| How to calculate gcd? Posted: 31 Oct 2021 07:56 PM PDT Note, In this question: $(a,b)$ means gcd $(a,b)$ Given $(a,b)=2$ Plus, $5a+7b=2$ I need to calculate: $(7a+14b, 14a+21b)$ I did the following: $$(7a+14b, 14a+21b) = (7(a+2b), 7(2a+3b)) = 7(a+2b, 2a+3b) = (4-3a, 6-a)$$ but stuck here...  |
| what is the purpose and consequences of Eigenvalues in graph theory? Posted: 31 Oct 2021 07:55 PM PDT I am trying to understand Eigenvalues and their repercussions in graph theory. I have read that Eigenvalues help describe certain parameters of graphs which provide information about the general structure of graphs. I know that we can take the adjacency matrix of a given graph, and obtain the eigenvalues, but I don't understand their consequences and what they tell us about a graph. Any help would be appreciate it.  |
| Is $\Omega$ the limit of all increasing sequence of sets $A_n$; Posted: 31 Oct 2021 07:53 PM PDT Let $\Omega$ be some set, and let $\mathcal{F}$ consist of all subsets of $\Omega$. Defina $\mu (A)=0$ if $A$ if finite, $\mu (A)=\infty$ if A if infinite. Is $\Omega$ the limit of an increasing sequence of sets $A_n$, with $\mu (A_n)=0$, but $\mu (\Omega)=\infty$. A solution to this question for $\Omega$ countably infinite is given here Show that $\Omega$ is the limit of an increasing sequence of sets $A_n$; My question - Is this always true or only when $\Omega$ countable infinite.
- When $\Omega$ countable infinite, Is $\Omega$ the limit of one or all increasing sequence of sets $A_n$, with $\mu (A_n)=0$, but $\mu (\Omega)=\infty$.
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| multiplication of functions Posted: 31 Oct 2021 07:50 PM PDT I've been multiplying functions for ages but I never really stopped and considered why I could multiply functions and have it "just work." Here's an example Say $f(1) = 9, g(1) = 7$. Then $(f(1)-g(1))^2 = (9-7)^2=2^2=4$. We could also do $(f-g)^2 = f^2-2fg+g^2$, and so $f^2(1)-2f(1)g(1)+g^2(1)=81-126+49=4$. Was this ever explained in grade school?? edit 1: Wait actually it makes sense. actually I think the larger question is why $(x+y)^2 = x^2+2xy+y^2$  |
| What is the volume of the region that is within a distance r outside of the surface of an n-dimensional hypersphere of radius R? Posted: 31 Oct 2021 07:44 PM PDT Suppose you have an n-dimensional hypersphere of radius R living within a d-dimensional space. Imagine the region in d-dimensional space consisting of all points that are a distance r outside the surface of that hypersphere. What is the volume of this region for any n, d, R and r? So take the special case of d=3 dimensions to begin with. For n=1 we have two hemispheres on either end of a flat line (the 1-d sphere). Their combined volume is just $2*(1/2)*(4/3)*\pi$$r^{3}$. For n=2 the region desired is the outer half of a donut. To find this volume I understand we can use the second centroid theorem of Pappus-Guldin by using the centroid of the semicircle of radius r, which is $\frac{4r}{3\pi}$. Then we find the volume of the desired region by taking the area of the semicircle of radius r and 'sweeping' it around the circumference of the circle of radius R + $\frac{4r}{3\pi}$. This gives the formula $2\pi(R+\frac{4r}{3\pi})*\frac{1}{2}\pi*r^{2}$. For n=3 you could obviously find the desired volume by taking the difference in volume of the sphere of radius R+r and the sphere of radius R, but I want a method that can generalise to a hypersphere of any dimension n in a space of any dimension d. Can the approach using the Pappus-Guldin theorem be generalised? Alternatively, I'm thinking a solution using calculus must be possible, but I don't know how to find it. So is there a neat formula for this? Thanks :)  |
