# Tag Info

2

Not exactly an solution to your question, but frankly it's too big to be fitting a comment. The distinction as to exactly what is being linear is very important. It is possible to approximate non-linear behaviours if we are allowed to make the linear space (as in linear algebra) large enough. Approximating solutions to smaller dimensional non-linear ...

0

I just recall a definition from calculus: Definition: A function $f : I \subseteq \mathbb{R} \to \mathbb{R}$ that is differentiable on the interval $I$ is linear if $\dfrac{\mathrm{d}f}{\mathrm{d}x} = c$ for a constant $c \in \mathbb{R}$.

1

Your last definition is (basically) as general as it gets (so far as I know). Definition. Let $S$ denote a semiring (with both $0$ and $1$.) Suppose $X$ and $Y$ are modules over $S$. Consider a function $f : X \rightarrow Y.$ Then: $f$ is linear iff for all $x,x' \in X$ and $s,s' \in S$, we have $f(sx+s'x') = sf(x)+s'f(x')$. $f$ is affine iff ...

0

Have you read the Wikipedia page for homotopy already? That might be helpful. In the picture you link, there are two paths with endpoints $x$ and $y$. The names of those paths happen to be $\gamma_0$ and $\gamma_1$. While $\gamma$ is a common variable for a path, I do not know where that convention comes from. The function $H$ is a homotopy between the two ...

3

Hyperbolic geometry is not really geometry on a hyperboloid. It's geometry on an infinite surface of constant negative Gaussian curvature, something which cannot be represented even in 3D. You can model it using a sheet of a hyperboloid, but the metric you get isn't the normal 3D metric you'd intuitively expect. Elliptic geometry is not the geometry on an ...

2

A $20\%$ decrease is $80\%$ of a value. Likewise, a $20\%$ increase is $120\%$ of a value.

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The terminology varies, but this description may help. You would say that the new value is "20% less than" the old value, or that it is "80% of" the old value. The first describes the amount of change compared to the original value (the "relative change"), and the second describes the new value compared to the old value (the "change factor"). Relative ...

1

In set theory, a set $x$ is transitive if for all $y$, if $y\in x$ then $y\subseteq x$. The (von Neumann) ordinals are defined to be the transitive sets that are well-ordered by $\in$. Thus the "$<$" relation between ordinals is simply $\in$. If $\alpha, \beta$ are ordinals with $\alpha\in \beta$, then $\alpha \subseteq \beta$. So any ordinal is a fine ...

0

Surds originated from the Latin word surdus which meant "mute". This muted sound is largely thought that it represents irrational numbers whereas rational numbers would be a pure, clear sound. Go to ...

2

Terminology In ordinary mathematics (outside formal set theory), it is only convention that determines whether something is called a "family, set, collection, or class". One reason for the varied terminology is that we often begin with a collection of basic objects (e.g., real numbers). Then we form "sets" of these basic objects (e.g. sets of real ...

2

One cannot really keep the answer to this separate from the question of what the words "set", "class", "family", etc mean -- because the question rests at least partially on a misconception. It looks like you think these are different things such that it makes sense to ask whether the thing you're looking at is one or the other. But that is often not the ...

0

It is the same symbol used for different concepts; unfortunatelty there are too many "products" in mathematics. The "extension" of the product operation from numbers to vectors, gives rise to two different operations; see : Josiah Willard Gibbs (1839 – 1903), Vector Analysis (textbook by E.B.Wilson, first published in 1901 and based on the lectures that ...

2

This is not a mathematical term but a figure of speech. One says that A is an honest B to assert that A satisfies the definition of B. Usually the reason to emphasize honest is that A was informally called "B" earlier in the text. This is associated with a somewhat conversational style of writing. Specifically for distance functions, this word would ...

0

No, unfortunately there is not name for these functions. In general, the integral of an non-fractional elementary function will result in another elementary function. Polynomials are the name for $ax^2 + bx + c$ sort of things with this being a three term polynomial. In general, I think the term you want is a non-fractional elementary function. Really, this ...

3

Given a ground set $X$ and an $n\in{\mathbb N}_{\geq1}$ a function ${\bf a}:\>[n]\to X$ is called an $n$-tuple and is presented in the form $(a_1,a_2,\ldots, a_n)$. A tuple is then an $n$-tuple for some $n\geq1$ – it's the same thing as what you call a list in your question. Now I have never seen a "tuple of tuples", but would be willing to consider a ...

1

Not really. Matrices are maps between spaces, where columns and rows are base vectors for their spaces. On them, a scalar product has to be definable - and this just works if one can exist in a space that the other "lives in". They therefore cannot be smaller or larger, therefore have to be the same length. So no, not as a matrix. But you can define ...

1

A monoid is supposed to have an identity element, which is with free generation considered as the empty string. Hence your example translates to the free monoid on the letter $x$.

