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1

It's an easy exercise to show that Rank(Def2) is equal to $\dim_K(K\otimes_RM)$, where $K$ is the field of fractions of $R$. This shows that Rank(Def2) is well-defined, and this is by definition the rank of $M$ in such context. "Is every cardinality of a $R$-linearly independent subset of a finitely generated $R$-module finite?" Yes, it is. In fact its ...


0

If you look careful at $M=t(M)\oplus F$ can notice that $F\simeq M/t(M)$. But finitely generated torsion-free modules over a PID are free, so $M/t(M)$ is free and thus has a (finite) rank. That's why Lang calls this the rank of $M$ which seems a reasonable choice. (Bourbaki also calls this the rank of $M$.) But as you can see there is no standard ...


0

I suspect what's happened is that Lang has used an odd choice of language and you've misread it. The definition is meant to be rank of $M :=$ rank of any $F$ s.t. $M \cong Tor(M) \oplus F$ He isn't trying to define the rank of $F$, though I can see why you might read it that way. Essentially, we're trying to extend the definition of rank for free modules ...


1

To elaborate on Thomas Andrews' comment: Addition is well-defined: For all $a,b,c,d\in\mathbb{R}$, if $a=b$ and $c=d$, then $a+c=b+d$. The well-defined nature of addition in $\mathbb{R}$ is taken as an axiom. Hence, if $a=b$ and $c=d$, we have $$ a+c=b+d\Longleftrightarrow a+c=b+c, $$ but I think there is something more interesting you could show, namely ...


1

You start with $$ a \oplus c = x \quad (*) $$ where $x$ is the resulting value, then note that $a = b$, so you replace $a$ with $b$ in equation $(*)$. This gives $$ b \oplus c = x $$ Of course this means $$ a \oplus c = x = b \oplus c $$ and thus $$ a = b \Rightarrow a \oplus c = b \oplus c $$ Let us redo this for variables $a, b, c \in \mathbb{R}$ and the ...


11

Since this question is about pronunciation of $\sinh$, or the hyperbolic sine, I thought it might be useful to see what the authoritative Oxford English Dictionary had to say: This reveals that there are actually three pronunciations often encountered. From left to right: sh-eye-n $\qquad$ (i.e., shine) s-i-n-t-sh $\qquad$ (i.e., cinch) s-eye-n-ay-tch ...


3

(Due to the nature of this question I've created a tongue-in-cheek response) I never short cut the pronunciation of my trig functions. Sine becomes Sin, Sine is most certainly not sinful. Cos becomes Cos, Tangent becomes tan, secant becomes sec Cosecant becomes csc!!! (that should be the real question here) To answer your question, your typical ...


2

In the literature on constructive and linear logic it is the other way around, although $\to$ doesn't strictly belong to either group because it isn't monotone. Troelstra's Contructivism in Mathematics defines the almost negative formulas of Heyting arithmetic as formulas that have no $\vee$ and limited use of $\exists$. In linear logic the terminology made ...


1

The only invertible elements in a polynomial ring over a field (or more generally a domain) are the constant non-zero polynomials. The multiplicative order of the polynomial cannot be the intended meaning. In fact, it is the order of its roots (in case it is irreducible). But let us be more systematic. In principle, it might be the degree (rare) or the ...


1

If you are dealing with this numerically, use expm1(n*log1p(x)) (if those functions are available in your math library) to get high precision results. expm1(x)=exp(x)-1 and log1p(x) = ln(1+x) both for small x without cancellation.


0

Definition of polyhedron, "A polyhedron is a close solid having certain no. of (regular or irregular) polygonal (flat) faces such that the sum of the solid angles subtended by all its faces at any point exactly inside it must be $4\pi$ Ste-radian". Mathematically, it is given as $$\bbox [4pt, border: 1px solid blue;] {\omega _{total}=4\pi \space sr} $$ ...


2

Let $\mathscr{A}$ be a family of sets. We say that the sets in $\mathscr{A}$ are mutually disjoint if no two of them have any elements in common. In other words, if $A,B\in\mathscr{A}$, and $A\ne B$, then $A\cap B=\varnothing$. Another term that means the same thing is pairwise disjoint. In pictorial terms, if you make a Venn diagram of the sets, you’ll ...


1

I'm not the authority on this, but this is how I interpret all of these words in math literature: Definition - This is an assignment of language and syntax to some property of a set, function, or other object. A definition is not something you prove, it is something someone assigns. Often you will want to prove that something satisfies a definition. ...


3

For a given group $G$ and field $F$, the category of group representations of $G$ (in the sense of group homomorphisms from $G$ into $GL(n,F)$) is equivalent to the category of finite dimensional modules over the group ring $F[G]$. Similarly, the category of $F$-algebra representations of an algebra $A$ (in the sense of ring homomorphisms from $A$ into ...


3

In representation theory people often use the word representation as a synonym for module, and for good reason. A representation of a Lie algebra is the same thing as a module for its universal enveloping algebra, a representation of a quiver is the same as a module over its path algebra, etc. One important example is that a representation of a finite ...


