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![]() We would like to show you a description here but the site won’t allow us. ![]() ![]() ![]() There's A Huge Ridiculous Twitter Fight Happening Right Now About A Shitty Tire. I wasn’t aware of who Laura Loomer was until we were emailed links to this Twitter- based tire- related mess by a number of readers, but it appears she considers herself an alt- right ‘guerrilla journalist’ and believes she had a tire slashed by some ne’er- do- wells who don’t like her. The problem is that her tire appears to be a poor, dry- rotted beast that died of neglect. Hencetheshitstorm. Now, without getting too far into the political minefield that surrounds all this, let’s just try and take an objective look at this tire to try and determine if it was, in fact, slashed. Here’s the tweets: And, here’s just the pictures of the real victim here, this innocent tire: Before we get into whether or not this tire was stabbed with malice, let’s just be clear about the condition of this tire: it’s terrible. This tire is an old, dry- rotted mess. Even if we were to assume that this tire had, in fact, been slashed with a blade, that would be closer to an act of euthanasia than vandalism. Look at the sidewall of that poor tire; it’s textured like gila monster skin, dry, brittle, and coming off in chunks. What this damage looks most like is a blowout related to the severely rotted and abraded sidewalls. Writer Linda Stone calls it email apnea, and while she recommends the scientifically dubious Buteyko breathing method to fight it, you don’t need to retrain your. The sky is blue because the atmosphere scatters the shorter wavelength (blue) end of the visible light spectrum more than the red end. As a result, more of this. Find the latest sports news and articles on the NFL, MLB, NBA, NHL, NCAA college football, NCAA college basketball and more at ABC News. Watch breaking news videos, viral videos and original video clips on CNN.com. ![]() ![]() A tire with a dry- rot- related blowout looks like this, and you can see a very similar pattern of damage as on the tire currently enjoying its time in the twitter spotlight. We should look at what tire- slash damage actually looks like, too. See how you get a clear and isolated cut? Now let’s look at the damage to Loomer’s tire: I’ve put some callouts in there to address some questions specifically. First, callout A, those are very strange and improbable locations for a person to slash a tire. I mean, even doing multiple slashes like that is unusual for tire slashing, since one does the job, why waste time adding so many? If I allow myself to enter the depraved mind of a brutal tire- slasher, I think my slashing time is better served by giving one slash to all four tires. Also, the location is problematic; most slashed tires find the damage in places where a hand wielding a sharp whatever could actually hit, usually on the upper sidewall of the tire. The damage to the tire at the bottom would require the slasher to be laying on the ground, digging their tire- ruining implement even under the tire at some points. That doesn’t seem likely. Also, the cuts seem to point radially from the tire’s center, like what happens in a blowout as opposed to someone hacking away at the tire. Okay, on to B; notice that big chunk of missing sidewall? Perhaps a dedicated tire- slasher dug out a chunk, but if bits of sidewall are coming off like that, the tire is pretty compromised. Which leads us to callout C, which shows the shitty condition of the tire, and a number of other cracks already forming. Loomer and her defenders have provided tweets with images of her tire repair shop receipts as ‘evidence’: While I sure like the illustration of what appears to be a Dodge Shadow/Plymouth Sundance on that yellow receipt, these are just handwritten (with a strikingly similar hand) work orders. Still, here they are, for what they’re worth. It should also be mentioned that Loomer has no patience for those of you who will not defend her tire. Now, I wasn’t there, standing by her Jeep, watching whatever nightmare ended the long, difficult life of her loyal tire. I reached out to a major tire manufacturer about this, and, while they in no way wanted to be associated with any of this mess, agreed to