still get repeated every few hours or so - i.e. you come to the end of the PRG(K)... This is where you are mistaken. Modern cryptographic PRGs simply do no repeat within any conceivable time frame. That is, starting from a seed, a well-constructed PRG (and this is true even when they are not so well constructed, like RC4) will simply never "...

11

A TRNG is never used instead of a CSPRNG. They serve different purposes. A TRNG is used to seed a CSPRNG. A CSPRNG alone isn't enough to generate random data since it's reproducible. A hardware entropy source alone isn't enough to generate random data because all entropy sources have biases. For any purpose that's related to security or cryptography, a ...

11

No, it doesn't need a random nonce. Yes, if you use an incrementing counter, that works. As the RFC says, the only requirement is uniqueness; as long as you make sure that each nonce you use is different, you have met the requirements - an incrementing counter does that quite nicely (and, in fact, is commonly used in practice)

10

Please bear in mind that this information is all secondhand. I have not looked closely at the original drafts of Hash DRBG (although you might find a draft that's early enough if you peruse the FOIA results in [1]). However, during conversations with folks at NIST I was told that there were certain weaknesses in early drafts of Hash DRBG that were very ...

10

Trying to distinguish a synchronous stream cipher from a CSPRNG seems to me a bit like trying to distinguish ice from frozen water. Any secure stream cipher is a CSPRNG, and any CSPRNG can be used as a stream cipher. Insofar as there is any difference, it mostly comes down to intended purpose and API design. A typical CSPRNG API might take an initial seed ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
Browse Questions

# Find the equations of the hyperbola satisfying the given conditions: Foci: $(\pm 5, 0)$, Transverse Axis is of length $8$.

Toolbox:
Given Foci: $(\pm 5, 0)$, Transverse Axis is of length $8 \rightarrow 2a = 8 \rightarrow a = 4$ and $c = 5$
Since $a^2+b^2 = c^2 \rightarrow b = \sqrt{25-16} = \sqrt {9} = 3$
Therefore, the equation of the hyperbola is as follows: $\;\large\frac{x^2}{16}$$-\large\frac{y^2}{9}$$=1$


text:
1. Posterior p.f.

Suppose that the proportion Theta of defective items in a large manufactured lot is known to be either .1 or .2 and the prior p.f. of Theta is as follows:
E(.1)=.7 and E(.2)=.3
^Those E's are a greek letter in the text, I'm guessing representing a prior distribution.
Suppose also that when 8 items are selected at random from the lot, it is found that exactly two of them are defective. Determine the posterior p.f. of Theta.

Any help?? This is our first day doing this so I am very lost...

2. First, let X denote the number of defectives in our sample of 8. Then, X|theta ~ Bin(8, theta), clearly; using Bayes rule, we get
$\displaystyle
\xi(\theta|x) = \frac{\xi(\theta)f(x|\theta)}{f(x)}
$

With this, we now solve the problem in two steps. First, we need f(2), which we get via
$\displaystyle
f(2) = \sum_{\theta \in \{.1, .2\}} \xi(\theta)f(2|\theta).
$

Once you have this, just calculate xi(.1|2) and xi(.2|2), and you are done.


text:
Question #458d1

Jan 30, 2017

Divide the circle into degrees!

Explanation:

There are $360$ degrees in a circle.

If $\frac{2}{5}$ are shaded, then $\frac{3}{5}$ are not.

Then find how many degrees $\frac{3}{5} \text{ths}$ of a circle is:

$\frac{3}{5} \times \text{360 degrees" = "216 degrees}$


text:
# Minimum Sampling frequency for this signal to avoid aliasing

I would like to compute the minimum sampling frequency for the signal $x(t)$ given below:

$x(t)= sinc ^3(300t) * sinc ^5(600t)$ and $*$ stands for convolution.

where the sinc function is defined as $sinc (w t) = sin (\pi wt)/(\pi wt)$.

I have tried like this : $sinc^3(300t)$ will correspond convolution of 3 rectangular functions in frequency domain which results in a frequency limited function to 3 times 150 Hz ie 450 Hz. And the other function corresponding to maximum of frequency 5 times 300 Hz which is 1500 Hz. Now finally convolution operator get transformed to multiplication in frequency domain , is the maximum frequency content limited to 450 Hz and minimum sampling frequency 900 Hz?

It would be helpful if someone shows me the derivation of fourier transform of sinc function for this purpose.

• Have you consulted your textbook or class notes? – John Dec 7 '13 at 17:07

## 1 Answer

There's several different ways you can go about solving this problem just in the time domain. Use trig identities or Eulers formula to reduce the equation. Both are simple enough and just require some time to work through. Feel free to use WolframAlpha to check your work. I find it sometimes helpful for tedious algebra.

Yes, the Fourier transform of a sinc function is a rectangular function, so you will have what you've described as convolved rectangular functions. This is more work than is necessary as after you've simplified the time-domain expression you will be able to find the maximum frequency, and subsequently the required sampling rate, for that signal.

• wolframalpha is really great. I really appreciate your help. – dexterdev Dec 9 '13 at 13:15