0

Stochastic calculus is to do with mathematics that operates on stochastic processes. The best known stochastic process is the Wiener process used for modelling Brownian motion. Other key components are Ito calculus & Malliavin calculus. Stochastic calculus is used in finance where prices can be modelled to follow SDEs. In the Black-Scholes model, ...


1

As pointed out in comments, the property that characterizes functions such that $$\forall s<t<u\qquad f(t)\leq\max\{f(s),f(u)\}$$ is Quasiconvexity. In particular a function satisfying the above inequality is called quasiconvex (or quasi-convex or quasi convex) and it is called strict quasiconvex if the inequality is strict. See article on ...


0

The comment by @James answers this question: a magma/groupoid $(G,+)$ where $$x+x=x\quad\forall x\in G,$$ this is where every element is idempotent, is called idempotent groupoid, or idempotent magma if you prefer. Examples in the titles (and bibliographies) of this article or this thesis.


0

I came here hoping for an elaboration on what I found in these publicly accessible slides, but actually I think it's stated more succinctly and satsifyingly (to my mind, and limited knowledge), so I'll restate it in case it's of use: Algebra - Procedural method/algorithm specifying how to obtain the results. Calculus - Description of what makes a result ...


0

To my knowledge, there is no specific name given to the inverses of any of the special functions, like the error function, the beta and $\Gamma$ functions, hypergeometric functions, etc., and Lambert's W function is no exception to the general rule.


1

There are two possible meanings: For any pullback square as below, $$\require{AMScd} \begin{CD} X' @>>> X \\ @VVV @VVV \\ Y' @>>> Y \end{CD}$$ if $X \to Y$ is a monomorphism, then $X' \to Y'$ is also a monomorphism. For any commutative diagram of the form below, $$\begin{CD} X' @>>> X \\ @VVV @VVV \\ Y' @>>> Y \\ @VVV ...


2

I know this question is more than a year old, but for the sake of posterity, the correct term for "the opposite of idempotent" (at least in computer science) is non-idempotent. For example, see section 9.1.2 of Hypertext Transfer Protocol -- HTTP/1.1 (RFC 2616): 9.1.2 Idempotent Methods Methods can also have the property of "idempotence" in that ...


0

I believe the value you are asking about is the $\textbf{separable degree}$. Read the comments and answers at this previous question for more information. This is a very important concept in field theory.


-1

Let me first write this a bit differently, you are probably familiar with this. If we consider gauge transformations $U(x)$ with an arbitrary gauge group $G$, the fields transforms as follows: $$ \psi \mapsto U \psi \;, \quad A_\mu \mapsto U A_\mu U^{-1} + i (\partial_\mu U) U^{-1} \;. $$ (For $G = U(1)$ if you insert $U = e^{ia}$, you get your formulae ...


3

The axiom of choice was something people had used without always noting that there is an assumption to be made in order to justify making infinitely many choices at once. On the other hand, Cantor felt that there shouldn't be intermediate cardinals between the naturals and the reals, so he hypothesized that this is the case and spent a considerable amount ...


1

I believe that such a matrix is said to be persymmetric.


1

Suppose you have a function $f:A \to C$, but you regard $A$ as a subset of some other set $i:A \hookrightarrow B$, and you do all your work from $B$. It is natural to ask if the function $f$ extends to a function $\tilde f:B \to C$ such that $f = \tilde f \circ i$. Now, when dealing with subsets, we like to abuse notation and talk about $x \in i(A)$ and ...


1

Given an algebra $\mathbf{A}=(A,F)$, the flat (one-point) extension of $\mathbf{A}$ is the algebra $\mathbf{A}^\flat=(A\cup\{0\},F\cup\{\wedge\})$ where each $f$ is extended to $A\cup\{0\}$ by setting all undefined values to 0 and $x\wedge y=0$ for distinct $x,y$ and $x\wedge x=x$. Since sets can be considered as algebras with no operations, your ...


0

I think the primary point of interest here is that many English (or really any natural language) words can have more than one meaning, and it can be frustrating. Like the comments say, the context is the only thing that can really help you identify what the words mean. In any given mathematical context, there is a canonical meaning for each term. When ...


0

Wikipedia's Inverse(Mathematics) has a lengthy list of interpretations as there is also negation that can be inverted on a Boolean value(turning true to false or false to true), inverting a permutation, inverting a linear transformation and other operations beyond addition and subtraction as logarithms can be inverses of exponential functions in some cases. ...


1

This is another grey area with mathematical terminology. Inversion can mean many different things in many different instances, but in general, it has to do with swapping two things, or creating something's "opposite." Each definition you give of inversion or finding an inverse can be made more precise with more words: $1/a$ is the multiplicative inverse of ...


2

For comparison's sake, we can try to build an "unnatural" group action. Given a group $G$ and a set $S$, we know that the identity $e \in G$ must act as the identity map and that the action of two elements is the same as the action of their composition. I propose as our "unnatural action" we take $S_n$ as our group and $S$ to be a set of $n$ labeled ...


