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0

The same idea as other answers, but a different explanation (which makes more sense to me, at least): there is no such thing as true, period. A statement can only be true or false in some model, which consists of a certain set of "original" statements whose truth is assumed, and everything provable from those statements. The "original" statements are the ...


21

You're treating the word "axiom" as you were probably taught in high school. That an axiom is something which is "simply true as an assumption". Modern mathematics has changed that definition to "an assumption made in a certain context". Not every axiom is called an axiom, some axioms are proved as theorems, and sometimes lemmas are used for axioms. And not ...


5

The existence of a model of a statement does not mean that statement is "true" (whatever that means; see below). For example, the Poincare disk is a model of Euclid's first four postulates plus the negation of the parallel postulate; this does not mean that the parallel postulate is "false." What having a model of a set of statements does mean, is: that set ...


0

Look at the fibers. Your condition implies that $f^{-1}(y) ⊆ cl(\{x\})$ for every $x ∈ f^{-1}(y)$, i.e. each fiber is indiscrete. So, such mapping is injective even if $X$ is $T_0$. Conversely, if each fiber is indiscrete, then the closure contains the “fiber-closure”. Every continuous map is a unique composition of a quotient map and an injective ...


2

It is typical, at least in my experience, that "random walks" unless otherwise stated are sums of iid random variables. You can form two random walks with iid copies of normal distributions with means $\mu, \tilde{\mu}$ and variances $\sigma^2$ respectively. Then by the law of large numbers, the averages of their partial sums will converge to the their ...


0

i think that indefinite integral and anti derivative are very much closely related things but definitely equal to each other. indefinite integral denoted by the symbol"∫" is the family of all the anti derivatives of the integrand f(x) and anti derivative is the many possible answers which may be evaluated from the indefinite integral. e.g: consider the ...


1

The term anti-invariant seems to be in use for what I believe is this general idea. The Wikipedia glossary of invariant theory defines an anti-invariant as A relative invariant transforming according to a character of order 2 of a group such as the symmetric group. which, if my understanding of the relevant terms is correct, is a special case of what I ...


1

It means the same thing. It's just for esthetic reasons that authors try to vary their vocabulary.


1

It is the same whether you say there exists or for some.


3

"There exists" does not mean there is only one, i.e. "There exists" is different from "There uniquely exists." So since there is no specification that there is exactly one such object (until proven otherwise), you can read the "there exists" as "for some" if you like.


0

Regarding everyday perception of number and the question of what "large" ought to mean, there is the field of Numerical cognition with many interesting experiments. E.g. when you flash a sheet with some dots for a second, people will be able to tell if there is only one dot, or two, or three. If there are three dots, people who see it for one second will not ...


3

The symbols $\gg$ and $\ll$ don't have a formal definition. Usually they are used to compare two extremely big numbers, for example $\mathrm{Graham's \; number} \ll \mathrm{TREE}(3)$ or something like that. They are only used because someone wants to make clear that one of them is so much greater. The numbers are indeed just small compared to everyday ...


0

It's a 2d/3d function where a ouside surface is defined as everywhere in the graph where the value crosses a certain point for example from 0.999 to 1.0001, 1 is the isometric value... <1 is inside the surface and >1 is outside of it. if the surface is the formula surface = y+1; then it is a flat plane parralel to the y axis. Search for isosurface ...


3

$\renewcommand{ket}[1]{|#1\rangle}$ $\renewcommand{bra}[1]{\langle#1|}$ There are several common terminologies: As you might guess based on the fact that $V^*$ is called the "dual" of $V$, in mathematics and mathematical physics the covector of a vector $v$ is called the "dual" of $v$. In physics, vectors are often denoted e.g. $\ket{v}$. The $v$ is a ...


3

Echoing @Lærne, $v^c$ is the dual of $v$. If you want a reference, how about Halmos's "Finite-Dimensional Vector Spaces"? In the second edition, sections 13, 14, and 15 along with sections 67, 68, and 69 may be what you're looking for.


16

They share a Latin root corresponding the the concept of 'wholeness'. In the context of integers, this would be interpreted as 'whole numbers' - i.e. numbers with no fractional part. In the context of integration, this would correspond to 'summing up to create a whole' in the sense of the integral representing a continuous sum or area. I think that's why ...


