Ansgar Esztermann
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 Mar11 comment Finding the basis of a vector space $\Bbb{W}$ Instead of starting with a family of vectors that might be a basis, try starting with a single vector, then look for a second, linearly independent, vector, and so on. See the link given by Umberto P. (it even contains a complete answer). Mar11 comment Finding the basis of a vector space $\Bbb{W}$ What have you tried so far? Are there any particular points you have trouble with? Nov13 comment Addition in linear vector spaces Addition of scalars is commutative even if much less than the full field axioms are given, e.g. in a quasi-field (multiplication neither commutative nor associative, only one distributive law), see math.stackexchange.com/questions/724122/…. Thus, to get non-commutative scalar addition, the requirements for the scalar structure would have to be relaxed considerably. Oct20 comment ZFC, NBG and Naive set theory @AndresCaicedo: I stand corrected. If you submit an answer of your own, I'll retract mine. Oct16 comment Inverse image of a closed interval is compact @jlang: No, the preimage of [0,1] under $f$ is a subset of $D$. Apr25 comment If G is a finite abelian group and $a_1,…,a_n$ are all its elements, show that $x=a_1a_2a_3…a_n$must satisfy $x^2=e$. What you have defined is obviously not a group. E.g. you've already seen that 22=6=42, but then (22)3=(42)3 and -- if associativity would hold -- 2=4. Apr9 comment Equivalence relation: prove that $(X \cap Y)$\ $E$ $\subset (X$ \ $E) \cap (Y$ \ $E)$ Are you sure that you have written the inclusion in the right direction? Apr9 comment How do I find arccos(-16.503) So, the problem seems to lie in an earlier step. How did you arrive at the expression $\arccos(-16.503)$? Apr4 comment cartesian product $A^2 = A$, possible? Am I missing something, or should it be $A_\omega\times A_\omega\subset A_\omega$? Apr3 comment Limit definition by ordinal numbers Principia Mathematica, *207: "A term x is said to be the "upper limit" of alpha in P if alpha has no maximum and x is the sequent of alpha. In this case, x immediately follows the class alpha, though there is no one member of alpha with x immediately follows." Apr2 comment Can two function $f$ and $g$ have same values through out a given interval and different values outside that interval? @user136561: This merely shows the difference (within a certain interval) is smaller than the resolution of the computer monitor. Apr1 comment Why does $f(x)=ax^2 + bx + c \ge 0\ \forall x \in \mathbb R$ imply $f$ has at most one real distinct root and discriminant $D \le 0$? @Sabyasachi: Take $a=1$, $b=c=0$. Mar28 comment Show from the axioms: Addition in a quasifield is abelian @azimut: Thanks, I've added the answer accordingly, fixing an incorrect eq reference along the way. Mar25 comment Is there any number $n$ such that $nm=0$, $n\neq 0$, and $m\neq 0$? @YiyuanLee: It is not a group but merely a monoid (there are no zero divisors in a group since they do not have an inverse). The group $(ℤ/6ℤ)^*$ exists, but it does not contain the elements 2 and 3. Feb14 comment Why is $(0,0,0)$ not acceptable as a co-ordinate on the projective plane? In other words, the points of the projective plane are the one-dimensional subspaces of a three-dimensional vector space. Any of these subspaces can be uniquely identified by a basis vector (so we can write $(x,y,z)$); but $(0,0,0)$ is not a basis vector, and it does not identify a one-dimensional subspace. Jan27 comment Exercise regarding boolean algebra? Hint: a) use X+(YZ)=(X+Y)(X+Z) Sep12 comment Blue eyes: a logic puzzle I do not think that the claim marked * is true. Consider the case where the guru announces "98 people who have blue eyes": everyone has known this before (universal knowledge), because everyone sees at least 99 blue-eyed people. Moreover, everyone knows that everyone knows: blue-eyed people see 99 other blue-eyed people, so these must each see at least 98 blue-eyed people. But there it stops. It is not true that everyone knows that everyone knows that everyone knows that the guru sees 98 blue-eyed people. This, however, is a necessary condition for "common knowledge". Feb20 comment Number of zeros in a number Shouldn't the input encoding be part of the problem statement? Jan28 comment Can an event be possible if its probability is zero? You do not even need look for repeated numbers -- the probability of generating any given number is zero (assuming a true random-number generator and infinite precision). Still it happens every time the generator is run. Jan15 comment Is $f = x^2$ or only $f(x) = x^2$ correct? It is certainly quite common in physics to omit function arguments; but note that the two notations you give for kinetic energy are not equivalent: E.g. for a rocket, the complete expression would be $E(t)=\frac{1}{2}m(t)v^2(t)$ since mass is not constant.