3

It's the action $Sym(S) \times S \to S$ given by $(f,x) \mapsto f(x)$. Every action $G \times S \to S$ defines a homomorphism $G \to Sym(S)$ and vice-versa. The natural action defined above corresponds the identity homomorphism. This makes it "natural". An "unnatural" group action $Sym(S) \times S \to S$ could for instance correspond to an nontrivial ...


0

Figured it out after a little Bit of google searching its called Gambler's fallacy


1

Undefined is a term used when a mathematical result has no meaning. More precisely, undefined "values" occur when an expression is evaluated for input values outside of its domain. $\sqrt{-9}$ (if no complex numbers) $\ln(-4)$ (if no complex numbers) $\tan(\pi/2)$ (units in radians, no complex infinity) n / 0 (if no complex infinity) In the above two ...


0

For convenience, let's restrict ourselves to real numbers. Multiplication of real numbers is a binary operation. It takes any two real numbers, x and y, and returns the real number $x \times y$, usually written as $xy$ or $x \cdot y$ (or, in the UK, $x.y$). One of the properties of multiplication of real numbers is that, for any non-zero real number $x$, ...


4

Something is undefined because none has defined it (yet). It is easy to define addition and multiplication of real numbers. One can then show the theorem that for arbitrary $a,b\in\mathbb R$, the equation $a+x=b$ has exactly one solution. By virtue of this theorem, one can define $b-a$ as this unique solution. One can also show that theorem that for ...


3

In this case, undefined means "not in the domain." The division operator, $div(numerator, denominator)$ is defined over the domain of real numbers except that y=0 is not in the domain. Now, that very specific wording aside, I think you might get an intuitive grasp of the meaning if I can draw from a non-mathematics topic, programming languages. ...


3

When referring to the result of a mathematical operation, undefined means there is no meaningful result. When referring to a mathematical object which satisfies some mathematical relationship, undefined means no object satisfies that relationship. The operation $f$(🍊) has no meaningful result. There is no $x$ that satisfies the relationship ...


1

tl; dr: For the result of the 'divide by zero' operation, undefined describes something that can be multiplied, but can otherwise be anything. Division is the inverse operation to multiplication; A number, $a \div b$, when multiplied by $b$ must return $a$; $b \cdot (a \div b) = a$. Suppose an object $c$, which can be multiplied, satisfies $a \div 0 = ...


6

There were a couple of issues with my original answer. Firstly, it generated quite a lot of controversy - I currently have 4 upvotes and 4 downvotes - despite that my answer was mathematically correct! I interpret this as meaning that it wasn't elementary enough to be useful (that is, it wasn't written at the right level). Secondly, the OP stated that the ...


6

From the algebraic point of view, the Real numbers form a field under multiplication and addition. If we look at the Reals as a field, there is no separate operation of "division", instead we multiply by the reciprocal. Since $0$ has no reciprocal in $\mathbb{R}$ (in fact the additive identity in a field never does), there exists no element available to ...


2

The answer by user38858 is very much to the point: It looks like you're seeing two different definitions on those various websites: Undefined = not yet defined. As in, this may have a value or meaning or definition, but we haven't gotten there yet. A lot of sites may do this with a variable X, saying it hasn't yet been defined, or we haven't ...


3

"a number divided by zero may be any real number" is a wrong statement and is never said. A number divided by zero is just not a real number. "zero divided by zero may be any real number" is an informal way to express that certain indeterminate expressions have a limit.


6

On some site of Mathematics, I read that it could be any number. and on some sites, I read that it may be anything. These are two major meanings of undefined but I think, they have a clash. I just wanted to address this because it looked like none of the other (very good) answers have. It looks like you're seeing two different definitions on those ...


7

What does the term "undefined" actually mean? In light of the already great answers provided by Carsten and Christian, I thought a more linguistic analysis of "undefined" may be in order. The following two terms are explained in the book Origins of Mathematical Words by Anthony Lo Bello: indeterminate$\quad$ The Latin noun terminus means the end of ...


45

To put matters straight: Division is a function $$q:\quad{\mathbb R}\times{\mathbb R}^*, \qquad(a,b)\mapsto q(a,b)=:{a\over b}\ ,$$ whereby $q(a,b)$ is the unique number $x\in{\mathbb R}$ such that $b \>x=a$. When we say that $\displaystyle{a\over0}$ is undefined then this means no more and no less than that the pair $(a,0)$ is not in the domain of the ...


66

Saying that 1 divided by 0 is undefined, does not mean that you can carry out the division and that the result is some strange entity with the property “undefined”, but simply that dividing 1 by 0 has no defined meaning. That is just like when you ask whether the number 1.9 is odd or even: That is not defined. Or when you ask what colour the number 7 has.


0

Most commonly the lower and upper bound of the summation index. See, for instance http://en.wikipedia.org/wiki/Summation.


1

From a remote sensing point of view, we usually refer to a bounded but unpredictable process as stochastic. If the process were unbounded and unpredictable I would tend to use random, but this case doesn't occur very much in my world! :)



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