Recent Questions - Mathematics Stack Exchange |
- The trace as an integral over the projective space
- How to guess a value given a data
- If X is $\overline{\mathbb{F}^W}$ - adapted then X is $\sigma(W)$ - measureable
- Find the value about Fibonacci Algebraic
- Question about the decay of $f\in W^{1,2}(\mathbb{R}^2)$
- Can intersecting circles intersect in any combinations?
- Requirements for totally graph-reducing moves
- Are there such things as 3-dimensional (and higher) analogues of matrices, and if so, do they have any applications?
- Derivative and integral of distribution valued functions
- Probability for a random variable to be greater than its mean
- Properties of Totally geodesic submanifolds
- Properness and compactness for rational points
- How to see Horizontal vector field as Lie bracket of vertical and geodesic vector field
- Confidence level 90%
- Local Global principle and Modules
- How to solve this second-order nonlinear ODE
- A question on finite $2$-groups
- Finding polynomial without constant term that commutes with $f(x)=x^3+3x$
- Relationship between angle of vectors and orthogonalization
- Does it suffice to show closure only for infinitesimal transformations in a Lie group?
- How to find the value of a joint density transform?
- Infinite-dimensional algebraic objects
- A net converges to a point iff every subnet accumulates in that point.
- What does the strong approximation theorem means for rational numbers?
- Why are the discriminants for a given polynomial & its resolvent polynomial congruent?
- How can you find the continuous digits of $g(x)=3 \lfloor x \rfloor^3$?
- Number of solutions to x^2 = a mod n with gcd(a,n) =1 is constant [closed]
- Existence and Uniqueness of Nonlinear ODE [closed]
- Intermediate Quadratic Equations
- The difference between log and ln
| The trace as an integral over the projective space Posted: 21 Dec 2021 01:37 AM PST Let $(E,h)$ be a Hermitian vector space of dimension $n$ and $u\in End(E)$. We have an expression of the trace of $u$ as the integral $$Tr(u)=\frac{n}{A}\int_{S}\langle v,uv\rangle d\mu$$ where $S$ is the unit sphere in $E$ and $A$ its volueme (see Integral around unit sphere of inner product). Is there a similar expression but using the projective space instead of the sphere? That is some equality that looks like $$Tr(u)=C\int_{\mathbb{P}E}\frac{\langle v,uv\rangle}{||v||^2} d\mu_{FS}$$ for some constant $C$ and $d\mu_{FS}$ being the Fubini-Study volume form. |
| How to guess a value given a data Posted: 21 Dec 2021 01:36 AM PST I got the following data: Its a lot larger than this. This is just a small sample. When visualized, this is what it looks like:
So there is a consistent correlation between rpm and vibration. My question is this: If a value is missing in |
| If X is $\overline{\mathbb{F}^W}$ - adapted then X is $\sigma(W)$ - measureable Posted: 21 Dec 2021 01:35 AM PST Let $(\Omega,\mathcal{F},\mathbb{F}=(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space, $W$ an $\mathbb{F}$-Brownian motion and $X$ a continuous $\overline{\mathbb{F}^W}$ - adapted process with $\overline{\mathbb{F}^W}$ being the completed filtration generated by $W$. We equip $C([0,\infty),\mathbb{R})$ with the sigma algebra generated by the projections. In the script my professor claims that $X:\Omega \rightarrow C([0,\infty),\mathbb{R})$ is $\sigma(W)$ - measureable, but I don't see why this should be true since $X$ is only $\overline{\mathbb{F}^W}$ - and not $\mathbb{F}^W$ - adapted. I would be grateful for any help. |
| Find the value about Fibonacci Algebraic Posted: 21 Dec 2021 01:36 AM PST
I can't solve this problem. Maybe no solution into this problem? I hope someone can assist me. Thank you! |
| Question about the decay of $f\in W^{1,2}(\mathbb{R}^2)$ Posted: 21 Dec 2021 01:25 AM PST I have an exercise which asks to prove or provide a counterexample to the following claim: Given $f\in W^{1,2}(\mathbb{R}^2)$, there exists a radius $R>0$ such that $\vert f(x)\vert \leq 1$ for every $x\in \mathbb{R}^2\setminus B_R(0)$, where $B_R(0)$ denotes the ball of radius $R$ and centered at the origin. We have seen that $W^{1,2}(\mathbb{R}^2)=W^{1,2}_0(\mathbb{R}^2)$, where the latter is the closure (w.r.t the $W^{1,2}(\mathbb{R}^2)$ norm) of smooth and compactly supported functions on $\mathbb{R}^2$, so intuition would suggest that functions in $W^{1,2}(\mathbb{R}^2)$ "decay at infinity", but I don't know how to formally prove that $L^p$-convergence of a sequence of compactly supported smooth function implies that the limit function is zero at infinity. Or if the claim is false, I can't find a counterexample. Thanks in advance for your help. |