give me an assessment on the condition of anonymity. So, okay. They said the tire appears to have been exposed to extreme weathering, or possibly exposure to an electrical motor that produced a lot of ozone. The cause of failure they said seems to be a “rim pinch” and then stated that “the cause of disablement does not indicate a cut in any way.”I’m not surprised they had no wish to wade into this contentious mess, and it’s probable I’ll be accused myself of pushing some agenda here, and I may very well be painted as a monster who’s views are just to the left of an anthropomorphised hammer and sickle that wants nothing more than to watch your mother die from exhaustion on a collective farm. The amount of attention this is getting is pretty surprising, and a lot of that attention has been mixed with a lot of unpleasant shit, from sites that revel in anti- semitism and similar garbage. All of this makes me feel a little conflicted about writing about this at all—it’s just a tire—but, that’s also sort of why we’re covering it, too. Even so, there’s a real lesson about tire maintenance here that we just can’t ignore; letting your tires get as bad as these is an actual safety issue, and bad tires absolutely cause accidents. Getting rid of that tire (and hopefully, the other three) is probably the best thing she can do for her car right now. I’m sure Loomer genuinely believes her tires were slashed. From what I can see here, though, the only way some vandal could have caused her tire damage is if the tire was stolen, taken into a time machine, gone back a few years to age the tire severely, then a cheese grater was taken to the sidewalls as the tire was frozen and baked, over and over, before returning to 2. That I’d believe. Cars - How To Information. ![]() ![]() TM Fix Your Ford Web Site Ford Explorer (2000.1998-2000 typically, but also other years before and after).Tech news and expert opinion from The Telegraph's technology team. Read articles and watch video on the tech giants and innovative startups. 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To construct a space of functions on which the action of the symmetry group can be globalized we consider the family of solvable Lie groups$$\begin{aligned} G_{\alpha , \beta }= \left\{ g_{q,t,x}= \begin{pmatrix} q^\beta & {}\quad 0 & {}\quad t \\ 0 & {}\quad q^\alpha & {}\quad x \\ 0 & {}\quad 0 & {}\quad 1 \end{pmatrix} \,: \, t,x\in \mathbb {R},\, q \in \mathbb {R}^{> 0} \right\} ,\qquad {\alpha , \beta }\in \mathbb {R}, \end{aligned}$$and the subgroup$$\begin{aligned} P_{\alpha , \beta }=\bigl \{g_{q,0,0}: q \in \mathbb {R}^{> 0} \bigr \},\qquad {\alpha , \beta }\in \mathbb {R}. On this subgroup, we define the family of characters parametrized by \(\gamma \in \mathbb {R}\), \(\chi _\gamma : P_{\alpha , \beta }\rightarrow \mathbb {C}^\times \) by$$\begin{aligned} \chi _\gamma (g_{q,0,0})=q^{- \gamma }\! Then, the induced representation of \(G_{\alpha , \beta }\) by this character on \(P_{\alpha , \beta }\) is given by$$\begin{aligned} I_{\alpha , \beta ,\gamma }& =\text {Ind}_{P}^{G_{\alpha ,\beta }}(\chi _\gamma )\\& =\bigl \{\varphi \in C^\infty (G_{\alpha ,\beta }) \,: \, \varphi (gp)=\chi (p)^{- 1} \varphi (g),\, g\in G_{\alpha ,\beta },\, p\in P_{\alpha ,\beta }\bigr \}, \end{aligned}$$where the action of \(G_{\alpha , \beta }\) on this space is given by left translation. The subgroup$$\begin{aligned} N_{\alpha , \beta }=\{g_{1,t,x} : t,x\in \mathbb {R}\} \end{aligned}$$is canonically isomorphic to \(\mathbb {R}^2\). Using this isomorphism, we can define a representation of \(G_{\alpha , \beta }\) isomorphic to \(I_{\alpha , \beta ,\gamma }\) by restricting the domain of the maps in \(I_{\alpha , \beta ,\gamma }\) to \(N_{\alpha , \beta }\). More specifically, we define the space$$\begin{aligned} I'_{\alpha , \beta ,\gamma }=\bigl \{ f\in C^\infty \bigl (\mathbb {R}^2\bigr ) : f(t,x)=\varphi (g_{1,t,x}),\, (t,x)\in {\mathbb {R}}^2, \text { for some } \varphi \in I_{\alpha , \beta ,\gamma } \bigr \}. To see that this restriction map defines an isomorphism, we notice the easily checked fact that$$\begin{aligned} g_{q,t,x}=g_{1,t,x}{\cdot } g_{q,0,0}. Thus,$$\begin{aligned} \varphi (g_{q,t,x})=\varphi \bigl (g_{1,t,x}{\cdot } g_{q,0,0}\bigr )=q^{\gamma } \varphi (g_{0,t,x})=q^{\gamma }f(t,x). Hence, the value of \(\varphi \) at \(g_{q,t,x}\) is completely determined by the value of \(f(t,x)\). Thus, the restriction map is surjective and it is clearly injective. To see that the isomorphism intertwines with the action of \(G_{\alpha , \beta }\) we define the action of \(G_{\alpha , \beta }\) on \(I_{\alpha , \beta ,\gamma }'\) by$$\begin{aligned} g_{q_1,t_1,x_1}{\varvec{.