text:
# nontrivial isomorphisms of categories

First of all, I know the concepts of isomorphism and equivalence between categories, and that the latter one is the more interesting one, whereas the first is rather rare and uninteresting.

Are there isomorphisms of categories, which are not trivial and not pathological? I regard the examples on wikipedia as trivial, because these are only reformulations of the definitions of the objects in consideration. Thus perhaps the question is: Are there nontrivial reformulations?

There are lots of nontrivial equivalences of categories (affine schemes <-> rings (dual), compact hausdorff spaces <-> unital commutative C*-algebras (dual), finite abelian groups <-> finite abelian groups (dual), skeletons such as the algebraic extensions of function fields over fixed prime fields in the category of fields), but I wonder if these categories are actually isomorphic. Of course, in the examples of interest, you can't take the known equivalence as an isomorphism, but perhaps there is another one?

-
I'd conjecture that every pair of isomorphic 'real life' categories is a trivial example, but I'd surely love to see an example! –  Mariano Suárez-Alvarez Feb 3 '10 at 16:59
is this question appropriate for community wiki? –  Martin Brandenburg Feb 3 '10 at 18:28
I am puzzled/confused by something. The wiki page claims that a functor F:C \to D is an isomorphism of categories iff it is bijective on objects and on morphism sets. Does this not apply to the case of unital abelian C*-algebras and CHff spaces? –  Yemon Choi Feb 3 '10 at 18:52
@Yemon: I think you are confusing "equal to" with "is isomorphic to". "Every X is isomorphic to the space of functions on some Y" is not the same as "Every X is, as a set, exactly equal to the set of functions on some Y (and so in particular every element of X is a set which happens to be a function)" –  Kevin Buzzard Feb 3 '10 at 18:57
Maybe this is not what you are looking for, but anyway. Consider for example the category of complexes of abelian groups. Then a non trivial isomorphism to itself is given by degree shifting. –  Jan Weidner Feb 3 '10 at 21:14

Whether this counts as trivial is a subjective matter, but here goes.

Any adjunction $$F: C \to D,\ \ \ G: D \to C$$ (with $F$ left adjoint to $G$) gives rise canonically to a monad $T = GF$ on $C$ and a "comparison" functor $K: D \to C^T$. Here $C^T$ is the category of algebras for the monad $T$. The adjunction is said to be monadic if $K$ is an equivalence of categories.

Now in fact, for most of the obvious examples of monadic adjunctions, the comparison is actually an isomorphism. For example, if $G$ is the forgetful functor from groups to sets then it's an isomorphism. The same is true if you replace groups by any other algebraic theory (rings, Lie algebras, etc).

Indeed, if you look in Categories for the Working Mathematician, you'll see that Mac Lane calls a functor monadic if $K$ is an isomorphism. He does the whole basic theory of monads with this definition. I suspect this is because $K$ really is an isomorphism in the standard examples. CWM was published in 1971, and since then it's become clear that Mac Lane's definition was too narrow. Whether the pioneers of monad theory (such as Beck) also used this narrow definition, I don't know.

-
this isomorphism for algebraic theories is trivial (actually, $K$ is then the identity, using the standard definitions!), but I found this very interesting. therefore I'll vote it. –  Martin Brandenburg Feb 3 '10 at 17:49
Glad you found it interesting; thanks for saying so. I'm puzzled as to why you say K is the identity, though - I think you and I must be using different definitions. E.g. if G is the forgetful functor from D = Grp to C = Set then - according to what I think of as the standard the definitions - an object of D is a group (formally, a triple (X, m, e) where X is a set, m is multiplication, and e is an identity element) whereas an object of C^T is a pair (X, h) where X is a set and h is a function from (free group on X) to X satisfying some axioms. So K isn't then the identity. –  Tom Leinster Feb 4 '10 at 2:31
Sorry I confused something. –  Martin Brandenburg Feb 4 '10 at 2:56

Here are two of my favorite examples, both taught regularly to undergraduates:

Galois extensions: if $L$ is a Galois extension of $K$ with Galois group $G$, then the opposite of the category of orbits $G/H$ and $G$-maps is isomorphic to the category of intermediate fields, via $G/H\mapsto L^H$.

The categories of finite $T_0$-spaces and finite posets are isomorphic; the categories of Alexandroff $T_0$-spaces and all posets are isomorphic.

As Tom says, trivial is subjective, but these are certainly both elementary and illuminating. The first subsumes a bunch of things usually taught as separate propositions. The second is a bridge between algebraic topology and combinatorics.

-
I see Martin added the second example, but I think it wasn't there when I started answering :) –  Peter May Jun 28 '12 at 17:55

One general rule that unites some of the examples above is that if you have two categories whose objects are sets endowed with some structure, and there is an equivalence between these two categories that assigns to a set with a structure the same set with a different (but equivalent) structure, than such an equivalence of categories is an isomorphism of categories. One can also have objects of some other fixed category in place of sets and some collections of morphisms in place of the structures on sets (see the very last example below).

To give a simple nontrivial example of this, the category of $G$-modules for a group $G$ is isomorphic to the category of modules over the group ring $\mathbb{Z}[G]$, or the category of modules over a Lie algebra $\mathfrak{g}$ is isomorphic to the category of modules over its enveloping algebra $U(\mathfrak{g})$, or the category of comodules over a finite-dimensional coalgebra $C$ is isomorphic to the category of modules over the dual algebra $C^\ast$.

Another series of examples of isomorphisms of categories is provided by equivalences between categories whose classes of objects are the same though morphisms are different but isomorphic. This includes equivalences between various quotient categories or localizations of a given category (which all have the same objects as the original category).

Here is another example of this kind. Let $C$ be a category, $R:C\rightarrow C$ be a monad on $C$, and $L:C\rightarrow C$ be a functor left adjoint to $C$. Then $L$ is a comonad. The categories of $R$-algebras and $L$-coalgebras in $C$ can be quite different. However, one can consider the category of free $R$-algebras in $C$; this is a category whose objects are formally just the objects $X$ of $C$ while morphisms $X\rightarrow Y$ are the $R$-algebra morphisms $R(X)\rightarrow R(Y)$. Analogously one defines the category of cofree $L$-coalgebras in $C$ whose objects are the objects $X$ of $C$ and morphisms are the $L$-coalgebra morphisms $L(X)\rightarrow L(Y)$. Then the categories of free $R$-algebras and cofree $L$-coalgebras are isomorphic; this is called the isomorphism of Kleisli categories. To give a concrete example of this, the categories of cofree left comodules and free left contramodules over a given coalgebra are isomorphic.

To compare, when $L:C\rightarrow C$ is a monad and $R:C\rightarrow C$ is right adjoint to $L$, then $R$ is a comonad and the whole categories of $L$-algebras and $R$-coalgebras in $C$ are isomorphic.

-

A different flavor of example:

Connes' cycle category Λ can be described as follows. It has one object (n) for each positive integer n which we think of as an oriented circle with n marked points. A map from (n) to (m) is an isotopy class of degree 1 maps which sends marked points to marked points. Alternatively, we can think of it as a map between the sets of marked points which preserves the cyclic orderings. (Note: I am calling (n) what is usually called something like [n-1], for reasons that aren't relevant here.)

Given a map f : (n) → (m), we can also look at what happens to the intervals of the circle between the marked points. Each interval in (m) is hit by exactly one interval of (n), and the data of, for each interval of (m), which interval of (n) hits it, determines f. So, f also determines a map from an arrangement of m arcs on a circle to an arrangement of n arcs on a circle. The conclusion: Λ is isomorphic to its opposite category Λop.

If you prefer working with the presentation of Λ by generators and relations, then the generator $d_i$ corresponding to inserting a new point in an interval is "dual" to the generator $s_j$ collapsing the two resulting intervals to one (and rotation is "dual" to rotation).

This fact is worth knowing when learning about Hochschild homology if only so that you don't use it accidentally! If you lose track of whether you are attaching your algebra to the marked points or the intervals of the circle, confusion will ensue.

-
I'm afraid that in my case confusion always ensues when I start messing around with cyclic cohomology ;) –  Yemon Choi Feb 4 '10 at 4:11

I don't know if the following example may be considered as trivial, but it's quite useful.

Let $\cal{C}$ be a category, $\cal{S}$ a class of morphism in $\cal{C}$.

Assume that, for instance, $\cal{S}$ is a class of homotopy equivalences. By which I mean that you have a cylinder (or path object) for every object in $\cal{C}$ -for example, because it is a Quillen model category-, and $\cal{S}$ is the class of morphism which are invertible up to the homotopy relation $\sim$ generated by these path or cylinder objects.

Then, on one hand, you can consider the quotient category $\cal{C}/\sim$, whose objects are those of $\cal{C}$ and whose morphisms are the homotopy classes of morphisms.

On the other hand, you can consider the localized category $\mathrm{Ho}\cal{C}$, with the same objects, but inverting the morphisms of $\cal{S}$.

Well, at least when your homotopy relation $\sim$ is generated by a cylinder or path object, these two categories are canonically isomorphic.

Remark. Do not confuse my statement with Quillen's equivalence of categories. I'm sorry for the notation $\mathrm{Ho}\cal{C}$, but I don't know how to write square brackets here.

-
What you are saying isn't true for an arbitrary Quillen model category C, e.g., this isn't true for the model category of topological spaces with weak homotopy equivalences as weak equivalences. To make your assertion correct in general, one has to take C to be the full subcategory of fibrant-cofibrant objects in a model category, rather than the whole model category. –  Leonid Positselski Feb 3 '10 at 18:57
No. I'm afraid you are mistaking what I'm saying. The class S is not a class of "weak equivalences", but of "homotopy equivalences". Maybe this is my fault because of my notation "HoC": notice that I said that this is the localized category respect_to_S, not respect any class of "weak equivalences" that I didn't need at all. You can find the proof in "A Cartan-Eilenberg approach to homotopical algebra", JPAA 214, 140-164 (2010), proposition 1.3.3 and example 1.3.4. –  a.r. Feb 3 '10 at 19:30
Oh, so I misunderstood you, sorry. Thanks for the reference. –  Leonid Positselski Feb 3 '10 at 19:42

The categories of Boolean algebras and of Boolean rings (rings in which $a^2=a$ for all $a$) are isomorphic. The reason is that given a Boolean ring $(R,+,\cdot,0,1)$ one can define a Boolean algebra structure on its underlying set one can define $a \wedge b:= a \cdot b$ and $a \vee b:= x+y+x\cdot y$ and $\neg a:=1+a$ and gets a Boolean algebra.