0

It seems to me that the structure is a pair of a monoid (though not a unital) and a monomorphism $(S,\verb|^|)$. A morphism $S\overset{\alpha}{\longrightarrow} S^\prime$ should be a (non unital) monoid morphism $\alpha$ such that the diagram commutes: $\require{AMScd}$ \begin{CD} S @>\alpha>> S^\prime\\ @V \verb|^| V V{\#} @VV \verb|^| V\\ ...


6

"The index of $N$ in $G$". That notation does not require normality (it can be seen as the number of left cosets of $N$, for instance), and the number it represents might of course be infinity.


0

According to wikipedia, an operator is a function whose domain and codomain are both vector spaces or modules. Since $\mathbb{R}, \mathbb{Q}, \mathbb{C}$ are all (one-dimensional) vector spaces, many familiar functions are also operators. However, a general function might be from a domain that is not a vector space, and hence not be an operator, e.g. ...


3

You might call it a vertex-signed graph. See e.g. this paper.


0

Descriptive statistic deals with describing given data, aiming to summarize a sample using measures of central tendency (mean, median etc.) and measures of variabilty (standard deviation, variance etc.). It "paints a picture" of the data where you can immediately see important information without having to work through thousands of numbers. Inferential ...


0

It is precisely an infinitesimal change. But an infinitesimal difference in what? A differential 1 form measures the flux across an infinitesimal line situated at some point. By extension a differerential 0 form evaluated at a point in space measures the flux across an infinitesimal point (an infinitiseimAl point is the same as a finite point is the same ...


3

As far as I've seen, the word cover is usually employed for this (no need for extra adjectives). That is, $\{E_\alpha\}_{\alpha\in A}\subset\mathcal{P}(X)$ is a cover of $X$ precisely when $X=\bigcup\limits_{\alpha\in A}E_\alpha$.


3

$(-\frac{\partial G}{\partial y}, \frac{\partial G}{\partial x})$ has a compact notation as $$ (z \times \nabla) G $$ But I don't know of any name for that expression. Note that $(z \times \nabla) G$ lies purely in the XY plane. Perhaps there is no name for this expression is because the situation for fields on a 3-D space is such that there is no simple ...


0

You can use the definition of simplical vertices. A simplical vertex $v$ is a vertex whose neighbors $N(v)$ form a clique. Thus, the set of vertices $V =\{v\} \cup N(v)$ is the set that you are interested in.


1

Don't take my answer as a reference, but from my experience, notably in French, "action of $G$ on $X$" usually refers to a set-action (i.e. an homomorphism $G \to \operatorname{Aut}_{\mathbf{Set}}(X)$) while "representation of $G$ on/in $X$" refers to an homomorphism $G \to \operatorname{Aut}_{\operatorname{Mod}(T)}(X)$. For example, one talks about "linear ...


1

Apparently the pattern they are referring to is $2^0\cdot 1=1$ is not prime $2^1\cdot10+2^0\cdot 1=21$ is not prime $2^2\cdot100+2^1\cdot10+2^0\cdot1=421$ is prime. Im not sure why this is the "logical order from left to right" but it is some logical order.


1

Numerator and denominator refer to a fraction. Ratios can be expressed as fractions, but they are not quite the same thing. For a ratio $a:b$, $a$ and $b$ are sometimes called the terms, with $a$ the antecedent and $b$ the consequent.


1

$\mathrm{ratio}=\frac{\mathrm{numerator}}{\mathrm{denominator}}$


0

The unit torus (in these cases) refers to a torus of major radius $R$ and minor radius $r$ of surface area $4\pi^2 R r=1$. As such it is the "unit square" with periodic boundary conditions. In random geometric graphs, the boundary of the domain plays an important role in much graph-theoretic behaviour. Sometimes it is useful to analyse graphs not inside the ...


2

It appears that when $X = \mathbb{R}$ this is referred to as being Quasilinear: In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. This utility function has the representation $u(x_1, x_2, \ldots, x_n) = x_1 + \theta (x_2, \ldots, x_n)$.


0

Yes. The torsion subgroup $T$ of $G$ is a direct sum of copies of $\mathbb{Z}_{p^\infty}$, for various primes $p$, and $G/T$ is a rational vector space, so it is isomorphic to a direct sum of copies of $\mathbb{Q}$. Since $T$ is a direct summand of $G$, therefore, $G$ is isomorphic to $T\oplus G/T$.


6

The difference is entirely a pedagogical one. Both approaches ultimately cover the same calculus. “Late transcendentals” is the traditional approach to teaching calculus where the treatment of logarithmic and exponential functions is postponed until after integration is introduced. In the traditional method, the natural logarithm is defined by ...