}}f(t,x)=q_1^{\gamma } f\bigl ((t- t_1)q_1^{- \beta }\!, (x- x_1)q_1^{- \alpha }\bigr ) \end{aligned}$$(3)and it is routine to check that the vector space isomorphism is indeed intertwining. At this point, we compare the action of the Lie algebra \(\mathfrak {g}_{\alpha , \beta }\) of \(G_{\alpha , \beta }\) with the actions of the symmetries of (1). The Lie algebra \(\mathfrak {g}_{\alpha , \beta }\) is spanned by the matrices \(\{\xi _1=e_{1,4}, \xi _2=e_{2,4} , \xi _3= e_{2,2}+e_{3,4}\}\), where \(e_{i,j}\) is the \(3{\times } 3\) matrix with zeros in all positions except for a one in the \(i,j\)- th position. Using (3), we can calculate the action of the one- parameter groups generated by \(\{\xi _1, \xi _2, \xi _3\}\). We obtain$$\begin{aligned} \begin{array}{l} \displaystyle \bigl (\exp (s\xi _1)\cdot f\bigr ) (t,x) = f(t- s,x), \\ \displaystyle \bigl (\exp (s\xi _2)\cdot f\bigr ) (t,x) = f(t,x- s), \\ \displaystyle \bigl (\exp (s\xi _3)\cdot f\bigr ) (t,x) = e^{s\gamma } f\bigl (e^{- s\beta }t,e^{- s\alpha }x\bigr ). We use these to show the following theorem. Theorem 2. 1. The action of the groups \(G_{\alpha ,\beta }\) on \(I'_{\alpha , \beta ,\gamma }\) globalizes the action of the one- parameter groups generated by the Lie symmetries of the equation$$\begin{aligned} u_{tt}+\varphi (u)u_{t}=\bigl (f(u)u_x\bigr ){}_x \end{aligned}$$in the following cases: where \(\sigma \ne - 4/3\). Proof. To obtain this result one needs to differentiate the actions (4) and note the fact that a vector field of the form$$\begin{aligned} V=\tau (t,x)\,\frac{\partial }{\partial t}+ \xi (t,x)\,\frac{\partial }{\partial x}+\phi (t,x,v(t,x))\,\frac{\partial }{\partial v} \end{aligned}$$generates a one- parameter group such that$$\begin{aligned} \frac{d}{ds}\biggl |_{s=0}\bigl (\exp (s. V)\cdot f\bigr ) (t,x)=- \tau (t,x)\,\frac{\partial f }{\partial t}(t,x)& -\,\xi (t,x)\,\frac{\partial f}{\partial x}(t,x)\\& +\,\phi (t,x,f(t,x)), \end{aligned}$$c. Then a case by case analysis of the table shows that the action of the one- parameter subgroup of \(\xi _i\) on \(I'_{\alpha , \beta ,\gamma }\) corresponds with the action of \(X_i\) in each particular case as desired.\(\square \)So far, we have just considered the cases when \(\gamma \ne 0\). Now, let us consider the cases for which the extension has \(\kappa \ne 0\) (and correspondingly \(\gamma =0\)). In this case the action of \(G_{\alpha , \beta }\) on \(I'_{\alpha , \beta , 0}\) becomes$$\begin{aligned} g_{q_1,t_1,x_1}{\varvec{.}}f(t,x)= f\bigl ((t- t_1)q_1^{- \beta }\!, (x- x_1)q_1^{- \alpha }\bigr ). Unfortunately, this action does not globalize the action of the one- parameter group generated by \(X_3\) whenever \(\kappa \ne 0\). To obtain the desired globalization, we define the map \(\theta _\kappa : G_{\alpha ,\beta } {\times } I'_{\alpha ,\beta , 0} \rightarrow I'_{\alpha ,\beta ,0}\) by$$\begin{aligned} \theta _\kappa (g_{q_1,t_1,x_1}){\cdot } f(t,x)=f(t,x)+\kappa \ln q_1. This map defines an action of \(G_{\alpha ,\beta }\) on \(I'_{\alpha ,\beta ,0}\), indeed$$\begin{aligned} \bigl (\theta _2\bigl (g_{q_1,t_1,x_1}{\cdot } g_{q_2,t_2,x_2}\bigr ){\cdot } f\bigr )(t,x)& =f(t,x)+\kappa \ln q_1 q_2\\& = f(t,x)+\kappa \ln q_1+\kappa \ln q_2\\& =\bigl (\theta _2(g_{q_1,t_1,x_1}){\cdot } \bigl ( \theta _2 (g_{q_2,t_2,x_2}){\cdot } f\bigr )\bigr )(t,x). Using this, we can define an action \(\gamma _\kappa : G_{\alpha ,\beta }{\times } I'_{\alpha ,\beta ,0} \rightarrow I_{\alpha ,\beta ,0}'\) by$$\begin{aligned} (\gamma _\kappa (g){\cdot } f)(t,x)=\bigl (\theta _\kappa (g){\cdot } (g{\cdot } f)\bigr )(t,x). This action can be explicitly calculated; we obtain$$\begin{aligned} \gamma _\kappa (g_{q_1,t_1,x_1}){\cdot } f(t,x)= f\bigl ((t- t_1)q_1^{- \beta }\!, (x- x_1)q_1^{- \alpha }\bigr )+\kappa \ln q_1. Now we can calculate the action of the one- parameter groups generated by \(\{\xi _1, \xi _2, \xi _3\}\); we obtain$$\begin{aligned} \bigl (\exp (s\xi _1){\cdot } f\bigr ) (t,x)& = f(t- s,x), \\ \bigl (\exp (s\xi _2){\cdot } f\bigr ) (t,x)& = f(t,x- s), \\ \bigl (\exp (s\xi _3){\cdot } f\bigr ) (t,x)& = f\bigl (e^{- \beta s}t,e^{- \alpha s}x\bigr ) +\kappa s. Differentiating these actions as in Theorem 2. Theorem 2. 