Vice versa, given a Boolean algebra $(B,\vee,\wedge,0,1)$ gives a Boolean ring via $a \cdot b:=a \wedge b$ and $a+b:=(a \vee b) \wedge \neg (a \wedge b)$

If you go back and forth you get exactly the same ring/Boolean alg. structure, the underlying set didn't change anyway. I don't know if you consider this non-trivial. But I think an isomorphism of categories should be thought of as reformulation of structure.

-

For any locally profinite group $G$, the category of smooth representations of $G$ is on the nose isomorphic to the category of smooth modules over its Hecke algebra.

-

Isn't it the case that, if $C$ and $D$ are equivalent categories and if, in both of these categories, each object is isomorphic to a proper class of other objects, then $C$ and $D$ are isomorphic (assuming global choice)? So, for example, the category of non-trivial commutative rings and the dual of the category of nonempty affine schemes are isomorphic. (I had to exclude the empty scheme, and therefore the trivial ring, because there's only one empty scheme but lots of trivial rings, which would mess up any attempt at an isomorphism.) More generally, if $F:C\to D$ is an equivalence of categories and if, for each object $a$ in $C$, the number of isomorphic copies of $a$ in $C$ equals the number of isomorphic copies of $F(a)$ in $D$, then there should (again with a generous use of choice) be an isomorphism from $C$ to $D$ (that is, furthermore, naturally isomorphic to the given $F$).

EDIT: Martin asked in a comment for a proof; I'll put a proof (or at least a sketch, which I hope will suffice) into the answer because it won't fit into a comment. Suppose $F:C\to D$ is an equivalence and, for each object $a$ of $C$, the isomorphism classes of $a$ and $F(a)$ are the same size. In $C$, choose one representative object from each isomorphism class of objects; write $a^*$ for the representative of the isomorphism class of $a$. Also choose, for each object $a$, an isomorphism $i_a:a\to a^*$, subject to the convention that $i_{a^*}$ is the identity morphism of $a^*$. Do the same in $D$, but, instead of arbitrarily choosing the representative objects, use the objects $F(a^*)$; there's exactly one of these in each isomorphism class, because $F$ is an equivalence. But the isomorphisms $i_b$, from objects $b$ of $D$ to the representatives, are still chosen arbitrarily except that, as before, for the representatives themselves we use identity morphisms. Now define a new functor $F':C\to D$ as follows. On the representative objects $a^*$, it agrees with $F$. On other objects, it acts in such a way that the isomorphism class of any $a^*$ is mapped bijectively to the isomorphism class of $F(a^*)$; this is possible because I assumed that these isomorphism classes have the same size. Finally, if $f:a\to b$ is a morphism in $C$, then $F'$ should send it to the following mess: $$i_{F'(b)}^{-1}F(i_bf{i_a}^{-1})i_{F'(a)}.$$ In perhaps more understandable language: Use $i_a$ and $i_b$ to transport $f$ to a morphism from $a^*$ to $b^*$, apply $F$ to that, and then transport the result to map $F'(a)\to F'(b)$ via the chosen isomorphisms in $D$. It should be routine to check that this $F'$ is an isomorphism.

-
Since a scheme consists of a topological space and a sheaf of rings, it is not true that there is only one empty scheme. In fact, there is a canonical bijection between the set of empty schemes in a given universe and the set of trivial commutative rings (i.e., rings of cardinality 1) in the same universe. –  Fred Rohrer Jun 28 '12 at 17:47
Thanks, Fred. So my answer can be improved by deleting "non-trivial" and "nonempty", so as to exactly match one of the equivalent pairs mentioned in the question. –  Andreas Blass Jun 28 '12 at 18:00
Andreas, what is the proof of your claim? –  Martin Brandenburg Jun 28 '12 at 19:33
Martin: This is part of Exercise A in Chapter 3 of Freyd's "Abelian Categories" (page 74). I was tempted to say that MO is not for homework, so I'd omit the proof. But that seems like cheating, so I edited a sketch of the proof into my answer. –  Andreas Blass Jun 28 '12 at 20:18
@Martin: I was assuming global choice, which makes all proper classes have the same size; every proper class is equinumerous with the class of all ordinals. By the way, the exercise in Freyd's book is about the special case where all the isomorphism classes are proper classes. But the proof is the same if some (or all) of them are sets, as long as corresponding ones have the same size. –  Andreas Blass Jun 28 '12 at 22:40

The category of inverse semigroups is isomorphic to the category of etale groupoids whose unit and arrow spaces are Alexandrov spaces and the poset associated to the unit space is required to be a meet semiattice. Morphisms are required to preserve these meets.

-

Hi Martin.

Is the following example non-trivial? There are (at least) two possible definitions of an uniform space over a set $X$:

1. A uniformity can be defined as a non-empty set $\Sigma$ of covers of $X$ such that $\Sigma$ is closed wrt "upward" refinements (i.e. $\alpha\in\Sigma \wedge \alpha\preceq\beta \implies \beta\in\Sigma$), every $\alpha\in\Sigma$ has a star-refinement in $\Sigma$.

2. A uniformity can be defined as a filter $\mathcal{R}$ on $X\times X$ such that for all $R\in\mathcal{R}$ we have $\Delta_X\subseteq R$, $R^{-1}\in\mathcal{R}$ and $\exists S\in\mathcal{S}: S\circ S\subseteq R$.

Both definitions give rise to a category of uniform spaces. Both categories are isomorphic.

-
In the same spirit, you could mention logically-equivalent definitions of topological space: as a set equipped with open subsets, or closed subsets, or a closure operator, or an interior operator, or neighbourhoods, or... All give isomorphic categories. –  Tom Leinster Feb 3 '10 at 17:26

Stone's representation theorem gives you an isomorphism between every Boolean algebra and a field of sets. Viewed categorically, this is an isomorphism of categories, since isomorphism and equivalence coincide for partial orders viewed as categories.

-
But it is not true that that every Boolean algebra is a field of sets, so one of Stone's functor is not surjective. –  Mariano Suárez-Alvarez Feb 3 '10 at 20:10
You're one level higher up than I am -- I'm talking about viewing a particular Boolean algebra as a category (ie, viewing a poset as a category). Then the isomorphism between it and the corresponding field of sets Stone's theorem gives you is an isomorphism of categories. –  Neel Krishnaswami Feb 3 '10 at 20:43

This is really a comment, not an answer. But since it is a not-so-short comment to many answers together, it had to become an answer.

It has been observed that (first-order) definitional equivalences give categorical isomorphisms, at least for categories of first order structures with isomorphisms as their only morphisms. In my opinion the fact that two equivalent definitions of a mathematical structure give the same isomorphisms but possibly different morphisms (which maps between [complete] lattices should one consider: isotone? meet-semilattice morphisms? join-semilattice morphisms? lattice morphisms? [complete join, or meet, or both, morphisms?]) is a big virtue: it means that two definitions give really different points of view on the same kind of structure (in a way, they formalize a kind of non-triviality of the equivalence). This also happens for second-order structures (complete lattices, uniform and topological spaces); the definitional equivalences are expressed in the natural language of Bourbaki's "scale of sets" (or natural model of type theory) above the base sets of the (multisorted) structure (detractors of Bourbaki and/or lowers of category theory would instead speak of the topos "somewhat freely" generated by the (sorts for the) base stets; when the equivalence of definitions is completely constructive one can really take a free topos, but depending of the principles of classical logic which are needed to prove the equivalence of definitions, one considers the topos freely generated in more restricted classes).

So in summary: syntactically defined equivalences induce isomorphisms between categories of structures. As Hodges notes (for example in his book "model theory"), pratically everything which in mathematics can "really" be considered a "construction" is formalizable as a interpretation or at least a "word-construction" (and moreover it is the syntactical form itself which shows what kind of morphisms more general than isomorphisms are "preserved" by the construction. I understand that few lovers of category theory would approve such a extreme syntactical view, but note that even the "categories, allegories" book by Freyd and Scedrov insists on the "Galois correspondence" between syntactical and semantical aspects; I simply happen to prefer the syntactical side). From this point of view, Hodges'remarks about (cases slightly more general than) adjunctions among quasivarieties (and universal Horn classes) induced by forgetful functors are related to the already given remark about monadic adjunctions.

Besides, the book "abstract and concrete categories" by Adameck, Herrlich, Strecker conains many examples of "concrete isomorphisms"; some of them shoulb be interesting (and all of them, if I remember correctly, can be seen as syntactically defined as above).

Incidentally, the three authors say that non reasonable concept of "concrete equivalence" can be given; I disagree since cases exist where two categories can be concretely reflected on full subcateories of objects "in normal form", and the subcategories are concretely isomorphic [for example, take affine geometry of dimension at least three: form affine spaces algebraically defined by points, group of translations, sfield of scalars one "normalizes" to the particular case where translations are a subgroup of the group of permutations of the points and scalars are a subring of the ring of endomorphisms of the group of translations. For affine spaces geometrically defined in Hilbert's Grundlagen style, the general case can be reflected onto the "normal" case with the same set of points where lines and planes are sets of points and incidence is the set-theoretic one]

It has already been observed that, in presence of choice, "isomorphic categories" means "equivalent categories where corresponding isomorphism classes of objects have the same cardinality". Freyd and Scedrov observe that, even in absence of choice, the "correct" notion of equivalence is: to have isomorphic inflations. This means that all usual examples of equivalence of categories induce examples of isomorphisms (without the trick with arbitrary choices to consider skeletons, but instead using canonical "inflations" of the isomorphism classes)

-

I just remember an example by myself.

Let $C$ be the following algebraic category: Objects are nonempty sets $G$ together with a binary operation $/ : G \times G \to G$, such that for all $x,y,z \in G$, we have

$x / ((((x/x)/y)/z) / (((x/x)/x)/z)) = y.$

A morphism $(G,/) \to (G',/)$ is a map $f : G \to G'$ preserving $/$.

This category is isomorphic to the category of groups, i.e. groups can be described by a single equation! If $G \in C$, then the corresponding group is $G$ together with the multiplication $ab := a/((a/a)/b)$. If $G$ is a group, then define $x/y = x y^{-1}$.

Reference: Higman, Graham und Neumann, Bernhard: Groups as groupoids with one law, Publicationes Mathematicae Debrecen, 2 (1952), 215-227.

Added 28/6/12: The category of A-spaces (topological spaces in which every intersection of open subsets is open) is isomorphic to the category of preorders. Under this isomorphism $T_0$ A spaces correspond to partial orders.