| Can intersecting circles intersect in any combinations? Posted: 21 Dec 2021 01:24 AM PST For every undirected graph G, with nodes $a_1,...,a_n$. Can we find n circles(of fixed radius 1), $c_1,...,c_n$ on the plane such that there exists an edge between ai and aj if and only if ci and cj intersect. Does it matter if we allow 1 point of intersection. This is a relaxation of generating all venn diagrams as I don't care about intersecting regions. |
| Requirements for totally graph-reducing moves Posted: 21 Dec 2021 01:23 AM PST This can be best explained with Kuperbergs $G_2$ paper. Ingredients:
Strangely, although skein equations are a very powerful concept of knot theory, I rarely saw them applied to graphs. For starters, it is straightforward to change "trivalent" to "tetravalent", $1-5$ to $1-3$ and get $B_2$ instead of $G_2$. (I did it.) Perhaps this was deemed unnecessary as a (much more elaborate) construction for a $B_2$ invariant was already given in the $G_2$ paper. And who has the complete overview over math literature of even a small field anyway... Thus:
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| Posted: 21 Dec 2021 01:37 AM PST A matrix is a group of numbers arranged in a rectangle. I wonder, has anyone studied 3-dimensional and higher analogues of matrices? For example, there could be such a thing as a 2 by 2 by 2 3d matrix, whose entries are all equal to 1. Has anyone else defined these entities, and more importantly, are they used in mathematics? |
| Derivative and integral of distribution valued functions Posted: 21 Dec 2021 01:43 AM PST Given a $C^1$ distribution valued function $f: \mathbb{R}^l \to \mathcal{D}'(\mathbb{R}^n)$ how does one defined its, say, j-th partial derivative? How does one define its integral over a compact set $K \subset \mathbb{R}^l$? My best guesses are,
Does anyone know if these are correct and/or could point to a reference giving these definitions ( the books that I have just say what it means for a distribution valued function to be continuous/differentiable, but not what the integral/ the derivatives is/are)? Many thanks! |
| Probability for a random variable to be greater than its mean Posted: 21 Dec 2021 01:20 AM PST I'm looking for a general lower bound on $\mathbb{P}(X \geq \mathbb{E}(X))$ (or, equivalently, on $\mathbb{P}(X \leq \mathbb{E}(X))$). My informal (and naive) reasonning so farI can think of 2 ways to make this probability arbitrarily small:
These examples suggest that the desired lower bound should depend positively on the variance of $X$ (example 1.) and negatively on the ``diameter'' of $X$ (example 2.), that is, $$ \text{diam}(X) = \text{sup}~ ( \text{supp}(X)) - \text{inf}~ ( \text{supp}(X)).$$ (To put it simply, assume that $X \in [a,b]$ almost surely and make the lower bound a function of $b-a$.) Are you aware of such result? If not, do you think that my hopes are justified, or did I miss a counter-example? |
| Properties of Totally geodesic submanifolds Posted: 21 Dec 2021 01:41 AM PST Suppose I have a totally geodesic manifold $M$ in a riemannian manifold $(N,g)$. Let $p\in M$ and consider the set $\exp_{p}^{-1}(M)$ where this exponential map is defined in $N$. Will we have that $\exp_{p}^{-1}(M)$ is a linear subspace of $T_pN$? That is if we take $y,z\in \exp_{p}^{-1}(M)$ then $\exp_p(y+z)\in M$? If not, what if we impose more conditions,i.e. that we have a riemannian submersion $\pi:N\rightarrow M$, can we get a better result here? This question came up cause I am trying to model a Banach manifold of curves using the exponential map and they need to satisfy a boundary condition with respect to a submanifold. Edit: What about if we restrict to a neighborhood where $\exp_p$ is a diffeomorphism. Say there exists an $r>0$ such that $\exp_p|_{B_{r}(0)}$ is a diffeomorphism. Therefore, since a geodesic in $M$ lifts to geodesic in $N$ we obtain that $\exp_p^{-1}(M)\subset T_pM$. Then we have $y+z\in T_pM$ and so $\exp(y+z)\in M$ due to the uniquess of geodesics and the fact that a geodesic in $M$ lifts to a geodesic in $N$. Any help and comments is appreciated, thanks in advance. |
| Properness and compactness for rational points Posted: 21 Dec 2021 01:16 AM PST I am reading Rational points on varieties by Poonen and I'm not sure about this proposition.
