Tag Info

New answers tagged

2

According to my dictionary Peturb To disturb greatly in mind; disquiet To throw in confusion; disorder To cause (a moving body, celestial object, etc.) to deviate from a theoretically (orbital) motion. I think we can ignore the first definition here but the other two are relevant . If you were to nudge a system slightly you are perturbing it and we may ...


0

The technical term for a set of values having no properties except that values are distinct is "set".


1

They're called abstract sets. Personally, I would simply call this a "set," or perhaps a "mere set" or "unstructured set" to emphasize that there's no further structure around.


0

A set is a collection of distinct elements in which order doesn't matter. A multi-set is a collection of elements (with possible repitition) in which order doesn't matter. Neither of these (or the next terms below) require or need any operations to be defined for them. For example, $\{ziggyswooglehorn, 5, \color{#C00}{\rm Red}\}$ is an example of a set ...


1

Googling for equalizer family morphisms (also in Google Books and Google Scholar) returned some hits. It seems that some people use the name multiple equalizer. Adámek, Herrlich, Strecker: Abstract and Concrete Categories. The Joy of Cats mention this notion in Example 11.4(2), p. 194. Another book using this notion is Castellini: Categorical Closure ...


3

Usually it's referred to as the triangle inequality.


1

A hypothesis is an educated guess, often a statement you want to show or prove (or disprove). A good example is the Riemann Hypothesis, which Riemann hypothesized in the mid 1800s. It says that the real parts of the nontrivial zeroes of the function $$\zeta(s) = \displaystyle \sum_{n \geq 1} \frac{1}{n^s}$$ all have real part $1/2$. We do not currently know ...


4

Theorems are things proved from a theory. Meta-theorems are things proved about the theory. The statement: If $\sf ZF$ is consistent, then $\sf ZF$ does not prove $\sf AC$. Is a meta-theorem about the theory $\sf ZF$. It quantifies over all proofs that we can write from the axioms of $\sf ZF$. If you like to think about it semantically, proofs are not ...


0

A counter-intuitive statement breaks with your intuition, a paradox breaks with your (possibly erroneous) reasoning.


2

A counter-intuitive statement is either true or false like a normal statement, but surprising in the sense that one would expect a different outcome. A paradox is undefined, in the sense that if one accepts it, it would be in contradiction to some other statement (an antinomy). It gives the dilemma to drop either the paradox or those statements which are ...


1

To converge in $C^\infty$ every derivative has to converge uniformly on each compact set. So for all $K$ compact, and multi-index $\alpha$ we have $$ \sup \limits_{z \in K} || D^\alpha (f_n - f) || \to 0 $$


1

Soap Film Problem - this is actually another term for "minimal surface problems", since soap bubles or other similar soap forms tend to minimize their surface. Links here and here.


0

Well, if you take an operator like $\subseteq$ you can view it as a map/function/relation. Let $\mathfrak{S}$ be the set of all sets, then you can view the operator as a function/mapping $\subseteq : \mathfrak{S} \times \mathfrak{S} \to \{0,1\}$, where $1$ can be interpreted as true and $0$ as false. The set of all functions mapping to $\{0,1\}$ is ...


0

Coordinate transform is a technical term. It refers to the process of finding out the new coordinates of a point fixed in space when the coordinate system is changed. "Change of coordinates" is not really a technical term. When a point $P$ has its coordinates changes from $(x_1,y_1)$ to $(x_2,y_2)$, it could be that the point is physically moved in the space ...


0

A sledgehammer is not just a special hammer, but a very large and heavy hammer. Normally one uses it for demolition, to break big things. You wouldn't want to use it to drive a nail, much less a small nail.


1

Trivial A solution or example that is ridiculously simple and of little interest. Often, solutions or examples involving the number 0 are considered trivial. Nonzero solutions or examples are considered nontrivial. For example, the equation x + 5y = 0 has the trivial solution x = 0, y = 0. Nontrivial solutions include x = 5, y = –1 and x = –2, y = 0.4. ...


0

There is, as is visible, no universally agreed upon convention. However, I would argue in favor of considering by an large the number of $n$-th roots of an element $a$ (in some structure) the cardinality of the set of solutions of $X^n = a$, so that in the complex and real numbers, and in any other field, $0$ is the unique root of $0$. The alternative ...