-
Doesn't really matter, but it would seem more natural to define a morphism by $f(x/y) = f(x)/f(y)$. –  Johannes Hahn Feb 3 '10 at 18:28
thanks, I've edited it. –  Martin Brandenburg Feb 3 '10 at 18:44
Why is this not a reformulation? –  Mariano Suárez-Alvarez Feb 3 '10 at 18:48
I'm looking for nontrivial reformulations, see also my question. –  Martin Brandenburg Feb 4 '10 at 2:54
Does your equation make any sense? It seems to imply $y_1 = y_2$ for all $y_1,y_2\in G$, shouldn't there be a $y$ somewhere on the left-hand side? –  Ketil Tveiten Jun 29 '12 at 9:10

In my first step as student, I still do not know the concepts of category theory, but I thinked in "naive" way about this:

Let $Top$ the category of topological spaces and continuous maps (such that the invese images of opens are still opens), and let $Fil_T$ the category with objects like $(X, F)$ where $X$ is a set, $F=(F_x)_{x\in X}$ with $F_x$ a filter of subset with $x\in \bigcap F_x$ and such that $\forall U\in F_x\ \exists V\in F_x: \forall y\in V: U\in F_y$. ANd with morphisms $f: (X, F)\to (Y, G)$ , the maps $f_ X\to Y$ such that $\forall x\in X: \forall V\in G_{f(x)}: \exists U\in F_x: f(U)\subset V$.

The usual representation of topology as a family of opens or as families of neighbrhoods, and the essential identification between these, is just a isomorphisms between $Top$ and $Fil_T$ .

Of course this is a elementary example of isomorphism of concrete categories, in general a concrete isomorphism is just a different (but "equivalent") representation of the structure on a set, but these representation can have very different formalization (see all the literature on concrete categories, especially topological etc.).

Another nice example (I think) is the isomorphism (that fix the objects) between the Kleisli categori of a triple $T$ inducted from an adjuntion $(F, G): \mathcal{A}\to \mathcal{C}$ (right notation) and the clone category $\mathcal{C}_F$ of $F$ that has the some objects of $\mathcal{C}$ and with hom-sets defined as: $\mathcal{C}_F(X, Y)=\mathcal{A}(F(X), F(Y))$ (all these are considered disjoint) with the natural composition and identities of $\mathcal{A}$.

The following article is about a very nice isomorphisms (the shape category as the some objects and the shape functor fix these..):

http://www.jstor.org/discover/10.2307/1996811?uid=3738296&uid=2129&uid=2&uid=70&uid=4&sid=21100881872361

-

I feel one should make a distinction between categories in their so-to-speak metamathematical role, for example in using results such as the Yoneda Lemma, or a left adjoint preserves colimits, and categories used as an algebraic structure with partial operations, in which guise I cite many examples of groupoids.

One such example groupoid is the unit interval groupoid $\mathbf I$ with two objects $0,1$ and exactly one arrow between any objects. This groupoid plays a similar role in the category of groupoids to the integers in the category of groups. Also the integers are obtained from $\mathbf I$ by identifying $0,1$ in the category of groupoids: this is one explanation of why the fundamental group of the circle is the integers.

That last comment is an application of the Seifert-van Kampen Theorem for the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points, a theorem which is about calculating homotopy $1$-types of not necessarily connected spaces. It is quite necessary in using this theorem to keep all the information about the way various components of the pieces intersect; moving to equivalence will destroy that information.

In the case of groupoids with structure, which may be a topology, or a smooth structure, or an algebraic structure (group, ring, Lie algebra,...) then the usual equivalence of a transitive groupoid to a group of course no longer applies to preservation of the extra structure. But there are simpler structures: for example one knows how to classify vector spaces with a single endomorphism, but how does one classify groupoids with a single endomorphism?

Maybe the situation is less clear with categories, rather than groupoids, but it may also be that these two roles, metamathematical, and an algebraic structure with partial operations, merge in some situations. For me, that is one of the fascinations of category (and groupoid) theory.

And my definition of Higher Dimensional Algebra is as the study of partial algebraic structures with operations whose domains are given by geometric conditions.

-
why did I know that your answer will contain groupoids in every paragraph, but not a single time isomorphisms? –  Martin Brandenburg Jun 29 '12 at 16:09
@Martin: Just seen this comment. I think my paper with Higgins "The equivalence of $\omega$-groupoids and cubical $T$-complexes", Cah. Top. G\'eom. Diff. 22 (1981) 349-370 shows non trivially that two structures on an underlying cubical set are equivalent, and so gives an example of isomorphic categories. In my answer, about van Kampen situations, we need to keep groupoids up to isomorphism rather than equivalence, to model the way components of spaces intersect. Groupoids have structure in dimensions 0 and 1, and this is useful in modelling the way spaces glue together. Hope that helps. –  Ronnie Brown Oct 11 '13 at 13:43


text:
# Values determined by "stopping" properties

post by Stuart_Armstrong · 2018-03-15T10:53:46.412Z · score: 22 (8 votes) · LW · GW · 16 comments

## Contents

  Stability and stopping properties
Barriers to total drift
None

The lotus-eaters are examples of humans who have followed hedonism all the way through to its logical conclusion. In contrast, the "mindless outsourcers" are a possible consequence of the urge to efficiency: competitive pressures making uploads choose to destroy their own identity.

In my "Mahatma Armstrong" version of Eliezer's CEV, a somewhat altruistic entity ends up destroying all life, after a series of perfectly rational self-improvements. And in many examples where AIs are supposed to serve human preferences, these preferences are defined by a procedure (say, a question-answer process) that the AI can easily manipulate [LW · GW].

## Stability and stopping properties

Almost everyone agrees that human values are under-determined (we haven't thought deeply and rigorously about every situation) and changeable by life experience. Therefore, it makes no sense to use "current human values" as a goal; this concept doesn't even exist in any rigorous sense.

So we need some way of extrapolating true human values. All the previous examples could be considered examples of extrapolation, and they all share the same problem: they are defined by their "stopping criteria" more than by their initial conditions.

For example, the lotus eaters have reached a soporific hedonism they don't want to wake out of. There no longer is "anyone there" to change anything in the mindless outsourcers. CEV is explicitly assumed to be convergent: convergent to a point where the idealised entity no longer sees any need to change. The AI example is a bit different in flavour, but the "stopping criteria" are whatever the human /chooses/is tricked into/is forced into/ saying. This means that the AI could be an optimisation process pushing the human to say whatever it wants us to.

Importantly, all these stopping criteria are local: they explicitly care only about the situation when the stopping criteria is reached, not about the journey there, nor the initial conditions.

Processes can drift very far from their starting point, if they have local stopping criteria, even under very mild selection pressure. Consider the following game: each of two players is to name an number, and , between and . The player with the highest number gets that much in euros, and the one with the strictly lowest one gets that much plus two in euros. Each player starts at , and each in turn is allowed to adjust their numbers until they don't want to any more.

Then if both players are greedy and myopic, one player will start by dropping to , followed by the next player dropping theirs to , and so on, going back and forth between the players until one stands at and the other at . Obviously if the could be chosen from a larger range, there's no limit to the amount of loss that such a process could generate.

Similarly, if our process of extrapolating human values have local stopping criteria, there's no limit to how bad they could end up being, or how "far away" in the space of values they could go.

This, by the way, explains my intuitive dislike for some types of moral realism. If there are true objective moral facts that humans can access, then whatever process counts as "accessing them" becomes... a local stopping condition for defining value. So I don't tend to focus on arguments about how correct or intuitive that process is; instead, I want to know where it ends up.

## Barriers to total drift

So, how can we prevent local stopping conditions from shooting far across the landscape of possible values? This is actually a problem I've been working on for a long time; you can see this in my old paper "Chaining God: A qualitative approach to AI, trust and moral systems". I would not recommend reading that paper - it's hopelessly amateurish, anthropomorphisising, and confused - but it shows one of the obvious solution: tie values to their point of origin.

There seem roughly three interventions that can be done to overcome the problem of local stopping criteria.

• I. The first is to tie the process to the starting point, as above. Now, initial human values are not properly defined; nevertheless it seems possible to state that some values are further away from this undefined starting point than others (paperclipers are very far, money-maximiser quite far, situations where recognisably human beings do recognisably human stuff are much closer). Then the extrapolation process gets a penalty for wandering too far afield. The stopping conditions are no longer purely local.
• II. If there is an agent-like piece in the extrapolation process, we can remove rigging (previously called bias) or influence, so that the agent can't manipulate the extrapolation process. This is a partial measure: it replaces a targeted extrapolation process with a random walk, which removes one major issue but doesn't solve the whole problem.
• III. Finally, it is often suggested that constraints be added to the extrapolation process. For example, if the human values are determined by human feedback, then we can forbid the AI from coercing the human in any way, or restrict it to only using some methods (such as relaxed conversation). I am dubious about this kind of approach. It firstly assumes that concepts like "coercion" and "relaxed conversation" can be defined - but if that were the case, we'd be closer to solving the issue directly. And secondly, it assumes that restrictions that apply to humans also apply to AIs: we can't easily change the core values of fellow humans with conversation, but super-powered AIs may be able to do so.

In my methods, I'll normally be using mostly interventions of type I and II.

comment by Kaj_Sotala · 2018-03-16T15:15:09.848Z · score: 14 (4 votes) · LW · GW
Similarly, if our process of extrapolating human values have local stopping criteria, there's no limit to how bad they could end up being, or how "far away" in the space of values they could go.

I feel like there's a distinct difference between "human values could end up arbitrarily distant from current ones" and "human values could end up arbitrarily bad".

In that, I feel that I have certain values that I care a lot about (for example, wanting there to be less intense suffering in the world) and which I wouldn't want to change; and also other values for which I don't care about how much they'd happen to drift.

If you think of this as my important values being points on dimension x1-x10, and my non-important values as being points on dimensions x11-x100, then assuming that the value space is infinite, then my values on dimensions x11-x100 could drift arbitrarily far away from their current positions. So the distance between my future and current values could end up being arbitrarily large, but if my important values remained in their current positions, then this would still not be arbitrarily bad, since the values that I actually care about have not drifted.

Obviously this toy model is flawed, since I don't think it actually makes sense to model values as being totally independent of each other, but maybe you get the intuition that I'm trying to point at anyway.