| Jordan canonical form of $T: \mathbb{Q}(\alpha) \rightarrow \mathbb{Q}(\alpha)$ where $\alpha \in \mathbb{C}$ and $T$ is multiplication by $\alpha$ Posted: 31 Oct 2021 07:39 PM PDT Suppose $p \in \mathbb{Q}[x]$ is irreducible and $\alpha \in \mathbb{C}$ is a root of $p$. Then consider the linear transformation $T : \mathbb{Q}(\alpha) \rightarrow \mathbb{Q}(\alpha)$, $x \mapsto \alpha x$. What is the Jordan canonical form of $T$, considering $\mathbb{Q}(\alpha)$ to be a $\mathbb{Q}$ vector space? Suppose $\deg(p) = n$ and $p = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$. Then $\mathbb{Q}(\alpha)$ has dimension $n$ with basis $\{1, \alpha, \alpha^2, \cdots, \alpha^{n}\}$. Moreover, obviously the transformation $T$ acts on the basis element as $\alpha^i \mapsto \alpha^{i+1}$ for $i < n-1$. But $\alpha^{n-1} \mapsto \alpha^n = -a_{n-1}\alpha^{n-1} - \cdots - a_1\alpha - a_0$ in $\mathbb{Q}(\alpha)$. Thus $T$ is represented by the companion matrix, $C_p$ Not sure what the next steps would be- I'm confusing myself a bit here. It is well known that the companion matrix $C_p$'s characteristic and minimal polynomial is $p$ itself. Moreover, we know that $\alpha$ is thus an eigenvalue, which means that $\bar{\alpha}$ is as well. Thus $(x-\alpha)(x-\bar{\alpha}) | p$. One cannot conclude at this point that since $p$ is irreducible, $(x-\alpha)(x-\bar{\alpha}) = p$, since it is possible that $(x- \alpha)(x-\bar{\alpha}) \in \mathbb{R}[x]\setminus \mathbb{Q}[x]$. Even if $p = (x-\alpha)(x-\bar{\alpha})$ after all, I'm still a bit confused. Now that we've expanded our scope to $\mathbb{C}$, it's pretty clear that $\alpha$ is the only eigenvalue, so would the JCF simply be $\alpha I$? I'm getting a bit confused with the JCF since we traditionally start with an algebraically closed field and $\mathbb{Q}$ isn't. Would appreciate some help - I believe the intent is to proceed as if $C_p \in \mathbb{C}^{n\times n}$. Would appreciate some help.  |
| The $E_k$ in the definition of the simple function on pg. 61 in Royden real analysis "4th edition". Posted: 31 Oct 2021 07:50 PM PDT I want to prove that the sum of 2 simple functions is a simple function and to do so I want to use the following facts: $$\chi_{A_i}=\sum_{j}\chi_{A_i\cap B_j} \text{ and } \chi_{B_j}=\sum_{i}\chi_{A_i\cap B_j}.$$ But say if I want to prove the first fact, upon fixing $i,$ I want $A_i$ to be subset of $E$ and $E = \cup_{j}B_j$ Here is the definition of the simple function stated in Royden: If $\varphi$ is simple, has domain $E$ and takes the distinct values $c_1, \dots, c_n,$ then $$\varphi = \sum_{i=1}^{n_1} c_k \chi_{E_k} \text{ where } E_k = \{x \in E| \varphi(x) = c_k \}.$$ My question is: Are we considering the $E_k's$ in the definition of the simple function to be a partition of $E$? Could someone clarify this to me please?  |