2

The most common name I've seen for the function that applies a function to an argument is "apply". However, I've also seen "eval" and several other names. In a programming languages context, the name "apply" is fairly standard, although its exact semantics varies significantly between languages. "eval" in this context is more commonly used to refer to a ...


0

I don't think equality of integrals over a single set ($\mathbb{R}$ in this case) endows the integrands with special names. In probability theory, we would say just that $X$ and $Y$ have the same first moments, but I think you're aware of that. OTOH since $X,Y$ are $\mathbb{R}$-valued, if you can show that $\int_a^b x\,f(x)\, dx = \int_a^b y\,g(y)\,dy$ for ...


3

Simply "$\subseteq$-incomparable" should suffice. Incongruent would bring connotations of some sort of congruency relation, but this is not the case, since $\subseteq$ is not a congruence relation. It's just a partial order.


0

Correct terminology for your problem would be "Single source Shortest path problem" Maybe you should try implementing the data structure (priority queuefor dijkstra) as fibonacci heap which has less running time Hope this helps


3

The use of "data" in mathematics is quite old. Oxford English Dictionary has a quote from 1645: T. Urquhart Trissotetras 53 The verticall Angles, according to the diversity of the three Cases being by the foresaid Datas thus obtained. I would suspect that its use in Latin is much older (as well as the use of the equivalent Greek term). What ...


1

Some people make a philosophical distinction, which is (somewhat fuzzily) mentioned in the Wikipedia article on 'a priori'. I agree with you that, for better or worse, the two terms are used almost interchangeably in Bayesian statistics. One difference in usage might be illustrated by the experiment of rolling a die. You might decide 'a priori' that the ...


2

Others have already explained the differences in usage of various conventions. My first course in topology was taught by Engelking so perhaps I can say something about his motivations for this choice. He had very specific standards for notation and terminology. In particular, if I recall correctly, one guideline was that notions that are used more often ...


4

There is no right or wrong in these cases. I believe that Bourbaki has “quasi compact” for the non Hausdorff case, but the terminology can be seen elsewhere. The book where I learned topology is Kelley’s, where the Hausdorff property has its importance, but is not assumed throughout. Perhaps Engelking has his reasons for including the Hausdorff property ...


7

There are many examples. The simplest is any space $X$ with more than one point that has the indiscrete topology, $\{\varnothing,X\}$. The simplest $T_1$ examples are any infinite set with the cofinite topology. The line with two origins is another $T_1$ example. Many of us topologists feel that there is no good reason to include Hausdorffness in the ...


7

I would argue that, overall in modern mathematics, the accepted meaning of "compact" is "every open cover has a finite subcover", i.e., it does not include the requirement of being Hausdorff. The approach of including Hausdorff-ness in the definition of compact, and instead using the word "quasi-compact" for the less restrictive condition, is traditionally ...


3

For 2, take for example: $$X=\{a,b\}$$ $$\tau=\{\emptyset,X\}$$ Every finite space is compact, so $X$ is compact (according to your second definition), but $X$ is not Hausdorff because there is no open set that contains $a$ but not $b$. To include a less trivial example, let $(X,\tau)$ be any compact and Hausdorff space with more than one point. Now ...


5

There is no difference: $\Bbb R = \left]-\infty,+\infty\right[$. Writing it like this serves to get you used with the symbol $\infty$, I guess (mostly psychological reasons?). Also, there will be a time when you'll need to use concepts dealing with the extended real line, so it will be natural to talk about: $$\left]-\infty,+\infty\right], \quad ...


1

As far as precalculus is concerned, there is no difference.


2

There is no difference. The notation $(-\infty, \infty)$ in calculus is used because it is convenient to write intervals like this in case not all real numbers are required, which is quite often the case. eg. $(-1,1)$ only the real numbers between -1 and 1 (excluding -1 and 1 themselves).


1

The full quote is: If $\partial\mathbb{D}=\{z\in\mathbb{C}:|z|=1\}$, let $B=$ the uniform closure of the polynomials in $C(\partial\mathbb{D})$. This means: consider the set of all continuous functions on $\partial\mathbb{D}$, equipped with the uniform norm $\|f\|=\sup_{\partial \mathbb{D}}|f|$. This space is denoted by $C(\partial\mathbb{D})$. ...


0

Another term that is sometimes used, especially in the context of topological spaces and related objects, is "self-map".



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