The author gives Serre's GAGA as a reference. Is it possible to prove it in a more "elementary" way? |
| How to see Horizontal vector field as Lie bracket of vertical and geodesic vector field Posted: 21 Dec 2021 01:07 AM PST I have seen two different definitions of Horizontal vector field. I am intersted to see how both these notions are same. Let $M$ be a smooth and compact Riemannian surface $(\operatorname{dim}(M)=2)$ with boundary and $S M$ its unit sphere bundle. Assume that every maximal geodesic has finite length. The unit sphere bundle $S M$ of $M$ is $$ S M=\left\{(x, v) ; x \in M, v \in T_{x} M,|v|=1\right\} $$ The horizontal vector field $X_{\perp}$ differentiates with respect to $x$ in a direction orthogonal to $v$ i.e. $$ X_{\perp} u(x, v)=\left.\frac{d}{d s}\left(u\left(\psi_{s}(x, v)\right)\right)\right|_{s=0} $$ where $\psi_{s}(x, v)=\left(\gamma_{x, v_{\perp}}(s), W(s)\right)$ and $W(s)$ is the parallel transport of $v$ along the curve $\gamma_{x, v_{\perp}}(s)$. Also sometimes it also defined as $X_{\perp}=[X, V]$ where $X$ and $V$ defined as follows. Let $X$ denote the derivative along the geodesic flow. That is, $$ X u(x, v)=\left.\frac{\mathrm{d}}{\mathrm{d} t} u\left(\phi_{t}(x, v)\right)\right|_{t=0} $$ for a function $u$ on $S M$. This $X$ is a vector field on $S M$. In two dimensions we can write the direction $v$ in local coordinates as $v_{\theta}=(\cos \theta, \sin \theta)$. We define the derivative $V$ as $$ V u\left(x, v_{\theta}\right)=\frac{\mathrm{d}}{\mathrm{d} \theta} u\left(x, v_{\theta}\right) . $$
Any help or hint or reference will be highly appreciated. Thank you so much for giving your valuable time. |
| Posted: 21 Dec 2021 01:05 AM PST Suppose that you are playing Humble-Nishiyama with a full deck of cards and sequences of length 3. Player A chooses the sequence RBR. How can I find all the winning strategies for player B with 90% confidence ? |
| Local Global principle and Modules Posted: 21 Dec 2021 01:13 AM PST Let $V$ be an A-module over a commutative ring $A$. (a) Let $x,y \in V$. Then $x=y$ $\Leftrightarrow \frac {x}{1}=\frac{y}{1}$ in $V_M$ for all $M\in Spm A$. If I have $x=y$ then for each $m\in M$ I will have $m(x-y)=0$. So, I am done. Conversly, if $\frac {x}{1}=\frac{y}{1} $ in $V_M$ then I have $m\in M $ such that $m (x-y)=0 $ for all $M\in Spm A$. But I am not sure how exactly I should proceed with the proof. (b) Prove that following statements are equivalent. (i) $V=0$ (ii) $V_P = 0 $ for all $P\in Spec A$ (iii) $V_M =0$ for all $M\in Spec A$. (iii) $\Rightarrow$(ii) holds as every maximal ideal is a prime ideal. (i) $\Rightarrow$(iii) is clear. (ii) $\Rightarrow$ (i) I am not getting any intuition on how exactly should I approach this part and would appreciate hints. Kindly help. |
| How to solve this second-order nonlinear ODE Posted: 21 Dec 2021 01:41 AM PST I'm trying to solve: $$ \ddot{x} + \frac{1}{2} \dot{x}^2 - n e^{-2x} ~ e^{c t} - 2 e^{-2t} =0, $$ where c and n are constants. I'm trying to find x as a function of t. My trial I started by letting: $x= log ~r \to e^{-2x} =r^{-2}$, then, $ x' = \frac{r'}{r}, ~~~ x'' = \frac{r''}{r}- \frac{r'^2}{r^2}$, substitute, we get: $$ \frac{\ddot{r}}{r} - \frac{1}{2} \frac{\dot{r}^2}{r^2} - n r^{-2} e^{ct} -2 e^{-2t} =0 $$ Any help on how to complete from here or from the original equation? |
| A question on finite $2$-groups Posted: 21 Dec 2021 01:31 AM PST Is there a classification of all finite $2$-groups $G$ such that if $m>0$ is the number of elements of any order greater than $2$, then $2m-1$ is the number of involutions in $G$? Also, the number of involutions in such $G$ is at most $\dfrac{\lvert G \rvert}{2}-1$. I know all $2$-groups of sizes $64, 128, 256$ and $512$ satisfying the properties above. But I am only checking if this is well-known in the literature. |