| T a diagonalizable linear operator on V $\implies$ By the Complex Spectral Theorem, T is normal Posted: 31 Oct 2021 07:56 PM PDT Prove if the following statement is true. Let V be a finite dimensional C-vector space with inner product and T a diagonalizable linear operator on V. Then there is a basis of eigenvectors of T for V. Applying the Gram-Schmidt orthogonalization process to this basis and then normalizing, an orthonormal basis of eigenvectors for V is obtained. By the Complex Spectral Theorem, T is normal. Attempt: T a diagonalizable linear operator on V if and only if T has a basis of n eigenvectors, in which case the diagonal entries are the eigenvalues for those eigenvectors. Let $\beta$ be that basis. By Gram-Schmidt, ${\beta}'$ is an orthonormal basis of V. Since T is diagonalizable, $[T]_{{\beta}'}$ is diagonalizable. There exists an orthonormal basis ${\beta}'$ of V such that $[T]_{{\beta}'}$ is diagonalizable if and only if T is normal. Therefore, the statement is true. Am I wrong?  |
| Prove $\lim_{x\to \infty}\frac{\sin x}{x^2} = 0$. Posted: 31 Oct 2021 07:36 PM PDT Prove using definition of a limit that $\lim_{x\to \infty}\frac{\sin x}{x^2} = 0$. Proof: Let $\epsilon > 0$. Note that $\left|\frac{\sin x}{x^2}\right| \leq \frac 1 {x^2}$ for $x\ne 0$. Then let $M= \frac 1{\sqrt{\epsilon}}$. Then for $x> M$ implies that $\left|\frac{\sin x}{x^2}\right| \leq \frac 1 {x^2} < \epsilon$. Am I allowed to give an $M$ the way I did?  |
| $U_{1} \subseteq U_{2}^{\perp}$ if and only if $P_{W}=P_{U_{1}}+P_{U_{2}}$ Posted: 31 Oct 2021 07:38 PM PDT Let $V$ be a finite dimensional vector $\mathbb{C}$-space with inner product. Consider two subspaces $U_{1}$ and $U_{2}$ of $V$. Define $W=\left \{ u_1 +u_{2}:u_{1} \in U_{1}, u_{2} \in U_{2}\right \}$. Prove $U_{1} \subseteq U_{2}^{\perp}$ if and only if $P_{W}=P_{U_{1}}+P_{U_{2}}$. $\bullet \Rightarrow$ Let $U_{1} \subseteq U_{2}^{\perp}$. Then $\left \langle u_{1},u_{2} \right \rangle=0$ and $U_{1}$ and $U_{2}$ are orthogonal. This implies there exists an orthonormal basis of $U_{1}$ that can be extended to an orthonormal basis of $U_{1} \cup U_{2}$ and then of $V$. Let $x \in V$. $x=\sum_{i=1}^{k}c_{i}u_{i}+\sum_{i=k+1}^{m}c_{i}u_{i}+\sum_{i=m+1}^{n}c_{i}u_{i}$ and $P_{W}x=\sum_{i=1}^{k}c_{i}u_{i}+\sum_{i=k+1}^{m}c_{i}u_{i}=P_{U_{1}}x+P_{U_{2}}x$. Therefore, $P_{W}=P_{U_{1}}+P_{U_{2}}$. $\bullet \Leftarrow$ Let $P_{W}=P_{U_{1}}+P_{U_{2}}$ and $x \in U_{1}$. $P_{W}x=P_{U_{1}}x+P_{U_{2}}x\implies x=x+P_{U_{2}}x\implies P_{U_{2}}x=0\implies x\in U_{2}^{\perp}$. Therefore, $U_{1} \subseteq U_{2}^{\perp}$. $\bullet\therefore$ $U_{1} \subseteq U_{2}^{\perp}$ if and only if $P_{W}=P_{U_{1}}+P_{U_{2}}$.  |
| Question on prop 0.18 by Hatcher Posted: 31 Oct 2021 07:29 PM PDT Here is prop 0.18 from Hatcher: if $(X_1,A)$ is CW pair and we have attaching maps $f,g:A\rightarrow X_0$ that are homotopic, then $X_0\sqcup_f X_1$ is homotopy equivalent to $X_0\sqcup_g X_1$ relative to $X_0$. Hatcher says if $F:A\times I\rightarrow X_0$ is a homotopy from $f$ to $g$, consider the space $X_0\sqcup_F(X_1\times I)$; this contains both $X_0\sqcup_f X_1$ and $X_0\sqcup_g X_1$ as subspaces. In the online errata by Hatcher, he explains that this follows from the fact if $q:X\rightarrow Y$ is any quotient map, $A\subset X$ is a closed saturated set, then $q|_A$ is a quotient map. My question is but $X_0\sqcup_f X_1$ is not saturated. I think instead Hatcher's observation gives that $X_0\sqcup_F(X_1\times\{0\}\bigcup A\times I)$ is a subspace. Can anyone please explain this to me? Thank you!  |