2. The action \(\gamma _\kappa : G_{\alpha ,\beta }{\times } I'_{\alpha , \beta ,0}\rightarrow I'_{\alpha , \beta ,0}\) globalizes the action of the one- parameter groups generated by the Lie symmetries of the equation$$\begin{aligned} u_{tt}+\varphi (u)u_{t}=\bigl (f(u)u_x\bigr ){}_x \end{aligned}$$in the following cases: where \(k\) is an arbitrary non- zero constant. This completes the globalization problem for the Lie symmetries of the nonlinear wave equations with dissipation for which the Lie symmetries form a solvable Lie algebra. Differential Equations. Differential Equations (Math 3. Here are my online notes for my differential equations. I teach here at Lamar University. Despite the fact that these are my “class notes”, they should be. I’ve tried to make these notes as self contained as possible. Calculus or Algebra class or contained in other sections of the notes. A couple of warnings to my students who may be here to get a. I wanted to make this a fairly complete set of notes for anyone wanting to. I have included some material that I do not. You will need to find one of your fellow class mates to see if. In. general I try to work problems in class that are different from my. However, with Differential. Equation many of the problems are difficult to make up on the spur of the. With that being said I will, on occasion, work problems off the top. I can to provide more examples than just those in my. Also, I often don’t have. Sometimes. questions in class will lead down paths that are not covered here. I try to anticipate as many of the. I can’t. anticipate all the questions. Sometimes a very good question gets asked in class that leads to. I’ve not included here. You should always talk to someone who was in class on the day you. THESE NOTES. ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using these notes as a substitute for class is liable to get you in. As already noted not everything in these notes is covered in. Here is a listing and brief description of the material in. Basic Concepts. Definitions Some of the common definitions and concepts in. Direction Fields An introduction to direction fields and what. Final Thoughts A couple of final thoughts on what we will be. First Order Differential Equations. Linear Equations Identifying and solving linear first order. Separable Equations Identifying and solving separable first order. We’ll also start. Exact Equations Identifying and solving exact differential. We’ll do a few more interval. Bernoulli Differential Equations In this section we’ll see how to solve the. Bernoulli Differential Equation. This. Substitutions We’ll pick up where the last section left off. Intervals of Validity Here we will give an in- depth look at. Modeling with First Order Differential. Equations Using first order differential equations to. The section. will show some very real applications of first order differential equations. Equilibrium Solutions We will look at the behavior of equilibrium. Euler’s Method In this section we’ll take a brief look at a. Second Order. Differential Equations. Basic Concepts Some of the basic concepts and ideas that are. Real Roots Solving differential equations whose. Complex Roots Solving differential equations whose. Repeated Roots Solving differential equations whose. Reduction of Order A brief look at the topic of reduction of. This will be one of the few times. Fundamental. Sets of Solutions A look at some of the theory behind the. Wronskian and fundamental sets of solutions. More on the Wronskian An application of the Wronskian and an. Nonhomogeneous Differential Equations A quick look into how to solve nonhomogeneous. Undetermined. Coefficients The first method for solving nonhomogeneous. Variation of Parameters Another method for solving nonhomogeneous. Mechanical Vibrations An application of second order differential. This section focuses on. Laplace Transforms. The Definition The definition of the Laplace. We will also compute a couple. Laplace