3

Sometimes, putting parentheses around logical statements can help make their meaning clearer if you are ever in doubt. In this instance: if (any infinite sequence in X has an adherent point in X) then (X is compact) What the first bracketed statement essentially means is "choose any infinite sequence in X and it will have an adherent point in X". Now, at ...

1

This is a comment rather than an answer. Suppose you have a quadratic equation, say $$x^2-x-6=0.$$ To solve this we first assume that $x$ is a solution. From this assumption we derive that $x=-2$ or $x=-3$. So $x$ a solution $\Rightarrow x=-2$ or $x=3$. Therefore the set of numbers that solve the equation are $\{-2,3\}$ --- and we can say that $-2$ and ...

0

It's the algorithm to write the digits of the number n in the numbersystem with base m I think. The $r_k$ are then the digits, but I do not know a name for the $q_k$. $q_0 = \lfloor n / m \rfloor$ ,$q_1 = \lfloor q_0 / m \rfloor$ ... Example: if $m=10$ One could ask: if $n=1234321$ what is the name for the part $q_0=123432$ ? or for the part $q_2=1234$? It's ...

-1

Sets $A_1, A_2, ...$ are mutually disjoint if $$\bigcap_{i=1}^{\infty} A_i = \emptyset$$ Sets $A_1, A_2, ...$ are pairwise disjoint if $$A_i \cap A_j = \emptyset \ \forall i \ne j$$ Apparently, most texts use 'disjoint' to refer to 'pairwise disjoint'. Whenever a text uses 'pairwise disjoint', we can assume 'disjoint' refers to 'mutually disjoint' ...

1

Such a magma is called either medial or entropic. From lines 1-2 of this paper: ... entropic groupoid (“medial” in the terminology of Ježek-Kepka) ... refers to Jaroslav Ježek and Tomáš Kepka. Furthermore, from page 27 of An Introduction to Quasigroups and Their Representations by J.D.H. Smith: and, from the bibliography of this ...

0

I don't think that there is any standard term. These are the quotients in the Euclidean algorithm. You might see terms such as "first quotient", "n-th quotient", or "final quotient", but I think that is about it.

1

The growth of $x^n$ is called linear ($n=1$), quadratic ($n=2$), cubic ($n=3$), quartic ($n=4$), quintic ($n=5$)... You can find here a few names after that, but I don't expect anyone to say "octavic/octic" with a straight face. In general if you don't know $n$ it's just called "polynomial growth".

0

For exponent $2$, you say quadratic, for $3$ and following, cubic, quartic, quintic... For general exponent, power function or power law. [In French you can use potentielle, but this can cause polysemic ambiguities.] The adverb quadratically can be freely used; I would abstain from "cubicly", "quarticly"... Also avoid the confusion with quadric, which ...

2

Writing this as an answer as I have exceeded the comment character limit. Gathering the comments made so far, I believe that "Proof Theory" as specified by @KevinQuirin is what you are looking for as a direct answer to the title question. However, if you want to learn more about cryptography and category theory, you should check out books that are ...

2

One aspect of the analogy involves what are sometimes called full structures. For any set $X$, the full structure on $X$ has universe $X$, and has all (finitary) relations and functions as part of the structure. (So it's a structure for a language with $2^{|X|}$ relation and function symbols if $X$ is infinite.) A 1-type over this structure amounts to an ...

0

$A$ is finer than $B$ and $B$ is coarser than $A$. $A$ is a refinement of $B$. This is borrowed from the language of partitions and covers, but there seems to be no reason it can't apply to arbitrary sets. (Besides, I'm pretty sure all sets can be considered to be covers, anyway.) Source (for partitions): ...

0

There is (sort of) but it's adapted from a more general setting. Given a preorder $(X,\preceq)$, a subset $B\subseteq X$ is cofinal in $X$ $\!\iff\!$ for all $x\in X$ there is $b\in B$ with $x\preceq b$. Given $A\subseteq X$, if for all $a\in A$ there is $b\in B$ with $a\preceq b$, then we can say $B$ is cofinal with respect to $A$; or, borrowing terms from ...

2

No one symbol that I know denotes this operation. A sort-of-two-symbol term denoting it is: $$\bigcup_{a\in A} \mathcal{P}(a).$$ If you need to use this operation a lot, introduce a definition for a symbol having that value.

1

$G$ is a constant. It's one number, not a variable ranging over numbers. You don't "treat it as a variable", and it's probably misleading or confusing (to yourself) to "assume that it's a varlable". Similarly, you can't "take the derivative of a constant with respect to itself" — it's borderline nonsense to say that or try to think it. But you can treat ...

2

In math, a tuple used to index a multi-dimensional array is often called a multi-index. In programming, an index that spans a range of values is often called a slice or a slice index. I would therefore suggest a term like multi-slice or slice multi-index. By the way, you're using the words "tensor" and "hypercube" in a way that could be confusing to many ...

4

You shouldn't use the term "hypercube" here, because that's not what you're talking about. Instead, you are talking about a multi-dimensional table, which you could think of as the vertex set of a refinement of the 1-dimensional skeleton of a hypercube. It is important to understand what you need this term for, though. For example, are you writing an ...

