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

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1answer
93 views

Elementary definition: what's a parallel volume-form?

This is a very elementary question, What is the definition for a volume form (or $n$-form) to be parallel with respect to the metric? To find out more about the concept, what kind of topic do I need ...
5
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1answer
598 views

Topology of uniform convergence?

Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X,R) of real-valued continuous functions on X, with the topology of uniform convergence I am having a hard ...
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1answer
101 views

Property of Division by vector for a field

Serge Lang in "Linear Algebra" on page 2 says that The essential thing about a field is that it is a set of elements which can be added and multiplied, in such a way that additon and ...
2
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1answer
154 views

how to write the process of decomposition of a graph into shortest closed sub graphs

If I want to decompose a graph in to possible shortest closed cycles (as shown in right side). then how can i describe this process with mathematical notations. to understand please refer below ...
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1answer
306 views

Mathematical notation of graph subdivision

If anyone can define a directed graph subdivision with mathematical notation, please post a response. My second question is: Irrespective from the planar embedded graph or not, is this definition ...
2
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2answers
135 views

In a groupoid, do any two objects have a morphism between them?

This is a question about the definition of a groupoid (in category theory). I've read the Wikipedia article, and it says that a groupoid is a small category in which every morphism is an isomorphism. ...
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1answer
111 views

Meaning of $L_A$?

Let $A$ be an m*n matrix with entries from a field $F$. $L_A: F^n \rightarrow F^m$ defined by $L_A=Ax$. I'm a bit confused about this definition. $L_A$ is a matrix representation of linear ...
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2answers
192 views

Definition of Unipotent in Positive Characteristic

Let $G$ be an affine algebraic group over an algebraically closed field $k$ whose characteristic is $p>0$. Can $\mathcal{U}(G)$, the set of unipotent elements of $G$, be characterized as all ...
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1answer
54 views

What is the graph $K^c_m$?

The book "Graph Theory with applications" by J.A. Bondy and U.S.R. Murty, which is available here. The Theorem $4.6$ of this book says that: If $G$ is a non-Hamiltonian simple graph with $n≥3$ ...
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0answers
204 views

I think a definition is wrong in “Model Categories” by Hovey.

I am working through the book "Model Categories", by Mark Hovey, and have a doubt about a definition given there. At the beginning of page 50 we read: Define a map $f:X\rightarrow Y$ to be a ...
5
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2answers
114 views

Is an “open system” just a topological space?

I think of an "open set" as being "roomy" or "spacious," in the sense that around every point, there is a little bit of room. This motivates the following definition. Definition. An "open system" ...
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2answers
223 views

Definition of “Up to” (homeomorphism,isotopy, etc), and Examples?

I've tried googling this usage and understanding the results but I'm struggling to make intuitive sense of it. So my question is, what is the phrase "up to" understood to mean, and what are some ...
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3answers
515 views

Positive constant scalar definition

In French when we say "$k$ est une constante positive", that means $k\geq 0$. But I remarked that using the same sentence in English, "$k$ is a positive constant", means that $k>0$. Can one explain ...
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2answers
39 views

Defined amount and value amount of a function

What is the defined amount and value amount for this function: $$f(x)=\sqrt{(x+7)(1-x)}?$$ The defined amount is all the x-values the function can be and the value amount is all the y-values the ...
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1answer
144 views

Differentiable manifold in dimension 1 and its critical point

Please, I want to know how to define a differentiable manifold in dimension 1, and if the circle is a differentiable manifold in dimension $1$, and what is its critical point. Thank you.
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1answer
47 views

If a ring $R$ is a field, must $R$ be a unitary ring?

If a ring $R$ is a field, then does it automatically imply that $R$ is a unitary ring? Thank you.
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2answers
63 views

Question on the definition of a ring.

A ring $\langle R,+, \cdot\rangle $ is a set $R$ with two binary operations such that: $\langle R,+\rangle $ is an abelian group. Multiplication is associative. Left and right distributive laws ...
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0answers
241 views

Different Definitions on the Differentiability of Functions on a closed set.

