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dg.differential geometry - if a triangle can be displaced without distortion, must the surface have constant curvature?

here's one approach to formalizing "moved around". let $g=(v,e)$ be a graph. let $p:v\rightarrow s$ be a placement of $g$, that is, a map from the vertex set of $g$ to $s$. let us call the data $(g,p)$ a framework on $s$. let us define a motion of $(g,p)$ to be a continuous family of placements $f:v\times[0,1]\rightarrow s$ such that:$f(v,t) = p(v)$ for all $v\in v$, that is, the motion begins at $p$.$d_s(f(u,t),f(v,t))=d_s(p(u),p(v))$ for all $uv\in e$ and all $t\in[0,1]$, where $d_s(\cdot,\cdot)$ is the distance function on $s$. this condition just states that the motion preserves the lengths of all edges of $g$.$\alpha_t(u,v,w)=\alpha_0(u,v,w)$ for all triples of vertices $uvw$ such that $uv\in e$ and $vw\in e$, where $\alpha_t(u,v,w)$ is the angle between the geodesic s...





what is on/in my lens?

it looks like fungus to me but another possibility – camera lenses, to correct lens aberrations are constructed using several individual lenses packed in the lens barrel. some are air-spaced, some are cemented together. the cement of choice is a super clear resin made from the sap of the canadian balsam pine tree. in recent years, this glue has been fabricated in a lab. glues made from sap make good food for beasties; the synthetic glue is somewhat immune. both are subject to crazing, a network of fine lines that resemble cracks. the only way to repair this is to disassemble and dissolve away the glue, then re-glue. finding someone to do this task will be difficult and expensive. the good news is, often these types of defect are mainly cosmetic; they likely only minimally degrade. a...





united states - should i pay loan balance down early even if i can't pay the whole thing off?

i have a home mortgage with an outstanding principal balance of $150,000 and a fixed interest rate of 3%. my loan was a 15 year loan, of which i have 13 years left. however, i plan to pay my loan off within the next 2-3 years, and have amassed a savings of $50k to put towards doing that (this savings amount grows by around $3k each month).i know that by paying the loan off early, i will save in interest. what i'm not sure is whether it matters if i wait until i can pay the loan off with a single large lump sum payment, or whether i should make large additional ongoing payments (i.e., put my $50k towards the loan right now). is there any interest advantage to putting my money towards the loan right now to pay down my principal balance? or would i end up paying the same amount in interest i...





science based - metal-feathered macaw viability

if it's pure steel? hell nah. but there are other options. the scaly-foot gastropod, also known by the considerably more badass-sounding names of iron snail and chrysomallon squamiferum, hails from the deep-sea thermal vents known as black smokers, deep-sea vents from which water gushes constantly. that water, by the way, originates from below the mantle.the sfg uses the chemosynthetic bacteria lurking in its glands to absorb and mineralize poisonous iron-sulphides in the water, making them non-poisonous for the snail. it then coats its shell with the minerals, constructing an unique three-layer structure. the outer layer, used to block the bulk of the attack, is made up of greigite (fe3s4), a ridiculously hard mineral. then comes a middle layer of squishy organic matter purposed to absor...





is this a useless group?

this should really be a comment, but it's too long.the group (for $x$ positive and $\not=1$) is just $(\mathbb{r}_{>0},\times)$, described in an unnecessarily messy way. however, there is a neat map involved here.for every positive $x$ other than $1$, the map $$\varphi_x: s\mapsto e^{s\over\log x}$$ is a group isomorphism from $(\mathbb{r},+)$ to $(\mathbb{r}_{>0},\times)$. so in a certain sense, what your notation is really doing is describing a continuously$^1$ varying group: to each positive $x$ other than $1$, you've assigned a copy $c_x$ of $(\mathbb{r}_{>0},\times)$, and they're intuitively "stretching" as we vary $x$. for example, $\varphi_e(1)=e$ and $\varphi_{e^2}(1)=\sqrt{e}$, so in some sense the $e$ of $c_e$ corresopnds to the $\sqrt{e}$ of $c_{e^2}$. indee...





what does it mean to "burn a zero day"?

quick answer: a 0day is burned when the exploit is used too often or haphazardly, resulting it in being discovered and patched. virtually every time a 0day is used, it risks being burnt. using a 0day more sparingly and cautiously can increase its shelf life. the metaphor intends to compare a 0day to a non-renewable resource like combustible fuel that loses its value when used up.what is a 0day?a 0day is an exploitable vulnerability that is not publicly known. when a 0day is discovered, it can be turned into a working, "weaponized" exploit. like all vulnerabilities, if it is discovered in public, it will usually be patched and fixed, making it so the exploit no longer works. every time you use an exploit, you necessarily transmit valuable information to a system that you do not control. i...





dnd 5e - can you get armor proficiency with the skilled feat?