0

It depends on how you define $L^2_w(a,b)$. The $L$ usually stands for "Lebesgue", which would imply a positive weight $w$ and Lebesgue measurable functions $f$ for which $$\|f\|_w^2=\int_{a}^{b}|f|^2wdx < \infty.$$ However, in texts that do not use Lebesgue integration, $L^2_w(a,b)$ could mean continuous, piecewise continuous, or ...

2

According to this note, a split $k$-algebra $A$ is a $k$-algebra all of whose simple modules $M$ are absolutely simple, in the sense that after extension of scalars $M \mapsto M \otimes_k L$ (where $k \to L$ is a field extension) they remain simple. If $A$ is semisimple, I believe this condition is equivalent to the condition that $A$ is a finite product ...

0

It's not that we define a $\sigma$-algebra "in order to avoid [the] problem" that non-(Lebesgue-)measurable sets exist. Rather, we define exactly what we mean by "measurable set, form the collection $\cal S$ of all such sets, and then prove that this collection is a \sigma$-algebra. Having done so, it's a triviality that every set in$\cal S$is ... 2 From John L. Kelley's General Topology (1955 edition, July 1957 printing, p. 65): A directed set is a pair$(D,\ge)$such that$\ge$directs$D.$[. . . .] A net$\{S_n,n\in D,\ge\}$is in a set$A$iff$S_n\in A$for all$n$; it is eventually in$A$iff there is an element$m$of$D$such that, if$n\in D$and$n\ge m,$then$S_n\in A.$The net is ... 0 I'm aware of only one way how to write this clearly -- in symbols: $$\exists_\infty n\in\mathbb{N} : P(n).$$ However, unless you need this a lot you shouldn't use this notation. If you need it a lot, you can introduce it: We suppose that$P(n)$holds for infinitely many$n$, i.e., that$\exists_\infty n\in\mathbb{N}:P(n)$. Some people use: "$P(n)$... 2 In some contexts (set theory, order theory, point set topology, though probably never in probability) you can say cofinally, or cofinally many, cofinally often. Given a preorder$(A,\preceq)$, a subset$X\subseteq A$is cofinal in$A \Leftrightarrow$for every$a\in A$there is$x\in X$with$a\preceq x$. A predicate$\varphi(x)$holds cofinally often, and ... 2 CW answer from comments to push it from unanswered queue: It is Género (form Portuguese Wikipedia.) 4 In descriptive set theory and logic one sometimes uses$\exists^\ast_n P(n)$for "there are infinitely many$n$such that$P(n)$holds", and$\forall^\ast_n P(n)$for "all but finitely many$n$satisfy$P(n)$". Then at least we have$\lnot \forall^\ast_n P(n) \leftrightarrow \exists^\ast_n \lnot P(n)$, like for normal quantifiers, e.g. 3 It seems like your problem is less with Hamiltonian circuits and more with circuits in general; you are right that that is a contradiction. The problem, however, is your definition of circuit. What you have is a closed walk. My definition of a circuit is a closed walk that repeats no edges. Unfortunately, this is a problem with graph theory in that the ... 0 Regardless of the definition you use for oblong, the answer ends up being "these lines don't have widely accepted names". Since you've defined an oblong as a non-square rectangle: In this case, I'm not aware of any particular name for the lines you describe. That's because there's nothing particularly fundamental about them as far as the geometry of the ... 0 I think you could describe$\int_{A}f\text{d}\mu$, where$f$is a$\mu$-measurable function and$A$a$\mu$-measurable set on which$f$is definite, as follows:$\int_{A}f\,\text{d}\mu$is "the integral of$f$on$A$with respect to the measure$\mu$, where$f$is called the integrand of this integral", so that the term "integrand" explicitly depends on a ... 0 Graph skewness is another measure of how planar graph a graph is. It follows the definition that you suggested: the minimal number of edges that have to be removed to make the graph planar. 6 No. Every single time a function is bijective, it will be said so, either explicitly or by using a qualifier that implies it, like isomorphism or automorphism. 1 They also called input-output coefficients, and they represents the amount of resource$ i$consumed per unit of variable$x_j$. 1$2^n$is a special function not only for the reason specified. It is the discrete analog of the continuous function$e^x$. Namely, it is the unique nontrivial function whose first difference is itself. $$\Delta 2^n = 2^{n+1}-2^n=2\cdot 2^n -2^n=2^n$$ (while in the continuous case$e^x$is the unique nontrivial function whose first derivative is itself) 1 I would call it the base$2$exponential function. If this doesn't work for you, perhaps you could call it the doubling function? 1 I don't think there is a short name for$2^n$. However, I think that "the number of subsets of a set with$n\$ elements" (or cardinal of the powerset, of course) is already nice enough. For instance, it fits very well with your exemple of +'es and -'es: just select the subset of positive terms.

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