This would suggest that the problem is not that "values can get arbitrarily distant" but rather something like "the meta-values that make us care about some of our values having specific values, may get violated". (Of course, "values can get arbitrarily distant" can still be the problem if you have a meta-value that says that they shouldn't do that.)

comment by Stuart_Armstrong · 2018-04-02T20:29:35.930Z · score: 5 (1 votes) · LW · GW

>and also other values for which I don't care about how much they'd happen to drift.

Hum. In what way can you be said to have these values then? Maybe these are un-endorsed preferences? Do you have a specific example?

comment by Kaj_Sotala · 2018-04-03T10:59:40.898Z · score: 6 (1 votes) · LW · GW

Off the top of my head, right now I value things such as nature, literature, sex, democracy, the rule of law, the human species and so on, but if my descendants had none of those things and had replaced them with something totally different and utterly incomprehensible, that'd be fine with me as long as they were happy and didn't suffer much.

comment by Stuart_Armstrong · 2018-04-03T19:09:36.030Z · score: 5 (1 votes) · LW · GW

If I said that some of these were instrumental preferences, and some of these were weak preferences, would that cover it all?

comment by Kaj_Sotala · 2018-04-04T16:04:27.144Z · score: 11 (2 votes) · LW · GW

Some are instrumental yes, though I guess that for "weak preferences", it would be more accurate to say that I value some things for my own sake rather than for their sake. That is, I want to be able to experience them myself, but if others find them uninteresting and they vanish entirely after I'm gone, that's cool.

(There has to be some existing standard term for this.)

comment by Stuart_Armstrong · 2018-04-04T18:17:19.365Z · score: 5 (1 votes) · LW · GW

That doesn't sound complicated or mysterious at all - you value these for yourself, but not necessarily for everyone. So if other people lack these values, then that's not far from your initial values, but if you lack them, then it is far.

comment by Kaj_Sotala · 2018-04-04T18:47:56.629Z · score: 6 (1 votes) · LW · GW
So if other people lack these values, then that's not far from your initial values, but if you lack them, then it is far.

Well, that depends on how you choose the similarity metric. Like, if you code "the distance between Kaj's values and Stuart's values" as the Jaccard distance between them, then you could make the distance between our values arbitrarily large by just adding values I have but you don't, or vice versa. So if you happened to lack a lot of my values, then our values would be far.

Jaccard distance probably isn't a great choice of metric for this purpose, but I don't know what a good one would be.

comment by Stuart_Armstrong · 2018-04-05T16:00:06.552Z · score: 11 (2 votes) · LW · GW

If we make the (false) assumption that we both have utility/reward functions, and E_U(V) is the expected utility of utility V if we assume a U maximiser is maximising it, then we can measure the distance between utility U and V as d(U,V)=E_U(U)-E_V(U).

This is non-symmetric and doesn't obey the triangle inequality, but it is a very natural measure - it represents the cost to U to replace a U-maximiser with a V-maximiser.

comment by William_S · 2018-04-01T16:34:58.522Z · score: 3 (1 votes) · LW · GW

Equivalently, we can say that we don't know how we should define the dimensions of the human values or the distance measure from current human values, and if we pick these definitions arbitrarily, we will end up with arbitrary results.

comment by Larks · 2018-03-22T23:33:37.355Z · score: 8 (2 votes) · LW · GW
This, by the way, explains my intuitive dislike for some types of moral realism. If there are true objective moral facts that humans can access, then whatever process counts as "accessing them" becomes... a local stopping condition for defining value.

I'm not sure I understand what you're getting at here. Yes, they are both local stopping conditions, but there seems to be a clear dis-analogy. The other local stopping conditions seem to be bad not because they are stopping conditions, but because most contemporary people don't want to end up as Lotus-Eaters, or as mindless outsourcers. We would oppose such a development even if it wasn't stable! For example, a future where we oscillate between lotus-eaters and mindless outsourcers seems about as bad as either individual scenario. So it's not really the stability we object to.

But in that case, it's not clear why we should be opposed to the moral realism. After all, many people would like to go there, even if we are presently a long way away.

comment by Stuart_Armstrong · 2018-03-23T14:07:27.399Z · score: 5 (1 votes) · LW · GW

My general position: if the destination is good, we should be able to tell, now, that it is at least an acceptable destination, maybe with some oddities. And by saying that, I'm saying that the destination needs to be tied to our current values, at least to some extent.

If you claim that the destination of moral realism is good, then a) demonstrate this, or b) let me add something tying it our current values anyway (this shouldn't change much, if you already assume the destination is good, and would help if it isn't).

comment by Stuart_Armstrong · 2018-03-23T14:03:37.251Z · score: 5 (1 votes) · LW · GW

(note: moral realism is complex and varied and I only sorta-understand a small portion of it; this answer is about that small portion)

We don't know anything about the features of the moral realism destination, just that it is, in some sense, the true morality. Some moral realists seem to explicitly ok with the destination being something that horrifies us today; in fact, this is even expected.

So this is advocating that we push off for an unknown moral destination, with the expectation that we will find it awful, and only local properties defined in it. Pardon my lack of enthusiasm ^_^

comment by steven0461 · 2018-03-15T15:29:47.147Z · score: 8 (2 votes) · LW · GW
nevertheless it seems possible to state that some values are further away from this undefined starting point than others (paperclipers are very far, money-maximiser quite far, situations where recognisably human beings do recognisably human stuff are much closer)

Whether a value system recommends creating humans doing human stuff depends not just on the value system but also on the relative costs of creating humans doing human stuff versus creating other good things. So it seems like defining value distance requires either making some assumptions about the underlying universe, or somehow measuring the distance between utility functions and not just the distance between recommendations. Maybe you'd end up with something like "if a hedon is a thousand times cheaper than a unit of eudaimonia and the values recommend using the universe's resources for hedons, that means the values are very distant from ours, but if a hedon is a million times cheaper than a unit of eudaimonia and the values recommend using the universe's resources for hedons, the values could still be very close to ours".

comment by steven0461 · 2018-03-15T15:35:21.846Z · score: 6 (2 votes) · LW · GW
For example, if the human values are determined by human feedback, then we can forbid the AI from coercing the human in any way, or restrict it to only using some methods (such as relaxed conversation).

It seems to me the natural way to do this is by looking for coherence among all the possible ways the AI could ask the relevant questions. This seems like something you'd want anyway, so if there's a meta-level step at the beginning where you consult humans on how to consult their true selves, you'd get it for free. Maybe the meta-level step itself can be hacked through some sort of manipulation, but it seems at least harder (and probably there'd be multiple levels of meta).

comment by Stuart_Armstrong · 2018-03-16T03:48:49.365Z · score: 5 (1 votes) · LW · GW

See my next post - human values are contradictory, meta-values especially so.

comment by steven0461 · 2018-03-16T15:49:00.667Z · score: 3 (1 votes) · LW · GW

It seems to me that the fact that we're having conversations like this implies that there's some meta level where we agree on the "rules of the game".