0

In a sense everything in $K^n$ is basis independent, because vector spaces of the very special kind $K^n$ don't actually need any basis in order to associate coordinates to vectors: vectors in $K^n$ are $n$-tuples of numbers, which components one can access directly. Yes, there is the standard basis, but it is given rather than chosen; moreover this is ...


8

The terminology varies a bit between people and fields, but what I would say is that $x^n=0$ has one root (namely $0$) of multiplicity $n$. If we explicitly say that we count that root "with multiplicity", of course, there are $n$ of it.


1

As many as roots of the polynomial equation $X^n=0$, that is, $n$.


4

Fundamentally, "equipped with" just means "and". A Hilbert space is an object which consists of two components: a vector space, and an inner product on that vector space. This means that strictly speaking, a Hilbert space and a vector space are different objects: a Hilbert space is not a vector space, and a vector space is not a Hilbert space. But the ...


3

Unlike vector spaces arising in other ways, ${\mathbb R}^n$ has a distinguished basis, namely the standard basis. Declaring this basis as orthonormal defines the distinguished scalar product $$\langle x,y\rangle:=\sum_{k=1}^n x_k y_k\tag{1}$$ on ${\mathbb R}^n$. This scalar product is a well defined function on ${\mathbb R}^n\times{\mathbb R}^n$ which does ...


0

The sigmoid squashing function is the same as the sigmoid function. The term sigmoid squashing function is favored in the neural net community. The logistic function is the classical squashing function. Other sigmoid functions include: arctangent, the hyperbolic tangent, the Gudermannian function, and the error function. It is called the squashing ...


3

You didn't say what was confusing, but I guess it is the word "smallest". In this context, $T$ is the smallest topology with property $P$ means that you should show: $T$ has property $P$ If $S$ has property $P$, then $T$ is a subset of $S$ For topologies, point 2 means that whenever $G$ is an open set in topology $T$, it is also an open set in ...


1

Vector space is an abelian group first: that is the group of vectors. But that is not enough. We also need to have a concept of scalar multiplication. These scalars are 'aliens': they are from another object, a field. ANd there are various conditions the alien field's addition and multiplication has to satisfy to interact 'compatibly' with the addition ...


2

Informally, let's consider a mathematical space, which we think of primarily as a set $A$ of points (or primitive elements) on which are defined various operations (e.g. addition, scalar multiplication with a field, scalar product, etc.). We can say that $A$ is "equipped" with these operations. That is, whenever we mention $A$, we have in mind these ...


11

I think the best way of explaining what it means to be equipped with something is by showing how the term is used. Mathematics has a very precise formalism, but in its pure form it is mostly not appropriate for daily use because then even the simplest problems would become very hard. Equipping a mathematical object with additional structures is one of the ...


46

The word "equipped" keeps notational pandemonium from breaking loose. For instance, if you were to be a bit more formal, you'd say A Hilbert space is a pair $(V, \left<\cdot,\cdot \right>)$, where $V$ is a vector space and $\left<\cdot,\cdot \right>\colon V\times V \to \mathbb{C}$ is an inner product. Additionally, all Cauchy sequences in ...


19

Generally, when we use the word "equipped", we mean not that it is possible to do an operation, but we have one in mind. That is on a vector space like $\mathbb R^2$, we could think of various inner products that would suffice. For instance the following two operations are both inner products: $$(x_1,x_2)\cdot(y_1,y_2)=x_1y_1+x_2y_2$$ ...


9

Well, sometimes we actually want to work with spaces were as few operations as possible are allowed. Often in math, we want to find the most general environment where a property holds, so we want to work in spaces with as few properties as possible. It's interesting to see just what we can do without needing to use an inner product, or a metric, or the lack ...


1

In Bourbaki's Commutative Algebra, one has the following definition. Suppose $A$ is a (commutative) ring and $S$ is a multiplicative subset containing no zero divisors. Let $B=S^{-1}A$, so that there is an injection $A\to B$. An $A$-submodule $M$ of $B$ is said to be non-degenrate (non-dégénéré) if $BM=B$.