I have encountered three different definitions on the differentiablity of functions on a closed set. In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The ...
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2answers
490 views

Formal Dirichlet-Bourbaki definition of function

What is the formal Dirichlet-Bourbaki definition of a function? I have come across this in this essay: http://www.k-12prep.math.ttu.edu/journal/contentknowledge/meel01/article.pdf on page 1. I know ...
0
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1answer
125 views

Operations with planar graphs

I think in the graph theory operations such as decomposition into cycles, union, intersection, difference, subdivision can be done. If I am given a planar graph (for e.g. see figure), then can the ...
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2answers
2k views

What is a direct correlation?

I have two contrary definitions of for the direct correlation between two variables $X$ and $Y$ Their correlation coefficient is close to $1$. There is a direct causal relationship between the ...
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4answers
282 views

Definition of subgroup generated by a subset

I'm confused about the definition of a subgroup $(W)$ generated by a subset $W$ of a group $G$. My textbook gives: Let $(W)$ be the set of all elements of $G$ representable as a product of ...
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1answer
362 views

Meanings of the terms “conjunct” and “disjunct” in a logic?

This sentential logic problem is stated as: Suppose that $A \models B$, where $A$ is a conjunction of literals and $B$ is a disjunction of literals. Show that $ \models \neg A$, $ \models B$, or a ...
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3answers
1k views

Definition of degree of finite morphism plus context

Let $f: X \rightarrow Y$ be a finite morphism of schemes, defined here, http://en.wikipedia.org/wiki/Finite_morphism I always assumed that the degree of $f$ was the degree of the induced field ...
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0answers
241 views

Understanding the roots of homomorphism and homeomorphism

I understand the formal (i.e. mathematical) definitions of the terms "homomorphism" and "homeomorphism" as they relate to functions, but I am curious as to the origin of these terms. I don't know ...
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3answers
3k views

What's the difference between direction, sense, and orientation?

I'm trying to understand the difference between the sense, orientation, and direction of a vector. According to this, sense is specified by two points on a line parallel to a vector. Orientation is ...
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2answers
163 views

Difference between Spanning set and Postitive Spanning Set

I do understand the difference as mentioned in the texts about spanning set and positive spanning set, im somehow missing how if $v_1.. v_r$ is a positive spanning set for $R^n$, then $v_2 ... v_r$ ...
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2answers
301 views

How to extend definition of n-tuple to the case $n=0$?

The classical definition of n-tuple $(x_i)_{i < n}$ starts at $n=2$. In this case $$(x_0,x_1) := \{\{x_0\},\{x_0,x_1\}\}$$(1). For $2<n=k+1$, $(x_i)_{i < n}:=((x_i)_{i < ...
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4answers
400 views

Is the free group on an empty set defined?

I'm guessing that the free group on an empty set is either the trivial group or isn't defined. Some clarification would be appreciated.
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1answer
46 views

What is an evaluation operator and what is its use?

In a lecture about numerical mathematics, mostly about ODEs, we were given the following definition: The two-parametric family $\mathbf \Phi^{s,t}$ of maps $\mathbf \Phi^{s,t}: D \mapsto D$ is ...
2
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1answer
321 views

What is a tangential gradient?

If $\mathbb{R}_{+}^{n}$ is the half space $\{x\in\mathbb{R}^n\vert\ x_n>0\}$ and $u$ is a twice differentiable function in $\mathbb{R}^{n}$. If we write: \begin{equation} ...
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6answers
8k views

Is the empty set a subset of itself?

Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty ...
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1answer
69 views

Harmonic conjugate of $u,v$ in $f=u+iv$

Do I understand correctly the definition of being harmonic conjugate if I understand it that: $v$ is the harmonic conjugate of $u$ but $u$ is not the harmonic conjugate of $v$, but rather $-u$ ?
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2answers
512 views

two notation: semi-metric and pesudometric

There are two notations: semi-metric and pesudometric make me unclear. Are they the same thing, or they are different? Thanks ahead.
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3answers
125 views

Precise differences in meaning of Power Series, Taylor Series

Being an physicist/artist, not a real mathematician, I often toss around the terms "Taylor Series" and "Power Series" without any concern. Are these terms be considered interchangeable by ...
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3answers
192 views

Formula for Product of Subgroups of $\mathbb Z$, Problem

What is the product of $\mathbb{Z}_2$ and $\mathbb{Z}_5$ as subgroups of $\mathbb{Z}_6$? Since $\mathbb{Z}_n$ is abelian, any subgroup should be normal. From my understanding of the subgroup product, ...
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3answers
1k views

What does it mean to have no proper non-trivial subgroup

I am reading a first course in abstract algebra and there is a claim that says a group $G$ with no proper non trivial subgroups is cyclic. But I don't understand what does it mean to have no proper ...
3
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2answers
141 views

Definition of complete in the context of Lebesgue measurable sets

I came across this statement on Lebesgue measurable sets. The Lebesgue measurable sets are said to be complete because every subset of a null set is again measurable and the lebesgue measurable ...
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3answers
125 views

Example of a (dis)continuous function

The following thought came to my mind: Given we have a function $f$, and for arbitrary $\varepsilon>0$, $f(a+\varepsilon)= 100\,000$ while $f(a) = 1$. Why is or isn't this function continuous? I ...
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1answer
104 views

Is uncountably summation defined?

We know that finite and countably summation is defined. But How about uncountably summation, say $$\sum_{i\in \mathbb{R}}0$$ Is it defined?
2
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1answer
167 views

Limit point definition

I have read the definition of a limit point of a set in Real Analysis. The definition goes like: A number $p$ is said to be a limit point of a set of reals, $S$, if every neigbourhood of $p$ has at ...
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3answers
135 views

Do dihedral groups $D_n$ for $n\geq 5$ exist?

I know we can generate dihedral group of order three ($D_3$) and four ($D_4$) but my question is whether we can generate dihedral group of order five?
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4answers
899 views

Congruent Modulo $n$: definition

In an Introduction to Abstract Algebra by Thomas Whitelaw, he gives examples of the congruence mod operation, such as $13 \equiv5 \pmod4$, and $9 \equiv -1 \pmod 5$. But when I first learned about ...
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2answers
143 views

What does relax mean in the mathematical context

Here is a direct citation from wikipedia: The assumptions were further relaxed in the works of Terence Tao and Van H. Vu, Friedrich Götze and Alexander Tikhomirov. Finally, in 2010 Tao and Vu ...
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0answers
71 views

Define composition of small cyles and making a big graph

I am having following sub graphs and wish to compose all and make a one bigger graph (say G). After that, I want to select the closed path where it is passing along the outer vertices of that ...
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2answers
131 views

Standard definition of group isomorphism

ProofWiki defines a group isomorphism as a bijective homomorphism. In Topics in Algebra 2$\varepsilon$, Herstein defines a group isomorphism as an injective homomorphism: Definition. A ...
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2answers
274 views

Basic question about analyticity vs. differentiability in complex analysis.

In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula," 3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to ...
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1answer
107 views

What is Absolute convergence?

Take: $$ (u*v)(k) = \sum_{i=-\infty}^\infty u(i)v(k-i). $$ $k$ is there, it's because you want to define $$ \ldots\ldots, (u*v)(-3), (u*v)(-2), (u*v)(-1), (u*v)(0), (u*v)(1), (u*v)(2), (u*v)(3), ...
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1answer
32 views

Change Along A Tangent Line

I am taking some time to review differentials. What I don't quite get is why the change along the tangent line is $f'(x) \Delta x$, and how it leads to $f'(x)dx$
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1answer
215 views

Different formulation of a Traveling Salesman Problem

Given a undirected, weighted, complete graph $(V,E,c)$ with $c \to \mathbb{N}$ and $v_0 \in V$ we are looking for a set $E' \subset E$ minimal with respect to $c$ with the following conditions: for ...