| Finding polynomial without constant term that commutes with $f(x)=x^3+3x$ Posted: 21 Dec 2021 01:06 AM PST Consider the polynomial $f(x)=x^3+3x$ over $\mathbb{Z}$. I am trying to find a polynomial $g(x)$ $(\neq f^{\circ n})$ of any degree (or series) without constant term which commutes with $f$ (or any iteration $f^{\circ n}, ~n \geq 1)$ under composition. Trivially, any $g(x)=x$, is another polynomial (or series) commutes with $f(x)$. By hand it seems to be laborious. Suppose I start with an investigation if there are degree $2$ polynomial $g(x)=ax+bx^2$ such that $f \circ g=g\circ f$. Then \begin{align} &f(g(x))=f(ax+bx^2)=3(ax+bx^2)+(ax+bx^2)^3=3ax+3bx^2+a^3x^3+3a^2bx^4+3ab^2x^5+b^3x^6, \\ &g(f(x))=g(x^3+3x)=a(x^3+3x)+b(x^3+3x)^2=3ax+ax^3+bx^6+6bx^4+9bx^2 \end{align} Comparing both equations, we get $b=0$ and $a=a^3 \Rightarrow a=\pm 1$. In this case $g(x)=\pm 1$, the trivial one. Suppose I start with an investigation if there are degree $3$ polynomial $g(x)=ax+bx^2+cx^3$ such that $f \circ g=g\circ f$. Then \begin{align} &f(g(x))=f(ax+bx^2+cx^3)=3(ax+bx^2+cx^3)+(ax+bx^2+cx^3)^3=3ax+3bx^2+3cx^3+c^3x^9+3bc^2x^8 \hspace{3cm}+(3ac^2+3b^2c)x^7+(6abc + b^3)x^6 + (3a^2c + 3ab^2)x^5 + 3a^2bx^4 + a^3x^3, \\ &g(f(x))=g(x^3+3x)=a(x^3+3x)+b(x^3+3x)^2+c(x^3+3x)^3=3ax+ax^3+2bx^6+6bx^4+9bx^2+cx^9+9cx^7+27cx^5+27cx^3 \end{align} Comparing both sides we get $b=0$ and the following equations: \begin{align} a^3-a=24c, \\ a^2c=9c, \\ 3ac^2=9c,\\ c^3=c. \end{align} Solving these, we see $c^3=c$ and $3ac^2=a^2c$. These two gives us $c=0$ or $c=\pm 1$. If $c \neq 0$, then $a=\pm 3$. Thus $g(x)=\pm (3x+ x^3)$, which is equivalent to $f(x)$ upto signs. Suppose I start with an investigation if there are degree $4$ polynomial $g(x)=ax+bx^2+cx^3+dx^4$ such that $f \circ g=g\circ f$. Then it becomes laborious. Is there any way to find non-trivial $g$ with the help of PARI/GP or SAGE ? Edit 1: According to the hints given by @achille hui, I have found that $g(x)=-5x-5x^3-x^5$ commutes with $x^3+3x$. However, I am looking for an polynomial whose first degree coefficient is $3$ or multiple of $3$. I would appreciate one such example. Edit 2: I have found one example I was looking for as my previous comment. Indeed $-6x-5x^3-x^5$ commutes with $x^3+3x$ |
| Relationship between angle of vectors and orthogonalization Posted: 21 Dec 2021 01:11 AM PST I'm trying to understand the following question: Let $u_1,...,u_d$ be a set of orthonormal vectors in $\mathbb{R}^n$. Let $a,b$ be unit vectors in $\mathbb{R}^n$ such that $\angle(a,b) < \epsilon$. If we orthogonalize $a$ and $b$ with respect to the $u_i's$ (with some Holder Transformation / Gram-Schmidt for both vectors) to $v_a,v_b$, is there a bound to $\angle(v_a,v_b)$ related to $\epsilon$? It seems like the angles should basically be the same, but I can't prove it nor can I find a good reference about what this bound should be, even if not $\epsilon$. EDIT: To give some context: The main question I'm looking to show that there are orthonormal bases $<u_1,...,u_d,v_a>$ and $<u_1,...,u_d,v_b>$ for the spaces spanned by $<u_1,...,u_d,a>, <u_1,...,u_d,b>$ respectively, such that $\angle(v_a,v_b) < \epsilon$ or some function of $\epsilon$, given that $u_1,...,u_d$ are orthonormal and $a,b$ are unit vectors with $\angle(a,b)<\epsilon$. I see why Gram-Schmidt may not work given Eric's example in the comments, but is there an answer to this question? ADDITIONAL EDIT: We assume the the subspaces mentioned in the above edit are not the same. Any help would be great! |