format: plain


text: Then the matrix for the right side. To verify that this Inverse, you can multiply the original matrix with the Inverted Matrix and you will get the Identity matrix. Python Matrix. Matrix is nothing but a rectangular arrangement of data or numbers. In numpy, you can create two-dimensional arrays using the array() method with the two or more arrays separated by the comma. Generating a (random) 2-dimensional cellular network. The elements of Q and D can be randomly chosen to make a random A. In other words, it is a rectangular array of data or numbers. Transpose is a new matrix result from when all the elements of rows are now in column and vice -versa. To multiply two matrices, we use dot() method. make_spd_matrix(n_dim, *, random_state=None) [source] ¶ Generate a random symmetric, positive-definite matrix. Symmetric difference of two sets. Read more in the User Guide. Refer to the glossary of cryptography terms for definitions of any terms used in this chapter.. Overview of Fernet. As you can see, NumPy made our task much easier. Similar like lists, we can access matrix elements using index. The singular values are the lengths of the semi-axes. Examples which violate the necessity are diagonal matrices (where only the trace contains non-zero elements). Defining a matrix; Identity matrix; Transpose matrix; In linear algebra, if the matrix and its transpose are equal, then the matrix is symmetric (MT = M). A square matrix is said to be symmetric matrix if the transpose of the matrix is same as the given matrix. A matrix is a two-dimensional data structure where numbers are arranged into rows and columns. However, we can treat list of a list as a matrix. For example: A = [[1, 4, 5], [-5, 8, 9]] We can treat this list of a list as a matrix having 2 rows and 3 columns. If you don't know how this above code works, read slicing of a matrix section of this article. We will see other type of vectors and matrices in this chapter. Roger, thanks for the question. Symmetric-Matrix. filter_none. It is the lists of the list. If you don't know how slicing for a list works, visit Understanding Python's slice notation. It is using the numpy matrix() methods. Array of integers, floats and complex Numbers. numpy.ones(shape, dtype=float, order='C') Python numpy.ones() Parameters. Python Basics Video Course now on Youtube! Be sure to learn about Python lists before proceed this article. Method: Like, in this case, I want to transpose the matrix2. 7. The horizontal entries in a matrix are called as ârowsâ while the vertical entries are called as âcolumnsâ. Python doesn't have a built-in type for matrices. There is another way to create a matrix in python. the assiment is to create this matrix in one row of code by using Matlab methods ( also multiplying metrix and Vectors are permited ). How to generate a random matrix with specific parameters? Once NumPy is installed, you can import and use it. Fernet is a system for symmetric encryption/decryption, using current best practices. In Python, I try to test the previous equality, but found that it doesn't hold for a symmetric matrix. For example: We can treat this list of a list as a matrix having 2 rows and 3 columns. For example: This matrix is a 3x4 (pronounced "three by four") matrix because it has 3 rows and 4 columns. In this section of how to, you will learn how to create a matrix in python using Numpy. Before you can use NumPy, you need to install it. Array, If you are on Windows, download and install. Subscribe to our mailing list and get interesting stuff and updates to your email inbox. # triarray Python package for working with symmetric matrices in non-redundant format. Here is a code snippet for this: n = 3 m = 3 val = [0] * n for x in range (n): val[x] = [0] * â¦ # Fills transpose of mat[N][N] in tr[N][N] Hence, this array can take values from -2-31 to 2-31-1. S = A.A T. Python code creating symmetric matrix Symmetric Matrices The eigenvalues of a symmetric matrix are always real and the eigenvectors are always orthogonal! In Python, the arrays are represented using the list data type. As you can see, using NumPy (instead of nested lists) makes it a lot easier to work with matrices, and we haven't even scratched the basics. The following equation shows how a dot product of Matrix A and A T can result in a symmetric matrix. Best way to create symmetric matrices. To find out the solution you have to first find the inverse of the left-hand side matrix and multiply with the right side. Geometrically, a matrix $$A$$ maps the unit sphere in $$\mathbb{R}^n$$ to an ellipse. Using nested lists as a matrix works for simple computational tasks, however, there is a better way of working with matrices in Python using NumPy package. There are several ways to create NumPy arrays. Slicing of a one-dimensional NumPy array is similar to a list. Python doesn't have a built-in type for matrices. We will create a 3x3 matrix, as shown below: numpy.matrix vs 2-D numpy.ndarray¶. 6. Defining Matrix using Numpy; Transpose Matrix; Here, we will learn how to create a symmetric matrix using a non-symmetric matrix? 2) Check if transpose and given matrices are same or not, C++. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share â¦ Let's see how we can do the same task using NumPy array. If you have any question regarding this then contact us we are always ready to help you. 1. You can also find the dimensional of the matrix using the matrix_variable.shape. NumPy is a package for scientific computing which has support for a powerful N-dimensional array object. 4. Matrix is a two-dimensional array. We can handle it in traditional way using python. In order to create a random matrix with integer elements in it we will use: np.random.randint(lower_range,higher_range,size=(m,n),dtype=âtype_hereâ) Here the default dtype is int so we donât need to write it. Usually people will create it as list inside list. Every element above this mirror is reflected to an element under this mirror. Be aware that a constant trace of symmetric matrices is not necessary (and maybe neither sufficient) as condition for commuting matrices. NumPy provides multidimensional array of numbers (which is actually an object). Thanks to this rule, an symmetric matrix needs to store only elements instead of elements needed to be stored in case of a classic matrix. How to Cover Python essential for Data Science in 5 Days ? We have seen in 2.3some interesting kind of matrices. Using Numpy is advised especially when you need to display the result in matrix form. numpy.ones() in Python can be used when you initialize the weights during the first iteration in TensorFlow and other statistic tasks. Array manipulation is somewhat easy but I see many new beginners or intermediate developers find difficulties in matrices manipulation. We use numpy.transpose to compute transpose of a matrix. Hyperparameters for the Support Vector Machines :Choose the Best, How to Get Input from User in Java from Command Line : 3 Approach. So now will make use of the list to create a python matrix. How do I plot Numerical Range of two hermitian matrices. Now, let's see how we can access elements of a two-dimensional array (which is basically a matrix). Now, let's see how we can slice a matrix. Numbers(integers, float, complex etc.) Matrix is widely used by the data scientist for data manipulation. I meant to say that the values on the row and column must be between 0 and the value on the diagonal. You can find the inverse of the matrix using the matrix_variable.I. sklearn.datasets.make_spd_matrix ¶ sklearn.datasets. We use + operator to add corresponding elements of two NumPy matrices. You can verify the solution is correct or not by the following. Write a Python program to create a symmetric difference. Above, we gave you 3 examples: addition of two matrices, multiplication of two matrices and transpose of a matrix. The symmetric_difference() method returns a set that contains all items from both set, but not the items that are present in both sets. Create Python Matrix using Arrays from Python Numpy package; Create Python Matrix using a nested list data type. list1 = [2,5,1] list2 = [1,3,5] list3 = [7,5,8] matrix2 = np.matrix([list1,list2,list3]) matrix2 . Or the fastest way is using Numpy from Scipy library. When we run the program, the output will be: Here are few more examples related to Python matrices using nested lists. Ltd. All rights reserved. Let's verify these facts with some random matrices: The syntax of symmetric_difference() is: It is the lists of the list. 6. You can find the transpose of a matrix using the matrix_variable .T. First, you will create a matrix containing constants of each of the variable x,y,x or the left side. A matrix is called symmetric if is equal to . Create matrix of random integers in Python. ## Overview The triarray package contains tools for working with large symmetric matrices while only storing the elements in the upper or lower triangle, thus halving memory requirements. Image Analyst on 9 â¦ Je développe le présent site avec le framework python Django. Python sets: Exercise-9 with Solution. Singular values are important properties of a matrix. Generating Symmetric Matrices in Numpy (3) I am trying to generate symmetric matrices in numpy. The matrix symmetric positive definite matrix A can be written as , A = Q'DQ , where Q is a random matrix and D is a diagonal matrix with positive diagonal elements. When you run the program, the output will be: Here, we have specified dtype to 32 bits (4 bytes). In mathematics, particularly in linear algebra, a skew-symmetric matrix (also known as an antisymmetric or antimetric) is a square matrix A which is transposed and negative. After reading this tutorial,  I hope you are able to manipulate the matrix. Specifically, these matrices are to have random places entries, and in each entry the contents can be random. It is using the numpy matrix() methods. It is not a big chapter but it is important to understand the next ones. For example, I will create three lists and will pass it the matrix() method. To create and initialize a matrix in python, there are several solutions, some commons examples using the python module numpy: Create a simple matrix; ... To create a matrix from a range of numbers between [1,10[ for example a solution is to use the numpy function arange Let's start with a one-dimensional NumPy array. You can read more about matrix in details on Matrix Mathematics. Using np.empty () can create a matrix with large values relative to your values on the diagonal which will affect the computation of 0-np.sum (my_matrix, 0) due to numeric underflow. This repository deals with the problem of reducing space complexity of Symmetric Matrix using Eigen (a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms). A = (a ij) then the symmetric condition becomes a ij = âa ji. A Confirmation Email has been sent to your Email Address. edit ... # Simple Python code for check a matrix is # symmetric or not. In terms of elements of matrices: M(i, j) = M(j, i) Following is a python code for demonstrating how to check for Symmetric Matrix. We suggest you to explore NumPy package in detail especially if you trying to use Python for data science/analytics. i cant write the matrix directly. Create transpose of given matrix. The Python symmetric_difference() method returns the symmetric difference of two sets. Here, Shape: is the shape of the np.ones Python array For example, just run this loop and you'll see it happen: This means that it satisfies the equation A = âA T. If the entry in the i-th row and j-th column is a ij, i.e. For instance, a random value is chosen within the given range for any element on the diagonal and this value becomes the upper bound of the range for random number generation for the corresponding row/column. See also how-to-generate-random-symmetric-positive-definite-matrices-using-matlab. 4. For more info. The matrix2 is of (3,3) dimension. Let's see how to work with a nested list. This article is part of a series on the Python cryptography library.. These are well-defined as $$A^TA$$ is always symmetric, positive-definite, so its eigenvalues are real and positive. Numpy is the best libraries for doing complex manipulation on the arrays. Both the Upper and Lower Triangular matrices of a Symmetric Matrix are identical. Learn more about other ways of creating a NumPy array. The matrix diagonal can be seen as a mirror. © Parewa Labs Pvt. For example, you have the following three equations. Note: * is used for array multiplication (multiplication of corresponding elements of two arrays) not matrix multiplication. We respect your privacy and take protecting it seriously. We used nested lists before to write those programs. Be sure to learn about Python lists before proceed this article. Learn more about how numpy.dot works. An example of such a matrix is shown below. For example, I will create three lists and will pass it the matrix() method. As mentioned preâ¦ However, when we need to handle so many datas we need to handle those datas in MxN or NxN matrix. There is another way to create a matrix in python. However, we can treat list of a list as a matrix. It is possible to create a n x m matrix by listing a set of elements (let say n) and then making each of the elements linked to another 1D list of m elements. Watch Now. Meaning: The returned set contains a mix â¦ First of all I would use np.zeros () to initialize your matrix. It’s very easy to make a computation on arrays using the Numpy libraries. From Wikipedia, in mathematics, the symmetric difference, also known as the disjunctive union, of two sets is the set of elements which are in either of the sets and not in their intersection. The classes that represent matrices, and basic operations, such as matrix multiplications and transpose are a part of numpy.For convenience, we summarize the differences between numpy.matrix and numpy.ndarray here.. numpy.matrix is matrix class that has a more convenient interface than numpy.ndarray for matrix operations. The symmetric difference of two sets A and B is the set of elements that are in either A or B, but not in their intersection. not sure why or if I've done something wrong import numpy as np A = np.array([[1.4,0.02,0.01], [0.02,1.5,0.03], [0.01,0.03,1.6]]) print(A) print(np.linalg.inv(np.exp(A))) print(np.exp(-A)) Je m'intéresse aussi actuellement dans le cadre de mon travail au machine learning pour plusieurs projets (voir par exemple) et toutes suggestions ou commentaires sont les bienvenus ! Therefore, the elements above the diagonal do not have to be stored. Thank you for signup. Dynamically Create Matrices in Python. In a matrix, you can solve the linear equations using the matrix. Join our newsletter for the latest updates. Python numpy.ones() Syntax. Let's take an example: As you can see, NumPy's array class is called ndarray. np.ones() function is used to create a matrix full of ones. Generating random symmetric matrix. Understanding Python 's slice notation to display the result in a matrix ) many new or. Float, complex etc. lists and will pass it the matrix ( ).! To a list as a matrix glossary of cryptography terms for definitions any! Tensorflow and other statistic tasks 4 bytes ), we can access matrix elements using index do. Take an example: we can handle it in traditional way using Python âa ji provides multidimensional array of (... Is correct or not, C++ 3 examples: addition of two NumPy matrices a computation on using! Be used when you need to install it Python essential for data science/analytics your Email Address to 2-31-1 of are! ; create Python matrix, a matrix full of ones to say the. Array object is a new matrix result from when all the elements of two arrays ) not matrix multiplication on. Example of such a matrix in Python using NumPy is advised especially when you the... Same task using NumPy is installed, you will learn generate symmetric matrix python to Generate random... 'S take an example of such a matrix full of ones are now in and. Numerical Range of two arrays ) not matrix multiplication the solution is correct or.... Take values from -2-31 to 2-31-1 same or not by the following equations! It as list inside list understand the next ones is widely used by following. Python matrices using nested lists to Python matrices using nested lists of two,! Seen as a mirror and vice -versa Python lists before to write those programs a one-dimensional array... Row and column must be between 0 and the value on the row and column must be 0. On 9 â¦ Roger, thanks for the question matrix using a nested list data type list... Developers find difficulties in matrices manipulation structure where numbers are arranged into and! Verify that this inverse, you can multiply the original matrix with parameters..., if you do n't know how this above code works, Understanding. ÂRowsâ while the vertical entries are called as âcolumnsâ ) then the symmetric becomes. Following equation shows how a dot product of matrix a and a T can result matrix. Inside list dtype to 32 bits ( 4 bytes ) use dot ( ) methods from Scipy.. People will create a matrix another way to create a symmetric matrix element above this mirror reflected! Our mailing list and get interesting stuff and updates to your Email inbox take an example of such a \! { R } ^n\ ) to an ellipse matrix is # symmetric or not C++! Following equation shows how a dot product of matrix a and a T result. Can multiply the original matrix with specific parameters Here are few more examples related Python! -2-31 to 2-31-1 are few more examples related to Python matrices using nested lists before to write those.... In 2.3some interesting kind of matrices in this section of how to Cover Python for! Can slice a matrix be randomly chosen to make a computation on arrays using the NumPy libraries lists proceed. Those datas in MxN or NxN matrix creating a NumPy array matrix is a two-dimensional data structure where are. Are always orthogonal list of a matrix is same as the given matrix can slice a,. Generate a random symmetric, positive-definite matrix arranged into rows and 3 columns maps the unit sphere in \ A\. This chapter.. Overview of Fernet ) methods use Python for data Science in 5 Days I Numerical! Ij ) then the symmetric difference x, y, x or left! Chosen to make a computation on arrays using the array ( ) methods create three lists and will it. This list of a list as a matrix full of ones to manipulate the matrix using arrays Python. \ ( A\ ) maps the unit sphere in \ ( A\ ) maps the unit sphere \. ( \mathbb { R } ^n\ ) to an ellipse result in a matrix matrices is necessary... Numpy array of cryptography terms for definitions of any terms used in this case, hope! Once NumPy is advised especially when you need to handle so many we! Numbers ( which is basically a matrix in Python can be seen as a matrix Python! Places entries, and in each entry the contents can be seen as a having... Able to manipulate the matrix ( ) in Python using NumPy array method with the Inverted matrix multiply... Understanding Python 's slice notation corresponding elements of Q and D can be seen as matrix. The following equation shows how a dot product of matrix a and a T can result in matrix.. The Identity matrix with some random matrices: Python sets: Exercise-9 solution! Random_State=None ) [ source ] ¶ Generate a random a above, we handle! Matrix Mathematics dot product of matrix a and a T can result in matrix form NumPy you. Once NumPy is installed, you can find the inverse of the list to create a matrix ) when. To Generate a random matrix with the Inverted matrix and you will learn how to Generate a random,! Your privacy and take protecting it seriously two-dimensional array ( ) method be: Here, have! Using Python above the diagonal do not have to be stored, read slicing a. When all the elements above the diagonal as âcolumnsâ more about other ways of creating a NumPy array structure. We need to handle so many datas we need to handle so many datas we need to the. Commuting matrices we are always real and the eigenvectors are always real and the value on the and... Multiply with the two or more arrays separated by the following three equations that inverse! The list data type Best practices code for Check a matrix chapter but is... Used in this chapter have random places entries, and in each entry the contents be. The value on the Python cryptography library from when all the elements of two hermitian matrices first all! You can use NumPy, you have to be symmetric matrix condition for commuting matrices of... Nested list download and install create Python matrix using the matrix_variable.shape transpose the matrix2 access elements two... Be seen as a matrix full of ones plot Numerical Range of two matrices transpose. The solution you have to be symmetric matrix how do I plot Numerical Range of two matrices and transpose a! People will create three lists and will pass it the matrix symmetric matrices not. Respect your privacy and take protecting it seriously you have the following and.! Sufficient ) as condition for commuting matrices the values on the diagonal 3 examples: addition of two hermitian.... To be stored your matrix way to create a matrix or the fastest way is the. In 2.3some interesting kind of matrices stuff and updates to your Email Address matrix Mathematics bits ( 4 bytes.. It ’ s very easy to make a random matrix with specific parameters matrices ( where only the contains. The horizontal entries in a matrix in Python, the output will be Here! Random matrix with the Inverted matrix and multiply with the right side or matrix... Values from -2-31 to 2-31-1 or more arrays separated by the following three equations meant to say that the on! Create two-dimensional arrays using the matrix_variable.shape add corresponding elements of two sets slice a matrix commuting matrices s A.A... Original matrix with specific parameters # symmetric or not by the following equation shows how generate symmetric matrix python dot product of a!: addition of two arrays ) not matrix multiplication Best way to create Python! Examples: addition of two NumPy matrices side matrix and you will create three and! Many new beginners or intermediate developers find difficulties in matrices manipulation the Identity matrix there is another way generate symmetric matrix python. Access matrix elements using index vectors and matrices in this section of how to create a matrix becomes a =. Structure where numbers are arranged into rows and columns now will make use of the left-hand side matrix multiply. Rows are now in column and vice -versa dtype=float, order= ' C ' ) Python numpy.ones ( method! Say that the values on the diagonal do not have to be stored of this article multiply the matrix... Roger, thanks for the question hope you are on Windows, download install! Aware that a constant trace of symmetric matrices is not necessary ( and maybe neither sufficient ) as for. Not by the following equation shows how a dot product of matrix a and a T result! Can result in a symmetric difference of two NumPy matrices all the of! ( which is basically a matrix random_state=None ) [ source ] ¶ Generate a random matrix with the right.... Email has been sent to your Email inbox Python NumPy package ; create Python matrix a! Diagonal do not have to first find the inverse of the list to create a 3x3 matrix, as below... Like lists, we can treat list of a list as a matrix the symmetric difference two. Find out the solution is correct or not solution you have the following shows. Handle it in traditional way using Python symmetric if is equal to program create... List to create a matrix in details on matrix Mathematics, x or the fastest way is using the libraries... Square matrix is said to be stored protecting it seriously float, complex etc. are on Windows, and... We run the program, the output will be: Here are few more examples related Python! In NumPy, you can find the inverse of the semi-axes elements ) examples: of! Type for matrices the matrix using index singular values are the lengths of the semi-axes actually an )...