transforms using the definition. Laplace Transforms As the previous section will demonstrate. Laplace transforms directly from the. In this section we introduce the way we usually compute Laplace transforms. Inverse Laplace Transforms In this section we ask the opposite. Here’s a Laplace. Step Functions This is one of the more important functions in. Laplace transforms. With the introduction of this function the. Laplace transforms starts to. Solving IVP’s with Laplace Transforms Here’s how we used Laplace. IVP’s. Nonconstant. Coefficient IVP’s We will see how Laplace. IVP’s. IVP’s with Step Functions Solving IVP’s that contain step. This is the section where the. Laplace transforms really. Dirac Delta Function One last function that often shows up in Laplace transform problems. Convolution Integral A brief introduction to the convolution. Laplace. transforms. Table of Laplace Transforms This is a small table of Laplace Transforms. Systems of. Differential Equations. Review : Systems of Equations The traditional starting point for a linear. We will use linear algebra. Review : Matrices and Vectors A brief introduction to matrices and. We will look at arithmetic. Review : Eigenvalues and Eigenvectors Finding the eigenvalues and eigenvectors of a. This topic will be key to. Systems of Differential Equations Here we will look at some of the basics of. Solutions to Systems We will take a look at what is involved in. Phase Plane A brief introduction to the phase plane and. Real Eigenvalues Solving systems of differential equations with. Complex Eigenvalues Solving systems of differential equations with. Repeated Eigenvalues Solving systems of differential equations with. Nonhomogeneous Systems Solving nonhomogeneous systems of differential. Laplace Transforms A very brief look at how Laplace transforms. Modeling In this section we’ll take a quick look at. Series Solutions. Review : Power Series A brief review of some of the basics of power. Review : Taylor Series A reminder on how to construct the Taylor series for a. Series Solutions In this section we will construct a series. Euler Equations We will look at solutions to Euler’s. Higher Order Differential Equations. Basic Concepts for nth Order Linear Equations We’ll start the chapter off with a quick look. Linear Homogeneous Differential Equations In this section we’ll take a look at extending. Undetermined Coefficients Here we’ll look at undetermined coefficients. Variation of Parameters We’ll look at variation of parameters for. Laplace Transforms In this section we’re just going to work an. Laplace transforms to solve a differential equation on a 3rd. Systems of Differential Equations Here we’ll take a quick look at extending the. Series Solutions This section serves the same purpose as the. Laplace Transform section. It is just. Boundary Value Problems & Fourier. Series. Boundary Value Problems In this section we’ll define the boundary. Eigenvalues and. Eigenfunctions Here we’ll take a look at the eigenvalues and. Periodic Functions and Orthogonal Functions We’ll take a look at periodic functions and. Fourier Sine Series In this section we’ll start looking at Fourier. Series by looking at a special case : Fourier Sine Series. Fourier Cosine Series We’ll continue looking at Fourier Series by. Fourier Cosine Series. Fourier Series Here we will look at the full Fourier series. Convergence. of Fourier Series Here we’ll take a look at some ideas involved. Fourier series converge to as well as. Fourier series. Partial Differential Equations. The Heat Equation We do a partial derivation of the heat. The Wave Equation Here we do a partial derivation of the wave. Terminology In this section we take a quick look at some. Separation. of Variables We take a look at the first step in the method. This first step is really the step that motivates the whole process. Solving the Heat Equation In this section we go through the complete. Heat Equation with Non- Zero Temperature. Boundaries Here we take a quick look at solving the heat. Laplace’s. Equation We discuss solving Laplace’s equation on both. Vibrating. String Here we solve the wave equation for a vibrating. Summary of. Separation of Variables In this final section we give a quick summary. |
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