| Does it suffice to show closure only for infinitesimal transformations in a Lie group? Posted: 21 Dec 2021 01:37 AM PST In order to prove something forms a group, you need to (among other things) show that it is closed under group multiplication. However, for a connected Lie group, is it sufficient to show that it is closed only for infinitesimal transformations? My logic behind this is as follows. If you have shown that the product of two infinitesimal transformations is also in the group, and any element (connected to the identity) can be written as an exponential which is simply the product of infinitely many infinitesimal transformations, then the exponential must also be in the group. The only thing making me doubt this is whether the fact that the exponential is a product of infinite infinitesimals causes problems. For example, say you have a subset of a connected Lie group and you need to show it is a subgroup w.r.t. the restriction of the group operation. The identity and inverse are just the identity matrix and inverse matrix so are already sorted. Thus it only remains to consider closure. An infinitesimal transformation is simply a matrix given by the identity plus an infinitesimal parameter times a generator. My question is, if you show that the product of two of these infinitesimal transformations is also in the subgroup, have you proved it for all finite transformations? |
| How to find the value of a joint density transform? Posted: 21 Dec 2021 01:07 AM PST We are given $X=(X_1,X_2)$ and the joint density function: \begin{align}f_{X_1,X_2}(x_1,x_2) =\begin{cases}4x_1x_2,&\quad0<x_1<1,0<x_2<1\\0&\quad\text{otherwise}.\end{cases}\end{align} If $Y_1 = X_1/X_2$ and $Y_2=X_1X_2$, find the value of $f_{Y_1,Y_2}(0.8,0.5)$. ——— I attempted to solve this question by first computing the inverse of $(Y_1,Y_2)$ which I found to be $$\left(\sqrt{Y_1Y_2}, \sqrt{Y_2/Y_1}\right).$$ I then computed its Jacobian to be $(Y_1+1)/(4Y_1^2)$. I then multiplied the Jacobian by $f_{X_1,X_2}((Y_1,Y_2)^{-1})$ using the inverse function as the parameters for $f_{X_1,X_2}$. I then finally substituted in the values $Y_1=0.8$ and $Y_2=0.5$. This got me an incorrect answer though, and was wondering what I did wrong here. |
| Infinite-dimensional algebraic objects Posted: 21 Dec 2021 01:37 AM PST Let $H$ be an infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its projectivization (the set of all $1$-dimensional complex linear subspace) endowed with quotient topology (thus a smooth Hilbert manifold). Question: In finite-dimensional case, the zero locus of $p_A$ is smooth if and only if $A$ is generic. This definition of smoothness can be extended to the infinite-dimensional case. Does Lefschetz hyperplane theorem still holds for these infinite dimensional 'smooth' objects? |
| A net converges to a point iff every subnet accumulates in that point. Posted: 21 Dec 2021 01:26 AM PST While working on a takehome for my functional analysis course I stumbled upon this small lemma A net $(x_i)_{i\in I}$ in a topological space $X$ converges to a point $x\in X$ if and only if every subnet has a accumulation point in $x$. This is a slightly stronger formulation of the following well known result in topology. A net $(x_i)_{i\in I}$ in a topological space $X$ converges to a point $x\in X$ if and only if every subnet converges to $x$. I managed to come up with the following proof, but I doubt my judgement because it seems a little unbelievable for me to come up with a stronger version of an existing mathematical result. Can you check my proof? the implication from left to right is trivial because if $(x_i)_{i\in I}$ converges to $x$ then so will any subnet. Convergence to $x$ implies that the subnet has a accumulation point in $x$ as well, because this is a weaker statement. Now if $(x_i)_{i \in I}$ does not converge to $x$, it has a subnet which does not converge to $x$, $(x_{\sigma(j)})_{j\in J}$. This means there is an open neighbourhood $U$ of $x$ such that for any $j_0 \in J$ there exists a $j \geq j_0$ such that $x_{\sigma(j)} \not\in U$. In other words, we found a subnet without an accumulation point in $x$. |