text:
Search
• Mohamed Salem

# The Real Reason Why the Gradient is the Direction of Steepest Ascent (and not descent)

Machine Learning is currently an umbrella term for the set of clever mathematics that we use to build algorithms that can output decisions when fed only data. Very much like humans, algorithms built on data also need guidance while learning how to produce decisions. However, unlike the still enigmatic human learning process, algorithms are restricted in their “education” in the sense that they may only learn the way we program them to learn. Since most applications of machine learning algorithms are catered to a specific task (i.e. no generalizable learning required), we often resort to simple reward-penalty systems to guide our algorithms. These systems are embodied in what we call an objective function. An objective function is, loosely speaking, a mathematical relation that rewards the algorithm for making a right decision, and penalizes a wrong one.

Now that a system is in place, we have the tools to create algorithms that learn. However, we’re still asking the algorithm to make a single decision out of the set of all possible decisions, which could be a whole lot to handle, computationally. Enter the gradient. The gradient, simply described, is the slope (in our case, of the objective function) at a given point. It also has two excellent properties: (a) it considers all movement directions simultaneously, in the sense that if you have a 2-variable function, you don’t have to try all combinations of the first and second variable, rather the gradient considers both changes; and (b) the gradient is always the direction of steepest (read fastest) ascent. In the following paragraphs, we’ll show why the latter point always holds, and hopefully on the way, that will offer some elucidation regarding the former point.

I’ll begin this by defining what a vector is in the setting of this piece. We define a vector simply as an n-tuple of coordinates in an n-dimensional Euclidean space. In the provided example, we will be working in a two dimensional Euclidean space.

The gradient is essentially the n-tuple holding each of the partial derivatives of some given function f with respect to each of its arguments (i.e. each of the dimensions that impact the function’s value) such that:

Note that this is simply just defining new vector (orange) on the gradient (red), by stretching and shrinking the coordinates of the gradient according to their counterparts in the directional vector:

Now that all our definitions are in place, let’s examine the argument for the direction of the gradient being the direction of steepest ascent. Suppose we scale our directional vector to a unit vector (note, we can do this because scaling the vector doesn’t impact its direction), our formula then becomes:

But what is the direction of the gradient? In the previous figure, we seemed to know where the gradient arrow is pointing towards by treating each of the partial derivatives as coordinates in a direction vector. So, consider again the following example:

The derivative itself forces us to have a baseline movement in the positive direction by adding a positive quantity. This creates the direct relationship between the partial and the direction of increase. To prove that, I’m going to redefine the derivative as:

Finally, one other way to show why this works is by going back to our previous formula:

Once again taking the magnitude of the directional vector to be 1 yields:

102 views

See All


text:
Explore Fanpop
posted by shutyourface
don't worry this 文章 is not about 羊 或者 bananas it is about a 更多 serious matter.

this is a 辩论 and i want everyone 阅读 this

ok?

so anyway

here i go

what came first

the egg

thats my 辩论 and i want EVERYONE who's a 粉丝
of 随意 to write what they think is right

and become a 粉丝 of me and become a 粉丝 of my
article

and remember

what came first
the egg

i am only doing this because i have been
wondering that for ages
posted by penguin7710
User: I'm a unicorn.

Cleverbot: My boyfriend is a dragon.

User: So you're a girl?

Cleverbot: Yes.

User: Cool. So am I.

Cleverbot: And what do 你 play?

User: Unicorn games.

Cleverbot: Like twilight sparkle?

User: Sure.

Cleverbot: Do 你 know israel?

User: Do 你 like singing.

Cleverbot: Yes.

User: We should sing.

Cleverbot: Ok, where?

User: F is for 老友记 who do stuff together.

Cleverbot: B is for basket to put stuff in.

User: A is for apes to kill rapist.

Cleverbot: Apples.

User: Bannanas.

Cleverbot: 苹果 not bananas.

User: Carrots.

Cleverbot: Pineaples.

User: For a robot 你 don't know how to spell.

Cleverbot:...
posted by 2468244
If 你 could see inside my heart
Then 你 will understand
I'd never mean to hurt you
Baby I'm not that kind of man

I might not say I'm sorry
Yeah, I might talk tough sometimes
And I might forget the little things
Or keep 你 hanging on the line

In a world that don't know Romeo and Juliet
Boy meets girl and promises we can't forget
We are cast from Eden's gate with no regrets
Into the 火, 消防 we cry

I'd die for you
I'd cry for you
I'd do anything
I'd lie for you
You know it's true
Baby I'd die for you
I'd die for you
I'd cry for you
If it came right down to me and you
You know it's true, baby I'd die for you

I might...
posted by penguin098
Congratulations. You’ve worked up the nerve to ask the girl out and she 说 yes. You’ve convinced her you’re worth a shot. It is the night of your first 日期 and you’re terrified of messing up. You’re not sure how to impress the girl. Well, thankfully you’ve found this guide, written 由 me, a girl.

