# Tagged Questions

For requesting, clarifying, and comparing definitions of mathematical terms.

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### Perfect powers by Oblath's result [duplicate]

What do you mean by this statement? Obl\'ath proved that the only perfect powers all of whose digits are equal to a fixed one $a \neq 1$ in decimal representation are 4, 8 and 9. This is equivalent ...
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### What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I wrong?...
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### Is there a meaning to the notation “\arg \sup”?

When $f$ is a function on a set $A$, the notation: $\arg\max_{x\in A} f(x)$ denotes the set of elements of $A$ for which $f$ attains its maximum value. This set may be empty, for example, if $f(x)=x$ ...
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### Arithmetic-definability of geometrically-defined arithmetic concepts

Arithmetic-definability of geometrically-defined arithmetic concepts For purposes of discussion take arithmetic to be the study of the [ positive real] numbers, sequences of numbers, etc. and take ...
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### Formal definition of “counterexample”.

What is the preferred formal definition of “counterexample” as in: zero is a counterexample for "every integer is either positive or negative". Where in the literature is the notion of “counterexample”...
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### Is there a concept of “Cross determinant”?

Suppose $A = \begin{bmatrix}a & b \\ c & d \\\end{bmatrix}$. The determinant of $A$ is $$\det A = ad - bc.$$ Suppose $B = \begin{bmatrix}e & f \\ g & h \\\end{bmatrix}$. Now one could ...
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Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula $$\... 0answers 36 views ### Advantages and disadvantages of a particular definition of rings and subrings My professor defines a ring as a triple (A, +,\ \cdot) such that (A, +, 0) is an abelian group, (A,\ \cdot) is a semigroup and \cdot distributes over +. Subsequently, B\subseteq A is said ... 2answers 37 views ### What is the difference between a polynomial and a function or can they be used interchangebly? I have been wondering over this basic question (seems rather trivial at first sight) for a long time- What is the difference between a polynomial and function? My confusion arises form the ... 0answers 14 views ### Definition of pseudo-differential operator I'm trying to understand the defintion of pseudo-differential operators (\psi do) on manifolds. According to Hörmander, The analysis of Linear Partial Differential Operators, v. III, 18.1.20 (1994), ... 1answer 55 views ### What does it mean in general to show something is well defined? [duplicate] There is another post that addresses this but quickly fix the problem to be something in arthmetics, and in turn what it means for that arithematics problem to be well defined. I have never ... 6answers 3k views ### Chain Rule Intuition We know that the chain rule is used to differentiate a composite function ,say$$f(x) = h(g(x))$$It's defined as the derivative of the outside function times the derivative of the inner function or ... 1answer 66 views ### What is the correct definition for positive operator and positive definite operator? As far as I know those operators are defined as follows: Positive operator is an operator L: H\rightarrow H such that \langle L\textbf u|\textbf u\rangle \geq0 for all \textbf u \in H and the ... 21answers 66k views ### What are imaginary numbers? At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ... 0answers 16 views ### set-theoretic definition of sum of sequences, difference of sequences, product of sequences,… I have f,g \in \Bbb R^\Bbb N, I define: (f+g):=\{(y,z)| z= f(y)+g(y) \} (-f):=\{(y,z)| z=-f(y) \} (f-g):=f+(-g) (f\cdot g):=\{(y,z)| z= f(y)\cdot g(y) \} Is it correct? I have problem ... 1answer 39 views ### Definition of Limits: x=c [closed] If 0 < |x-a| < \varepsilon, and it's possible for x=a, then wouldn't the equation become 0 < 0 < \varepsilon (which is technically impossible)? 1answer 24 views ### What is actually the standard definition for Radon measure? I see that there are various definitions for Radon measure and they are NOT equivalent, but they are equivalent on locally compact Hausdorff spaces. I think this is the reason why Radon measure has ... 0answers 15 views ### Finitely generated module: terminology. What's the meaning of the expression: S is a subring of \mathbb{C} finitely generated as \mathbb{Z}-module? Maybe that the additive group of the ring S is a finitely generated abelian group? ... 4answers 667 views ### What is the “Principle of permanence”? While reading the book "The Number-System of Algebra (2nd edition)." term "Principle of permanence" occurred to me. I remember I had read this in the book "Beginning algebra for college students.". I ... 2answers 30 views ### How do you rigorously explain the fact that u \in L^p can be non defined over sets of measure 0? In all the definitions of L^p(\Omega) spaces I have been given these are defined to be the set of functions f: \Omega \to \mathbb{R} whose norm ||\cdot||_{L^p} is finite. We define is as the ... 