| Is ℒ{ t·γ(t-1) } = ℒ{ (t-1)·γ(t-1)+γ(t-1)} ? I'm confused if these two are the same or if I can even write the second one like this . Posted: 31 Oct 2021 07:21 PM PDT The step function γ(t-1) is 0 for all values less then 1. Thus, my reasoning is that multiplying (t-1) by γ(t-1) and adding γ(t-1) should be the same to the original (ℒ{ t·γ(t-1) }). In addition, I was wondering is there is any property for transforms of the form ℒ{ f(t)·γ(t-a) }? If so, how can we use these? I know that [ ℒ{ f(t-a)·γ(t-a) } = e^(-as)F(s) ], but I do not think this will be helpful for transforms in the form ℒ{ f(t)·γ(t-a) }. I just started learning Laplace transforms a day ago and my professor has not provided us anything in regards to this.  |
| Find polynomials with rational coefficients whose roots are sin^2(pi/4m), sin^2(3pi/4m),sin^2(5pi/4m),.....,sin^2(((2m-1)pi)/4m) Posted: 31 Oct 2021 07:18 PM PDT I am stuck on this problem. Find polynomials with rational coefficients whose roots are sin^2(pi/4m), sin^2(3pi/4m),sin^2(5pi/4m),.....,sin^2(((2m-1)pi)/4m) I know cos2mx=(1-sin^2x)^m-(2m 2)((1-sin^2x)^(m-1))(sin^2x)+(2m 4)((1-sin^2x)^(m-2))(sin^4x)-... and we use that to get the roots however im having trouble getting there.  |
| Finite group with sequence of index 2 subgroups, such that the smallest one has odd order Posted: 31 Oct 2021 07:17 PM PDT This is a homework question so please only provide me with some small hints. For a finite group $G$ suppose that $G_k \subset \dots \subset G_0 = G$ such that each $G_i$ has index 2 in $G_{i-1}$. The claim is that if $G_k$ has odd order then it is normal in $G$. By the index assumption each $G_{i}$ is normal in the group above it. Thus I would like to transitively conclude $G_k$ is also normal in $G$. For this I would need to show that $G_k$ is characteristic in $G_{k-1}$, and then I could proceed by induction. Does this seem like a good first step? Can you give me a hint as to how to proceed?  |
| Lemma & Proof of Independent indicator random variables Posted: 31 Oct 2021 07:16 PM PDT |
| How do I go about modifying Euler Forward matlab code for differential equations Initial Value Problem? Posted: 31 Oct 2021 07:21 PM PDT Euler Code: function [x,y]=euler_forward(f,xinit,yinit,xfinal,n) % Euler approximation for ODE initial value problem % Euler forward method % Calculation of h from xinit, xfinal, and n h=(xfinal-xinit)/n; % Initialization of x and y as column vectors x=[xinit zeros(1,n)]; y=[yinit zeros(1,n)]; % Calculation of x and y for i=1:n x(i+1)=x(i)+h; y(i+1)=y(i)+h*f(x(i),y(i)); end end
Supporting File (to be modified). (Note, I am confused what I need to add): A=50; f=@(x,y) A*y-2; % Calculate exact solution g=@(x) exp(A*x); %this should be a function of x and A (since A is a parameter) xe=0:0.05:1; ye=g(xe); % Call functions [x1,y1]=euler_forward(f,0,1+2/50,0.1,2); % Plot plot(xe,ye,'k-',x1,y1,'k-.') xlabel('x') ylabel('y') legend('Analytical','Forward') Absolute_Error = abs(ye(end)-y1(end))
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| Integrating wrt a measure that is a collection of probability measures Posted: 31 Oct 2021 07:10 PM PDT Let $\mu$ be a probability measure defined on $\mathcal{A} \times \mathbb{R}$ where $\mathcal{A}$ is some measurable set and $\mathbb{R}$ is the set of real numbers. We have that $\mu(a, \cdot)$ is a probability measure so that $\int_{\mathbb{R}} \mathrm{d}\mu(a, x) = 1$, for all $a \in \mathcal{A}$. Now, let $f$ be a measurable function that I am currently not imposing any constraints on. I would like to have some sort of result where we have $$ \int_{(a,x) \in \mathcal{A} \times \mathbb{R}} f(a) \mathrm{d} \mu(a,x) = \int_{a \in \mathcal{A}} f(a) \mathrm{d} a. $$ I was wondering what kind of assumptions do we need to conclude something like this equality? And in what conditions can we extend it to $f$ being a finite measure so that the same expression equals to something like $\int_{a \in \mathcal{A}}\mathrm{d} f(a)$?  |