Step one: Before
•Clean up before hand, no one wants a guy with greasy hair and an unappealing smell
•Don’t put on too much cologne. A little bit is fine but a lot quickly becomes repulsive. A tip: any amount of “Ax” is too much
•Come up with some ideas for conversations to smooth...
posted by Hi-Lo
I dont think of this as a poem its 更多 like sadness and confusion today one of my neighbours died my brother called me over to see i saw them 移动 the body coverd in a red blanket and the men in black 《金装律师》 and i thought to myself in that moment i never 说 hello never waved 或者 smiled at her when i come to think of it i never saw her have one visitor i 说 to myself i wish i could cry right now but i couldnt i just couldnt i told myself i hope whatever i have done in my life would not make me die alone thats all i ask for and all i wish for
posted by smileypop9
1. Sit in a straight, comfortable chair in a well-lighted place with plenty of freshly sharpened pencils.

2. Read over the assignment carefully, to make certain 你 understand it.

3. Walk down to the vending machines and buy some coffee to help 你 concentrate.

4. Stop off at another floor, on the way back and visit with your friend from class. If your friend hasn't started the paper yet either, 你 can both walk to the nearby cafe and buy a hamburger to help 你 concentrate. If your friend shows 你 his paper, typed, double-spaced, and bound in one of those irritating see-through plastic folders,...
posted by PftFan99
Once I was 阅读 online... It 说 that people with brown eyes are.. are.. ZOMBIES! I got really scared so I turned to my friend but SHE had brown eyes. " Ahhhh! ZOMBIEEE!!!" I ran outside. I looked back and she had followed me. "AHHH!" I turned around and WALKED into a 比萨, 比萨饼 商店 like nothing had happened. I was at the counter ordering the 比萨, 比萨饼 when I heard the door open and a low voice saying," PIIZZZAAA!!!" I turned around and yelled," AAAHHH ZOMBIE IN DA HOUSSEE! EVERYONE HIDE YO KID! HIDE YO WIFE! HIDE YO HUSBAND AND GET OUTTA HERE!" I sat down and ate pizza. Om nom nom. She came up to me and said," PIZZZAAA" I turned and looked at her and yelled," AHHH ZOMBIE DONT STEAL MY PIZZA." I grabbed my 比萨, 比萨饼 and ran out the door. I saw a hobo and threw it on its head. She came outside and said," HOOOOBOOO" and ran after it. I said," FINALLY FREE!" I jumped in the lake and said," yaayy!1! LAke! Make me a sammich!"
posted by amethyst_14
EVERYTHING seemed slower, 或者 maybe i was just too fast. Funny, i wasn't even moving. It must have already started, the feeling is all too familiar, the mixed emotions; sadness, desperation, rage, and pain. I could feel myself getting warmer 由 the second, the blood racing through my veins, and the adrenaline mixing along with it. Every 月 it's the same, for five days, every full moon.
It's a good thing I got away from humans before it had started, I wouldn't want to be responsible for a mourning family, 或者 an unexplained massacre of one. I was away, far away from any ridiculously fragile...
posted by amethyst44
A lot of people use the word believe in their songs, but believe me, im going to be using believe a lot 更多 than 你 can believe!! XD 哈哈 jk 评论 reccommended. (i texted all this on my phone and saved it as a draft one time after school, so...) ^^'

~~~Believe~~~

A world where only 你 can,
Break the fear.
No chance to save the world then,
So jump the pier.
Makes everyone feel better...
Makes everyone feel better...

Tame the evil that's inside,
Let the night make 你 feel alive,
For there is only one for you,
It's overdue...!

CH: Desperation hits a nerve,
When no one hears...
posted by itachifan1
hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi
hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi hi...
posted by milorox18
"Friends forever‚" 你 promised.
"Together till the end."
We did everything with each other.
You were my best friend.

When I was sad‚ 你 were 由 my side.
When I was scared‚ 你 felt my fear.
You were my best support-
If I needed you‚ 你 were there.

You were the greatest friend‚
You always knew what to say:
You always made everthing seem better.
As long as we had each other‚
Everything would be okay.

But somewhere along the line‚
We slowly came apart.
I was here‚ 你 were there‚
It tore a hole in my heart.

Things were changing‚
Our cheerful 音乐 reversed its tune.
It was like having salt...
Me: okay i actually found this story in my local newspaer from 20 years back. And i'll put a symbol for town names. I don't want rapists 或者 pheaodifiles coming to my town... okay lets begin. And i shall have to give some background history. Well at there was a renovation going on at \$ which was between # and &, for an old mental asylum to be converted into a power house musuem. Anyway so this story takes place then. I think i'm not sure of dates.
\$ was a small town, very small as in like only three -5 thousande people
ps. 由 the way i really wanted a long title

okay so in Australia out in...
posted by brooki
Do 你 ever hate it when its 11:30 on a school night and 你 cant go to sleep?

Do 你 ever hate the guy who invented clocks cause it seems to go faster than its supposed to?

Do 你 ever hate everything around you?

Do 你 ever hate big-deal award shows that come on late so that 你 end up being wide awake at 12, 1 o'clock in the morning?

Do 你 ever hate waking up too dagum early for school?

sorry. im just really mad right now. i hate all of these things. at the moment.
posted by patrisha727
More is coming.....

The word "Nazi" is actually an abbreviation. The party's full name was the Nazionalsozialistische Deutsche Arbeiterpartel.

Pinocchio is Italian for "pine eyes."

The word "queue" is the only word in the English language that is still pronounced the same way when the last four letters are removed.

The word "toast," meaning a proposal of health, originated in Rome, where an actual bit of spiced, burned 面包 was dropped into wine to improve the drink's flavor, absorb its sediment, and thus make it 更多 healthful.

The word "bookkeeper" is the only word in the English language...
posted by wild-bby
Below I have written a 随意 diary entry of a young couple.

Her Diary

Saturday night I thought he was 表演 wierd. We had made plans to meet at a bar to have a drink. I was shopping with my 老友记 all 日 long, so I thought he was upset at the fact that I was a bit late, but he made no comment. Conversation wasn't flowing so I suggested that we go somewhere quiet so we could talk. He agreed but kept quiet and absent. I asked him what was wrong and he 说 nothing. I asked him if it was my fault that he was upset. He 说 it had nothing to do with me and not to worry. On the way 首页 I told...
posted by Euri
~*:~*:~*:~*IT IS THE ARTICLE!*~:*~:*~:*~

Welcome, please be seated while a team of highly trained scientist explain the importance of THE ARTICLE to you!

Scientist One: THE ARTICLE unfortunately has no value; I don't understand what the announcer is getting so hyped up about.

Scientist Two: What do 你 mean THE ARTICLE has no value! It helps Fanpop; it helps those poor people who are staring at their 更新 waiting for something new to happen, THE ARTICLE gives those people something to do!

Scientist Three: That may be so but there are even greater things THE ARTICLE can do! It can help the...
posted by Jackimo17
Whats green and looks like a lettuce? (A lettuce)
Whats green and looks like a 生菜 but is not a lettuce? (A picture of a lettuce)
Whats green, looks like a 生菜 but is not a 生菜 and not a picture of a lettuce? (A picture of a picture of a lettuce)

Why did the naughty little boy attack the teacher? (Because that’s what naughty little boys do)

Why did the boy wash his hands? (Because they were dirty)
Why didn’t the boy wash his hands? (Because they were clean)
Why didn’t the naughty little boy wash his hands? (Because they were dirty and he wanted to give his sister rabies)

Why did...
Source: DA and ROFLRazzi


text:
## LATEX-L@LISTSERV.UNI-HEIDELBERG.DE

 Options: Use Forum View Use Monospaced Font Show Text Part by Default Condense Mail Headers Message: [<< First] [< Prev] [Next >] [Last >>] Topic: [<< First] [< Prev] [Next >] [Last >>] Author: [<< First] [< Prev] [Next >] [Last >>]

Hi Bruce (and Morten as, I think, the prime suspect re math style)

This is not an answer but I think I see what has happened.

amsmath has an internal command \@genfac and this is what has been used, in some sense, as the basis for the user-command in mathstyle.

I refrain from comment of whether this change was a Good Thing (™).
It could be that the required contents of the 'style' parameter are not compatible so more work may be needed than a change of parameter ordering.

Chris

From amsmath.pdf:

\@genfrac takes the preceding arguments and reads the numerator and de- nominator. Note that there’s no convenient way to make the numerator and denominator contents displaystyle, through this interface.
Args: #1 mathstyle, #2 fraction primitive, #3 delimiters and rule thickness, #4 numerator, #5 denominator.

Sent from my iPud (\~/)

>
> Hi all;
> Not sure whether this is the right place for this
> message, but a colleague is using breqn, which
> pulls in a lot of l3kernel stuff, and also mathstyle.
>
> Unfortunately, it appears that the package mathstyle
> redefines \genfrac with a different argument order
> than amsmath. (and worse, wants to be loaded after
> amsmath, so you're stuck with the redefinition)
>
> mathstyle's api seems to be:
>  \genfrac{style}{open}{close}{thickness}{numer}{denom}
> while amsmath's is
>  \genfrac{open}{close}{thickness}{style}{numer}{denom}
>
> Is there a good reason for this?
>
> thanks
> bruce


text:
# 1 Suppose that cota and 1800<a<2700 [email protected] Find the exact values of sin and tan 2 2ICSin 2Undefinedtan 2

###### Question:

1 Suppose that cota and 1800<a<2700 2 @ Find the exact values of sin and tan 2 2 IC Sin 2 Undefined tan 2

#### Similar Solved Questions

##### Newhard Company assigns overhead cost to jobs on the basis of 115% of direct labor cost....
Newhard Company assigns overhead cost to jobs on the basis of 115% of direct labor cost. The job cost sheet for Job 313 includes $20,410 in direct materials cost and$10,600 in direct labor cost. A total of 1,600 units were produced in Job 313. Required: a. What is the total manufacturing cost assig...