1answer 24 views ### Formal construction of free groups and objections in arguments For simplicity, consider X=\{a,b\}. Let Y be another set in bijection with X, and write its elements to be a^{-1},b^{-1}. Let W(X) be the collection of all words in a,b,a^{-1},b^{-1}, ... 1answer 44 views ### Stabilization of embedding? In D. Freed's lecture notes he mentions "stabilization of embedding" in theorem 4.48. Does anyone know the definition? I can't find it online. 1answer 27 views ### Is a relation between A and B the same as a mapping from elements of A to subsets of B? The way I always saw it was that a relation is a subset of A \times B, or a collection of ordered pairs (a,b), where a \in A and b \in B. Is there any meaningful distinction between the two ... 1answer 20 views ### What should I change in the definition of a function to make it time dependent f \to f(t)? Suppose I have a function that takes from sets X \subset \mathbb{R}, Y \subset\mathbb{R} to \mathbb{R}$$d: X \times Y \to \mathbb{R}$$The elements are identified by d(x,y) = xy \in \mathbb{... 1answer 26 views ### formula for defining terms in a finite set Suppose there's a finite set, S of terms in \mathbb{R} which have the property P(x). Suppose we know how to define the maximum value of the set by the relation, max(x). We also have the ... 1answer 117 views ### Relation between open sentences and sets (conceptual question) Hi I'm a college student getting into the more proof oriented side of math. I was reviewing Mathematical Proofs, A Transition to Advanced Mathematics 2nd edition and after thinking about chapters 1 ... 2answers 46 views ### When 2 functions are equal? Are 2 functions equal when they have same domain, same codomain and same law ? EXAMPLE 1 f: \mathbb{R} \to \mathbb{R} x \to x^2 and g: \mathbb{R} \to \mathbb{R^+_0} (set of positive ... 1answer 25 views ### Is there a difference between arc-wise connectivity or path-wise connectivity? When authors refer to arc-wise connectivity, do they mean path-wise connectivity? I am studying space filling curves and when reading books, I either come across the concept of arc-wise connectivity ... 3answers 113 views ### Strange question about free abelian group I was given the following question for homework, but it makes no sense to me. Let F be a free abelian group over a set S with respect to the function \varphi \colon S \to F. Identify the set ... 0answers 20 views ### For a graded poset, why do we only consider the characteristic polynomial defined in terms of \mu? When we have a graded poset P with 0 and 1 we can define the characteristic polynomial f_P(t) of P:$$f_P(t)=\sum_{x\in P}\mu(0,x)t^{r(1)-r(x)}$$However, given a poset, have two functions, ... 3answers 64 views ### Confusion Regarding Munkres's Definition of Basis for a Topology The definition of Basis for a Topology as given in Munkres's book is as follows, If X is a set, a basis for a topology on X is a collection \mathcal{B} of subsets of X (called basis ... 3answers 129 views ### Why there is no value for x if |x| = -1? [duplicate] According to the definition of absolute value negative values are forbidden. But what if I tried to solve a equation and the final result came like this: |x|=-1 One can say there is no value for x... 2answers 32 views ### Clarification about the definition of surjectivity Related to the question Proof that Laplacian is surjective \mathcal{P}^n\to\mathcal{P}^{n-2}, I know in general that the surjectivity is defined to be : \forall f \in \mathcal{P}^{n-2}, \exists \... 4answers 530 views ### Topology, closure definition - well defined? I came upon the following definition for closure, Given a subset of a topological space X, the closure of A is defined as the intersection of all closed sets containing A. How is this ... 1answer 30 views ### Equivalent definition of Lebesgue measurability for sets? When introducing measurability, we noted that we wanted the following property to hold for disjoint A, B \in \mathcal{P}(\mathbb{R}) m(A \cup B) = m(A)+m(B) (additivity) We then defined a set A ... 0answers 33 views ### What does it mean for a tangent bundle to be parallel? I am looking at page 8 of the paper here, just below definition 2.13. I have never come across a phrase such as 'T\mathfrak{C}\subset{TM} is parallel' - what does it mean for a tangent bundle to be ... 1answer 20 views ### Diameter of a Topological Manifold I know that for a Riemannian Manifold is defined the concept of diameter. I wuold know if it's defined a similar concept for a most general Topological Manifold. Thanks in advance. 2answers 52 views ### Is there a way to denote a repeated operation? For instance, if you have say: 4 + x = 10 We instantly calculate 10 - 4 to derive x, which is 6. So you could say there is only 1 operation, subtraction, where 4 ... 1answer 20 views ### What is the connection between positive definite hessian and metric? In heard from a someone in verbatim that if you take the taylor series of a certain function, if the Hessian is positive definite, then it is a metric. This quote is without context and therefore ... 1answer 26 views ### Why do we call this transformation non-singular? In linear algebra books, the authors call the linear transformation T with the property$$T(\alpha)=0\implies \alpha=0 non-singular. What's the motivation behind the term "non-singular"?
Several years ago I was bored and so for amusement I wrote out a proof that $\dfrac00$ does not equal $1$. I began by assuming that $\dfrac00$ does equal $1$ and then was eventually able to deduce ...