| What does the strong approximation theorem means for rational numbers? Posted: 21 Dec 2021 01:05 AM PST In the field of rational numbers, the only valuations are the p-adic valuations. Here is an interpretation of the weak approximation theorem for the field of rational numbers: Let me say that a (reduced) rational number $q$ is a multiple of $n$ if its numerator is, and its denominator is coprime to n. I note $q \equiv a\ \mod n$ if $q-a$ is multiple of $n$ (one can check that the basic principles of the congruence theory work with this setting, modulo some adaptations). Now, the weak approximation theorem is equivalent to the following statement: If $n_1$, $n_2$, ... $n_k$ are coprime natural numbers, and if $q_1$, ... $q_k$ are $k$ rational numbers, then there exists a rational $q$ such that $q \equiv q_i\ \mod n_i$ for every $i$. It's a "field" version of the Chinese remainder theorem. But I have some trouble in understanding the meaning of the strong approximation theorem in this setting. Some insight? Do you know a simple interpretation of the strong approximation theorem for $\mathbb Q$? |
| Why are the discriminants for a given polynomial & its resolvent polynomial congruent? Posted: 21 Dec 2021 01:24 AM PST I have been looking into the formulae for the cubic & quartic polynomials, using the resolvent method to come up a corresponding resolvent polynomial of a lesser degree. It seems that it for the case of the given cubic & quartic, the discriminant for the given polynomial is equal to some scale (i.e., my definition here of "congruent") of the discriminant for the corresponding resolvent polynomial (there is no such situation for the quadratic as the resolvent polynomial is linear). And of course, since there is no solution to the quintic polynomial, there is no resolvent. An example of this is the cubic having the solutions { 0 , +3 , -3 }, thus with the discriminant of 2916. The resolvents are a quadratic solution { +/- i [27^(1/2)] }, and thus with the discriminant of -108, which is -(1/27) of 2916. This -(1/27) scale factor exists for all cubic polynomials, and indeed is the reason why for the case of 3 real solutions, an "intermediate" imaginary solution needs to be calculated (i.e., the resolvent is this "intermediate" solution). Similarly for a quartic, there is a cubic solution for the resolvents, but in this case, the discriminants are the same. In general, it could be said that the discriminants differ only be a scale factor (which could be the degenerate scale factor of 1). Since there is no solution to the quintic, there is no such thing as a resolvent, and so the consideration of such is moot. I was just wondering if there is some deeper mathematical reason why these discriminants are scale factors of each other, other than simply saying, "it is this way for the only 2 cases in which it exists". |
| How can you find the continuous digits of $g(x)=3 \lfloor x \rfloor^3$? Posted: 21 Dec 2021 01:16 AM PST I have a task that I'm just not getting anywhere with: Let $ g: \mathbb{R} \rightarrow \mathbb{R} $ given by $g(x)=3 \lfloor x \rfloor^3$ for all $ x \in \mathbb{R} $.