| Show that a certain set of $n \times n$ matrices is a subspace of vector space $\mathbb{R}^{n\times n}$ Posted: 31 Oct 2021 07:20 PM PDT how do I prove (or disprove) that the set $$\left\{X \in \mathbb{R}^{n \times n} \mid A X+X B+X^{T} C+D X^{T}=0\right\}$$ is a subspace of $\mathbb{R}^{n \times n}\,$? I only know that the zero vector exists, but I don't know how to proceed further. Any help appreciated.  |
| How to determine regions of positive and negative concavity of $h(g)= \frac{(g-a)(g-b)}{(g+a)(g+b)}$? Posted: 31 Oct 2021 07:24 PM PDT We may assume that $a<0,b<0, |b|>|a|$ How to check where the concavity of the function $h(g)= \frac{(g-a)(g-b)}{(g+a)(g+b)}$ is positive or negative? $h''(g)=\frac{4 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3} - {\left(a + b\right)} g^{3} + 3 \, {\left(a^{2} b + a b^{2}\right)} g\right)}}{a^{3} b^{3} + 3 \, {\left(a + b\right)} g^{5} + g^{6} + 3 \, {\left(a^{2} + 3 \, a b + b^{2}\right)} g^{4} + {\left(a^{3} + 9 \, a^{2} b + 9 \, a b^{2} + b^{3}\right)} g^{3} + 3 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} g^{2} + 3 \, {\left(a^{3} b^{2} + a^{2} b^{3}\right)} g}$ The second derivative seems unhelpful. Asymptote at $g=-a$, and $g=-b$  |
| Question about Real and Complex primes (Algebraic number theory) Posted: 31 Oct 2021 07:07 PM PDT I posted same question ago, but there was no answer. I'm reading Jurgen Neukirch, Algebraic number theory, p.183 
Now I can't understand the underlined statement. Why that $\mathfrak{p}$ is real or complex embedding depends whether the $K_{\mathfrak{p}}$ is isomorphic to $\mathbb{R}$ or to $\mathbb{C}$? (By the Ostrowski theorem, $K_{\mathfrak{p}}$ is isomorphic to $\mathbb{R}$ or $\mathbb{C}$)  |
| A text message plan costs $2$ per month plus $0.12$ per text. Find the monthly cost for $x$ text messages. Posted: 31 Oct 2021 07:17 PM PDT A text message plan costs $2$ per month, plus $0.12$ per text. Find the monthly cost for $x$ text messages.  |
| Prove that the number of prime factors of an integer n greater than 1 is at most log$_2$(n). Posted: 31 Oct 2021 07:47 PM PDT Prove that the number of prime factors of an integer n greater than 1 is at most log$_2$(n). For example 28 = 2 x 2 x 7 has 3 prime factors, and 3 < log$_2$28 = 4.80735.  |
| Signed Borel Measures and Functions of Bounded Variation Posted: 31 Oct 2021 07:51 PM PDT Let $\nu$ be a finite signed Borel measure on the closed interval $[a,b]$. We can define a function $F_\nu : [a,b]$ by $$F_\nu(x) = \nu([a,x]).$$It can be shown that $F_\nu$ has bounded variation and is right-continuous. An exercise in my class notes asks to prove that $|\nu|([a,b]) = V(F_\nu, [a,b])$, where the right-hand side denotes the total variation of $F_\nu$. Immediately after this, the notes state the following theorem: "The map $\nu \mapsto F_\nu$ is a bijection from the set of finite signed Borel measures on $[a,b]$ to the set of functions of bounded variation that take value $0$ at $a$ and that are right-continuous." My problem is that neither of these statements are true. Both break when we consider the measure $\delta$ given by $$\delta(A) = \begin{cases} 1, &\quad a \in A \\ 0, &\quad a \notin A \end{cases}.$$ In this case, $F_\delta$ is identically $1$, and so $V(F_\delta,[a,b]) = 0$ while $|\delta|([a,b]) = \delta([a,b]) = 1$. Moreover, $F_\delta$ does not take the value $0$ at $a$, so the second statement is false too. How do we fix this? Can we just limit ourselves to the meausures $\nu$ for which $\nu(\{a\}) = 0$? Or is there a change we can make so that the results still apply to all signed measures?  |