2

In the Banach space theory there is a property called local unconditional structure, which is l.u.st for short. Another property is the Dunford-Pettis property which is DP for short.


2

Game theory has a trembling hand, some cheap talk, and, collectively, an El Farol Bar problem.


-1

The difference between an operator and a function is simply that we've decided to call the operator an operator and we've decided to call the function a function. From a formal point of view, that's all there is to your question. Any deeper meaning to the choice of words is merely a matter of convention -- when someone calls something an operator in some ...


0

To answer this question, may I suggest that everyone think like this: instead of asking "IS an operator a function?" or "What IS an operator?" etc. we reframe the question like this: "How are the words 'operator' and 'function' (and, if you like, 'mapping', 'transformation') used when we are talking about mathematics?" [With the understanding that, despite ...


3

Although the term may not really be in common use: A paper about "Generalized staircases: Recurrence and symmetry" refers to a figure showing a certain surface at page 10, and calls it "The eierlegende Wollmilchsau surface" The term eierlegende Wollmilchsau literally means "egg-laying wool-milk-sow", and refers to any (usually imaginary) thing that "can ...


5

There are topological spaces called hedgehog spaces. According to the linked Wikipedia article, a $K$-hedgehog space is sometimes said to have "spininess $K$." And let's not forget the process of blowing up points on a plane.


1

The Sieve of Eratosthenes is an abstract thing given a mundane (concrete) name, not unlike the "snowflake".  And I've heard that the term googol was chosen specifically because it sound funny.


2

Given that $f\circ f\circ\ldots\circ f\ $ ($n$ times) is called the $n$th iterate and the functional square root is often called half iterate, I'd call $f\circ f$ the second iterate of $f$.


2

The closest you come in words is probably to speak about "$f$ iterated twice" or something like that.


1

Generalizing the question to functions, note that $\circ$ is just a binary function $\circ(x,y)=z$ but expressed with infixed notation $x\circ y$, in my opinion the common terminology is: given a $n$*-ary function* $f:X_0\times X_1\times...\times X_n\rightarrow Y$ $$f(x_0,x_1,...x_n)=y$$ arguments : $x_0,x_1,...x_n$ are the elements af the domains ...


1

You show "sum" as the result of the function "sum." If you do not insist on the sought-after word beginning with "oper-," how about -- "result?"


3

Speaking of the "value" or "result" of the operation would by far be the most understandable.


1

As mentioned in the comments, a point $x$ that satisfies $f(x) = x$ is called a fixed point of $f$. Fixed points are often of practical interest as they can guarantee that an algorithm that repeatedly uses the function $f$ will stop at some point. Therefore many fixed point theorems exist, which describe sufficient conditions for when a fixed point exists ...


2

A local minimizer beats all competitors in some neighborhood. In a stronger norm, neighborhoods are smaller, and therefore a local minimum is easier to have. Hence, being a local minimizer in a stronger norm is a weaker property than being a local minimizer in a weaker norm. Such terminological switches happen all the time. For example, if topology ...


1

I have not run across a strict definition of "term", which may be because it is similar to "element" in that it cannot be defined without a circular definition, but here goes anyways. A term of a mathematical object (such as a polynomial, sequence, equation, etc) is one simple "piece" distinguishable from the rest of the "pieces" in some way (i.e., in ...


0

It is the simplest example of third degree curve in 3D space. It ought to be perhaps called a "bent and twisted" cubic as $\kappa$ and $\tau$ scalars are non-zero. Curvature and Torsion are equally important for space curves to describe the way they bend and twist in 3D space.


2

Telling a story on myself. When as a graduate student I first heard about noetherian rings (before I saw a definition) I wanted to know what an ether was, so I could think about a ring that didn't have any of them. I later taught for a while at Bryn Mawr College, where a colleague used Emmy Noether's desk.


5

Not exactly math, but physics is close enough. There are higher derivative of velocity called jerk, jounce, snap, crackle and pop. And there's also screw theory.


3

The Wiener Sausage is what the nbhd's of a Brownian motion trace out. One might argue things named after Norbert Wiener or Mark Kac are not unusual since they were relatively famous mathematicians. But its still funny.



Top 50 recent answers are included