I am trying to find the values of $m,n$ that makes $U_{m,n}$ graphic. I am guessing that it is only graphic if $n = m + 1,$ I tried $U_{2,4},U_{2,5},U_{3,6}, U_{2,3},U_{4,5}$ and I think I am correct. Am I correct or I am missing something? EDIT: Here is the definition of regular: A regular matro...
Consider the complete bipartite graph $K_{3,3}$ in plane such that all its vertices lie on a circle. Is this framework locally rigid in plane (which I believe is the case) and if so, how to prove this? I know that the above framework is not infinitesimally rigid. Moreover, it is known (due to Bol...
The classical theorem of Asimow and Roth says that for a generic framework (i.e., coordinates of the nodes are algebraically independent), local rigidity and infinitesimal rigidity are equivalent. I was wondering how this theorem fails if we replace "generic'' by "(affine) general position". So m...
I am reading Algebraic Topology by Allen Hatcher. I come to know for any group $G$ ,we can make an universal cover $X$ of $X/G$ by properly discontinuous action . There is a paragraph mentioned in Hatcher's book about Caley Complex. Though it is mentioned ,but I couldn't figure out when I want to...
In first-order logic, the notion of functional completeness is well-defined. But in higher-order logics, where we can quantify over predicates and not just individuals, the notion of functional completeness may be more complex.
I am reading about Functional Completeness in Wikipedia. In the "Formal Definition: "Since every Boolean function of at least one variable can be expressed in terms of binary Boolean functions, F is functionally complete if and only if every binary Boolean function can be expressed in terms of t...
If I know that the set of operators {∨, & , ¬} is functionally complete, how do I go about proving/disproving the functional completeness of the following set of operators? a) $\{\vee,\neg\}$ b) $\{\to,\neg\}$ c) $\{\to\}$ I have looked at the answer here for (b) : Prove that the set {→, ¬} ...
I need to prove the functional completeness of $\{\text{or},\text{ xor},\text{ xnor}\}$ with the help of $\{\text{not},\text{ or},\text{ and}\}$ (which have been already proven to be functional complete). My attempt is that I only have to show that $\{\text{or},\text{ xor}\}$ is functional comple...
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