| $\beta \in \mathbb{Q}(\alpha)$ such that $\mathbb{Q}(\alpha) = \mathbb{Q}(\beta)$ and $\beta^3 \in \mathbb{Q}$ does not exist. Posted: 31 Oct 2021 07:09 PM PDT I'm stuck on proving that. Let $f \in \mathbb{Q}[X]$ is irreducible cubic polynomial which has three real roots, and $f(\alpha)=0$, then there does not exist $\beta \in \mathbb{Q}(\alpha)$ such that $\mathbb{Q}(\alpha) = \mathbb{Q}(\beta)$ and $\beta^3 \in \mathbb{Q}$. I'll write down what I noticed. - $1, \alpha, \alpha^2$ is base of $\mathbb{Q}(\alpha)$
- Let $g$ is minimal polynomial of $\beta$ on $\mathbb{Q}$, $\mathrm{deg}(g) = 3$.
- Let $x, y, z \in \mathbb{R}$ are roots of $f$ ($\alpha$ is one of them), then $xyz, x+y+z, xy+xz+yz \in \mathbb{Q}$
please some hint or solution. Thank you!  |
| How do I prove that this linear transformation is bijective? Posted: 31 Oct 2021 07:48 PM PDT Let $V$ be a finite-dimensional inner product space (either real or complex) and $V^*=\mathrm{Hom}(V,k)$ be the dual space. Let $v\in V$. Define a mapping $L_v$ from $V$ to the ground field $k$ by $L_v(u)=\langle u,v\rangle$. Note that $L_v$ is linear. Define $$\begin{align}L:V&\to V^*\\v&\mapsto L_v\end{align}$$ Now, $L$ is linear if $V$ is a real inner product space and anti-linear if $V$ is a complex inner product space, i.e., $L(av+bw)=aL(v)+bL(w)$ in the real case, and $L(av+bw)=\bar aL(v)+\bar bL(w)$ in the complex case. I would like to prove that $L$ is bijective. Suppose $L(v)=L(w)$. Then, for all $u\in V$, $$\begin{align}\langle u,v\rangle&=\langle u,w\rangle\\\langle u,v\rangle-\langle u,w\rangle&=0\\\langle u,v-w\rangle&=0\end{align}$$ The only vector that gives zero inner product with all vectors is the zero vector. Therefore, $v=w$, and $L$ is injective. It seems to me that surjectivity holds by construction. However, in the problem, there is a caution to "take care to account for the not-quite linearity of $L$ in the complex case". Am I missing something?  |
| Heun's method for non-homogeneous LTI Posted: 31 Oct 2021 07:26 PM PDT Heun's method for homogeneous system is simple enough: $$\bar{x}_{k+1} = x_k + hf(x_k)$$ $$x_{k+1} = x_k + \frac{h}{2}\left( f(x_k) + f(\bar{x}_{k+1}) \right)$$ I have a non-homogeneous LTI system: $$\dot{x} = Ax + Bu$$ where $A=\begin{bmatrix} 0 & 1 \\ -3730.2 & -26.15 \end{bmatrix}$, $B = \begin{bmatrix} 0 \\ 26.04 \end{bmatrix}$, $u = 0.5\sin{3t}+0.4$ With Forward Euler's method, I have the following discretized state space: $$A_d = I + hA$$ $$B_d = hB$$ With Heun's method, I have the following discretized state space: $$A_d = I + hA + \frac{h^2}{2}A^2$$ $$B_d = hB + \frac{h^2}{2}AB$$ It's a simple enough system that I can get an exact analytical solution. When I compare the three, I notice that Heun's always does worse than forward Euler's after the initial transient dies out. For reference, $h=0.005$ and the initial conditions are all zero. Is there a mistake with my derivation of Heun's for this LTI? 