From the picture you can see that the function is not continuous on the whole of R, but only on these single horizontal areas. If my intuition is correct, this function should be continuous at all points $a \in \mathbb{R} \setminus \mathbb{Z}$ and discontinuous at points $a \in \mathbb{Z}$. But how can I now show these two cases? I found a proof of the normal floor function in an old script: Is it possible to do the proof like this for my given function? (f) The function $ \mathbb{R} \rightarrow \mathbb{R}, x \mapsto\lfloor x\rfloor $ is continuous at all points $ a \in \mathbb{R} \backslash \mathbb{Z} $ and discontinuous (i.e., not continuous) at the points $ a \in \mathbb{Z} $. Proof. For let $ a \in \mathbb{Z} $. Then for $ x_{n}=a+\frac{1}{2 n} $ that $ x_{n} \rightarrow a $ and $ \left\lfloor x_{n}\right\rfloor=a $, also $ \lim \limits_{n \rightarrow \infty}\left\lfloor x_{n}\right\rfloor=a $. However, for $ x_{n}=a-\frac{1}{2 n} $ that $ x_{n} \rightarrow a $ and $ \left\lfloor x_{n}\right\rfloor=a-1 $, also $ \lim \limits_{n \rightarrow \infty}\left\lfloor x_{n}\right\rfloor=a-1 $. The limit also does not exist for $ a \in \mathbb{Z} $. Let $ a \in \mathbb{R} \backslash \mathbb{Z} $. Then $ \lfloor a\rfloor<a<\lfloor a+1\rfloor $ holds. Moreover, $ \lfloor x\rfloor=\lfloor a\rfloor $ holds for all $ x \in(\lfloor a\rfloor,\lfloor a+1\rfloor) $, which is an open interval containing $ a $. If $ x_{n} \rightarrow a $ also holds for $ n \rightarrow \infty $, then $ x_{n} $ is finally contained in $ (\lfloor a\rfloor, \lfloor a+1\rfloor) $. Thus $ \lim \limits_{n \rightarrow \infty}\left\lfloor x_{n}\right\rfloor=\lfloor a\rfloor $ follows. This shows continuity in $ a $. |
| Number of solutions to x^2 = a mod n with gcd(a,n) =1 is constant [closed] Posted: 21 Dec 2021 01:09 AM PST I have come across this question from the book, Elementary Number theory By David Burton, during independent self study. Any help will be appreciated! Q) For a fixed n>1, show that all solvable congruences x^2 = a (mod n) with. gcd(a,n)=1 have the same number of solutions. I have tried using Lagrange's Theorem to establish that the number of solutions, if x is not equal 0, are that there exists a minimum of 2 incongruent solutions for x^2=a (mod n), namely if x_0 is a solution then n - x_0 is also a solution. Now, how do I show that the number of solutions for such solvable congruences is the same for all such congruences. For example: x^2 = 1 (mod 8) has 4 incongruent solutions, namely, x= 1,3,5,7. Therefore, now, How do I determine that the total number of incongruent solutions is constant for any such congruence relationship where (a,n) = 1, that is 'a' and 'n' are co-prime. |
| Existence and Uniqueness of Nonlinear ODE [closed] Posted: 21 Dec 2021 01:15 AM PST What are the conditions on the non-linear functions $f$ and $\gamma$, for the system $$\ddot{x}+\gamma(t)\dot{x}+f(x) = 0, $$ to have global solutions for all $t\geq 0$? |
| Intermediate Quadratic Equations Posted: 21 Dec 2021 01:19 AM PST If $n$ is a constant and if there exists a unique value of $m$ for which the quadratic equation $x^2 + mx + (m+n) = 0$ has one real solution, then find $n$. Let the roots of the quadratic be $r,s.$ Vieta gives $-m=r+s, m+n=rs.$ Thus, $n=rs+r+s \implies n+1=(n+1)(m+1).$ From here, I don't know what to do. I have a feeling that the factoring trick was unnecessary. Any quick (and slick) solutions? Thanks. |
| The difference between log and ln Posted: 21 Dec 2021 01:13 AM PST $$\dfrac{1}{2}\ln(x+7)-(2 \ln x+3 \ln y)$$ Our professor let's us solve this but i do not understand how $\ln$ works. He says it has same properties with $\log$ but i still don't get it. What's the difference of the two? |
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