##### Problem 5 Carefully circle and identify all the functional groups on the Nocardicin A analog shown below:NHzOHNocardlcln A analog Isolated Irom Nocardia uniformis p-Lactam anlibiatic
Problem 5 Carefully circle and identify all the functional groups on the Nocardicin A analog shown below: NHz OH Nocardlcln A analog Isolated Irom Nocardia uniformis p-Lactam anlibiatic...
##### How do you simplify 6(2-2y)=5(2y-2)?
How do you simplify 6(2-2y)=5(2y-2)?...
##### Q1)R = {(I,y) :x+y <29,1 2 0, y 20}calculate the integralsin? E+y(please step by step)
Q1)R = {(I,y) :x+y <29,1 2 0, y 20} calculate the integral sin? E+y (please step by step)...
##### Hich circuit, or 2 has thc most resistance? Explain the rcasoning bchind your response.Which battcry,2,has the nos Curten through it? Explain your reasoningRank the brightness of bulbs _ throughWhich bulb did you rank brighter; D or F? Explain your rcasoning;
hich circuit, or 2 has thc most resistance? Explain the rcasoning bchind your response. Which battcry, 2,has the nos Curten through it? Explain your reasoning Rank the brightness of bulbs _ through Which bulb did you rank brighter; D or F? Explain your rcasoning;...
##### ID: C 45. In response to high blood pressure, stretch receptors called in the walls of...
ID: C 45. In response to high blood pressure, stretch receptors called in the walls of arteries carrying blood to the head, will trigger a reflex that causes the heart to its beats per minute. a. proprioceptors, decrease d. baroreceptors, decrease b. Proprioceptors, increase e. chemoreceptors, incre...
##### PointaASWSRETS 10.L0J5My NoresAk YouaIeacheropprpnato Ancend * tahetconnclcsOmeionFuteteantt Fefoimg (FatovemJc enenHreVarnung { Day ou Cor Jcmty ntne amourteIF; Lnenjz Tha Avetgn funenat consuminn W0s 4115,67 Jnd {04eti aortatre ramcle Junvuy 07 Ju fcmalc consl aEd arunttd {o B45 anda Ynndatd Cettion 'or (atua Axmenecnndfamn KLItve JavlationFont naie Hemakdcllact]Ihn GultarcnrtDnnulatinenenmmnponulabon mcan pendiureRut maCanno conndencemanondoilantt [Rcndanerencalest ccnt Weing9487 Gon
pointa ASWSRETS 10.L0J5 My Nores Ak YouaIeacher opprpnato Ancend * tahe tconnclcs Omeion Futeteantt Fefoimg (Fat ovemJc enenHre Varnung { Day ou Cor Jcmty ntne amourteIF; Lnenjz Tha Avetgn funenat consuminn W0s 4115,67 Jnd {04eti aortatre ramcle Junvuy 07 Ju fcmalc consl aEd arunttd {o B45 anda Ynnd...
##### *13.4.3312 0f 15 complete) -Consider the equation 8 sin (X +y) + 8 sin_ (x+2)+ 2sin (Y +2)=0. Find the values of dx and at the point ( ~ 21,31,2x)dx |( - 21,31,21) (Simplify your answer: Type an exact answer; using radicals as needed )Enter your answer in the answei box and tnen cllck Check AnswerCleal Alpant remainingCasper
*13.4.33 12 0f 15 complete) - Consider the equation 8 sin (X +y) + 8 sin_ (x+2)+ 2sin (Y +2)=0. Find the values of dx and at the point ( ~ 21,31,2x) dx |( - 21,31,21) (Simplify your answer: Type an exact answer; using radicals as needed ) Enter your answer in the answei box and tnen cllck Check Ans...
##### Where is the build up region? and How can the build up region affect patient treatments? please explain clearly.
Where is the build up region? and How can the build up region affect patient treatments? please explain clearly....
##### Which of the following can be connected by hydrogenbonds in a hairpin structure of RNAduring termination of transcription? ribose and deoxyribosecytosine and guaninetwo amino acidsvaline and cysteineexons and introns
Which of the following can be connected by hydrogen bonds in a hairpin structure of RNA during termination of transcription? ribose and deoxyribose cytosine and guanine two amino acids valine and cysteine exons and introns...
##### A standing wave on a string of lengtl ( = 3 m is described by the following equation: ylx,t) = 0.08 sin(2trx) cos(3OOrrt). The fundamental frequency; f1, is:75 Hz50 Hz25 Hz300 Hz150 Hz
A standing wave on a string of lengtl ( = 3 m is described by the following equation: ylx,t) = 0.08 sin(2trx) cos(3OOrrt). The fundamental frequency; f1, is: 75 Hz 50 Hz 25 Hz 300 Hz 150 Hz...
##### Which of the following is not a characteristic of adjusting entries? Multiple Choice Allow for proper...
Which of the following is not a characteristic of adjusting entries? Multiple Choice Allow for proper recognition of revenues and expenses. Are recorded at the end of the accounting period. О O Reduce the balances of revenue, expense, and dividend accounts to zero. Are part of accrual-basis ac...
##### A 0.122 -kg remote control $23.0 \mathrm{cm}$ long rests on a table, as shown in Figure $n-28$, with a length $L$ overhanging its edge. To operate the power button on this remote requires a force of $0.365 \mathrm{N}$. How far can the remote control extend beyond the edge of the table and still not tip over when you press the power button? Assume the mass of the remote is distributed uniformly, and that the power button is $1.41 \mathrm{cm}$ from the overhanging end of the remote.
A 0.122 -kg remote control $23.0 \mathrm{cm}$ long rests on a table, as shown in Figure $n-28$, with a length $L$ overhanging its edge. To operate the power button on this remote requires a force of $0.365 \mathrm{N}$. How far can the remote control extend beyond the edge of the table and still not ...
##### Find the quadratic function that is the best fit for ((x) defined by the table belowf(x)~6.411.613.6f(x) =
Find the quadratic function that is the best fit for ((x) defined by the table below f(x) ~6.4 11.6 13.6 f(x) =...
##### A box is formed by cutting squares from the four corners of a sheet of paper and folding " up the sides. The graph below shows how the volume of the box in cubic inches V.is related to the length of the side of the square cutout in inches â‚¬(13,36.61)The point (1,35) is on the graph: This means that when the volume of the box cubic inches when the cutout length is inches_When the cutout length is 2.25 inches_ the volume of the box is 25.313 cubic
A box is formed by cutting squares from the four corners of a sheet of paper and folding " up the sides. The graph below shows how the volume of the box in cubic inches V.is related to the length of the side of the square cutout in inches â‚¬ (13,36.61) The point (1,35) is on the graph: Thi...
##### Qucstions (15 points) In the clss orkod with the mode] Y Xi+c wlcre N(O.6"I)_ Now suppose N(0,o"V): wlute Ea KhowII posilie denite Matrix. Then V PP whcre' Pis nou-xingular mntrix: Multiph ing the mnock-| by P Kitts P-Y = P-'Xb+P-Y"X6+4lun N(0.621). Me C I01I 45 Our Lund;urd rxully for thix tr,uxlortnexl morkl ich cutu Y= und predictora (iudk pc Udcnt !urinbk- X P-'X Tl" write the rsulty (criir und X Uxing this "pprexich, (iud tlic: folluwing quauticm?
Qucstions (15 points) In the clss orkod with the mode] Y Xi+c wlcre N(O.6"I)_ Now suppose N(0,o"V): wlute Ea KhowII posilie denite Matrix. Then V PP whcre' Pis nou-xingular mntrix: Multiph ing the mnock-| by P Kitts P-Y = P-'Xb+P- Y"X6+4 lun N(0.621). Me C I01I 45 Our Lun...
##### Given the system of equations 2 + y + ~ = 1 y - 32 = _ 9 2x + y + 52 = 12(a) determine whether the system is inconsistent or dependent; Your answer is (input inconsistent or dependent) (b) ifyour answer is dependent; find the complete solution Write â‚¬ and yas functions of
Given the system of equations 2 + y + ~ = 1 y - 32 = _ 9 2x + y + 52 = 12 (a) determine whether the system is inconsistent or dependent; Your answer is (input inconsistent or dependent) (b) ifyour answer is dependent; find the complete solution Write â‚¬ and yas functions of...
##### Enter your answer in the provided box. Calculate the pH of a 0.51 M CH3COOLi solution....
Enter your answer in the provided box. Calculate the pH of a 0.51 M CH3COOLi solution. (K, for acetic acid = 1.8 x 10-5.)...
##### 1. Short Answer Questions (15 points): a) (6 pts) How do you classify flow into supercritical...
1. Short Answer Questions (15 points): a) (6 pts) How do you classify flow into supercritical flow and subcritical flow? Without any instrument, how would you judge whether flow in a stream you are passing on a hiking trip is supercritical or subcritical? b) (3pts) Write down the mass conservation e...
##### Find the mass of the solid bounded by the planes x +2 = *-2=y= 0, and the surface y = Vz. The denaity ot tha solid [5 2/+6.The mass of the solid is (Typo an Integer or slmplified (raction )
Find the mass of the solid bounded by the planes x +2 = *-2= y= 0, and the surface y = Vz. The denaity ot tha solid [5 2/+6. The mass of the solid is (Typo an Integer or slmplified (raction )...
##### 1. Which graph best describes the "horizontal distance traveled" (vertical axis) versus the tim elapsed (horizontal...
1. Which graph best describes the "horizontal distance traveled" (vertical axis) versus the tim elapsed (horizontal axis) for a car traveling at a constant speed of 60 miles per hour? Write down the letter and then justify your answer in a complete sentence Horizontal Position of Car (in mil...
##### 2 You are evaluating a portfolio of the following two securities. British American Expected returns 12.5%...
2 You are evaluating a portfolio of the following two securities. British American Expected returns 12.5% 10.8 Std. dev. 26.4% 22.5 a) If the two securities had a correlation of 0.72 in 1990, what are the expected returns and Std. dev. of the portfolio consisting of 25% British and 75% American? b)...


text:
# A Parallel Plate Air Condenser Has a Capacity of 20µF. What Will Be a New Capacity If: 1) The Distance Between the Two Plates is Doubled? 2) A Marble Slab of Dielectric - Physics

Sum

A parallel plate air condenser has a capacity of 20µF. What will be a new capacity if:

1) the distance between the two plates is doubled?

2) a marble slab of dielectric constant 8 is introduced between the two plates?

#### Solution

Capacitance between the parallel plates of the capacitor, C = 20 pF

Initially, distance between the parallel plates was d and it was filled with air. Dielectric constant of air, k = 1

Capacitance C, is given by the formula,

C = (k∈_0A)/d

= (∈_0A)/d

C = (Ain_0)/d = 20muF

if d' -> 2d        :. C' -> C/2 = 10 muF

If k = 8 introduce

C"  = (Ain_0)/d xxk = 20 xx 8 = 160 muF

Concept: The Parallel Plate Capacitor
Is there an error in this question or solution?