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| Proof of the equivalence of validity of formula and its closure in a structure (Shoenfield's book) Posted: 31 Oct 2021 07:50 PM PDT Shoenfield's Mathematical Logic contains the following theorem and its corollary: Closure Theorem. If ${\bf A}'$ is the closure of ${\bf A}$, then $\vdash {\bf A}'$ iff $\vdash {\bf A}$. Corollary. If ${\bf A}'$ is the closure of ${\bf A}$, then ${\bf A}$ is valid in a structure ${\cal A}$ iff ${\bf A}'$ is valid in ${\cal A}$. The proof of the corollary is presented as follows: Suppose that ${\bf A}$ is valid in ${\cal A}$. If $T$ has ${\bf A}$ as its only nonlogical axiom, then ${\cal A}$ is a model of $T$. By the closure theorem, $\vdash_T {\bf A}'$; so ${\bf A}'$ is valid in ${\cal A}$ by the validity theorem. The converse is proved similarly. $\square$ I can see intuitively that this is true, since the definition of validity in a structure is essentially asserting that the universally quantified version of ${\bf A}$ is valid (i.e. every instance of ${\bf A}$ is valid). Now, I don't completely understand why the case when ${\bf A}$ is the only nonlogical axiom of $T$ is sufficient for the proof. Note that at this point in the book, the Completeness Theorem is not yet proved and thus we can only infer that if a formula is a theorem, then it is valid (called the Validity Theorem in the book), but not the converse. Say that you had a theory $T$ with multiple nonlogical axioms or where ${\bf A}$ is not a theorem: how does the proof cover this case? Is it because the structure is applied to the language, rather than the theory itself?  |
| How ti prove the sum of squares degrees $d^2_{1}+d^2_{2}+\cdots+d^2_{95}\le 300000$ in a graph Posted: 31 Oct 2021 07:21 PM PDT Let $G$ be a simple graph with $95$ vertices,$2021$ edges,and vertex degrees $d_{1},d_{2},\cdots,d_{95}$,show that $$d^2_{1}+d^2_{2}+\cdots+d^2_{95}\le 300000$$ My attempt: Using Euler's theorem $$d_{1}+d_{2}+\cdots+d_{95}=2\cdot 2021=4042$$,and note $d_{i}\in N^{+}$.By the nature of the inequality, as if by an adjustment, but then How to prove $$d^2_{1}+d^2_{2}+\cdots+d^2_{95}\le 300000$$  |
| Proving coefficient $a_n = O(1/n) $ and $b_n =O(1/n)$ in fourier series Posted: 31 Oct 2021 07:20 PM PDT This question was asked in my real analysis quiz and I was unable to solve it. So, I am asking for help here. Question : If $f(x) \sim a_0 /2 + \sum_{n=1}^{\infty} \left(a_n \cos(nx) + b_n \sin (nx)\right) $ and if f is of bounded variation on $[0,2\pi]$ show that $a_n =O(1/|n |) $ and $b_n =O(1/|n|)$. Using the condition of bounded variation i wrote $f =g-h$ where $g$ and $h$ are increasing on $[0,2 \pi]$. But I am not able to use any result of fourier series to proceed. Do you mind giving me some hints on how to prove it? Thanks for your time.  |
| When we have the power set $2^S$, does the 2 actually mean anything? Posted: 31 Oct 2021 07:44 PM PDT I have seen that most math books refer to the power set as $2^S$, usually in a cursory manner and without much detail. I was wondering if the 2 meant anything, because I normally just interpret it as a cardinality thing, like if S has two elements, then the power set has $2^2 = 4$ elements. Thanks!  |