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draw a demonstration of pythagorean theorem with tikz - tex

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mg.metric geometry - steiner's inequality reference request

if $\omega\subset\mathbb{r}^2$ is a bounded regular domain and $\omega_r$ is defined as in the question, then $$ \operatorname{a}(\omega_r)= \operatorname{a}(\omega)+l(\partial\omega)r+\pi r^2\chi(\omega), $$ where $\chi(\omega)$ is the euler characteristic of $\omega$, see pp. 10-11 in [3]. if $\omega$ has no holes, then $\chi(\omega)=1.$ for example if $\omega=b(r)$ is a disc of radius $r$, then $\chi(b)=1$ and the above equality reads as: $$ \pi(r+r)^2=\pi r^2+2\pi rr+\pi r^2 $$ which is trivially true.the book [3] contains many generalizations of the formula, including higher dimensional ones. another reference is [2]. see also the introduction in [1]. the paper [1] is available on arxiv and if you know how to look for boo...





what is on/in my lens?

it looks like fungus to me but another possibility – camera lenses, to correct lens aberrations are constructed using several individual lenses packed in the lens barrel. some are air-spaced, some are cemented together. the cement of choice is a super clear resin made from the sap of the canadian balsam pine tree. in recent years, this glue has been fabricated in a lab. glues made from sap make good food for beasties; the synthetic glue is somewhat immune. both are subject to crazing, a network of fine lines that resemble cracks. the only way to repair this is to disassemble and dissolve away the glue, then re-glue. finding someone to do this task will be difficult and expensive. the good news is, often these types of defect are mainly cosmetic; they likely only minimally degrade. a...





creative writing - how to write a story without conflict, like "my neighbour totoro"?

remember back to elementary school. there are, three kinds of stories. man v.s. man (let's call this capt. america v.s. iron man)man v.s. nature (how about, the perfect storm or alive)man v.s. self (castaway, for example)now many "real" stories have some of each category, but every story relies on one more than the other. for your story (with no antagonist) your going to do more of the man v.s. self or man v.s. nature. for example, you could tell a very heart-wrenching story of a family separated due to a hurricane. the family members can meet awesome people.the children can stay with "the older couple next door", the parents with friends in another city. all while the roads, and infrastructure are being rebuilt after the storm. the children, while never in any danger of any kind can ...





existence - being alive today: the most improbable coïncidence?

the first time i recall encountering this argument was in alan moore’s watchmen, where the probability of what you describe is likened to “events with odds so astronomical they’re effectively impossible, like oxygen spontaneously becoming gold”.i find the argument similar to the gambler’s fallacy. an example:in a fair coin toss, the probability of landing on heads (or tails) is 50%. before we start tossing the coin, how likely do you think it is that the coin will land 20 consecutive times on heads? as you’d expect, they are astronomically low. also as expected, while making the calculations every consecutive toss we add lowers the probability — landing 10 consecutive heads is less likely than landing on it twice.but now let’s say we’ve already made 19 tosses that all lan...





c++ - can a declaration affect the std namespace?

the language specification allows implementations to implement by declaring (and defining) the standard functions in global namespace and then bringing them into namespace std by means of using-declarations. it is unspecified whether this approach is used20.5.1.2 headers4 [...] in the c++ standard library, however, the declarations (except for names which are defined as macros in c) are within namespace scope (6.3.6) of the namespace std. it is unspecified whether these names (including any overloads added in clauses 21 through 33 and annex d) are first declared within the global namespace scope and are then injected into namespace std by explicit using-declarations (10.3.3).apparently, you are dealing with one of implementations that decided to follow this approach (e.g. gcc). i.e. yo...





what does it mean to "burn a zero day"?

what is a 0day?a 0day is an exploitable vulnerability that is not publicly known. when a 0day is discovered, it can be turned into a working, "weaponized" exploit. like all vulnerabilities, if it is discovered in public, it will usually be patched and fixed, making it so the exploit no longer works. every time you use an exploit, you necessarily transmit valuable information to a system that you do not control. if the system is being extensively monitored, the exploit technique may be discovered and with it, the necessarily knowledge to fix the vulnerability and roll out patches to all affected systems.what does it mean to "burn" one?i was the one who wrote that comment. to "burn" a 0day is slang for using it either too often or using it in a high-risk situation where it is likely th...





phd student failing - academia stack exchange

having worked through an ma and phd myself while not only starting a family but also simultaneously working freelance and undertaking sessional teaching, i know that life is not easy in this situation. so many things can and do go wrong from the birth forwards. mothers need the support of fathers, and when childcare falls through, or a child is ill then it is the person studying and working from home, with the most flexible schedule upon whom everything falls, because they can catch up the work. or so it is supposed, but in reality having had a first baby who struggled to feed and therefore to sleep, i found that exhaustion very quickly took over and that the extra hours available to me were very few. i had to work through the night to keep everything going, and every few hours our child w...





what is on/in my lens?

it looks like fungus to me but another possibility – camera lenses, to correct lens aberrations are constructed using several individual lenses packed in the lens barrel. some are air-spaced, some are cemented together. the cement of choice is a super clear resin made from the sap of the canadian balsam pine tree. in recent years, this glue has been fabricated in a lab. glues made from sap make good food for beasties; the synthetic glue is somewhat immune. both are subject to crazing, a network of fine lines that resemble cracks. the only way to repair this is to disassemble and dissolve away the glue, then re-glue. finding someone to do this task will be difficult and expensive. the good news is, often these types of defect are mainly cosmetic; they likely only minimally degrade. a...





linear algebra - why doesn't this example of basis change work?

i'm learning about the change of basis in linear algebra, and trying to come up with an example to understand it. but somehow my example below doesn't make sense.let $b_1 = ((1,0),(0,1))$ be the standard basis in $r^2$, and $v = (2,3)$ be a vector.let $b_2$ be another basis of $r^2$, $b_2 = ((1,-1),(1,1))$. my understanding is that, in this system, the same $v$ above has the coordinate $v = (-1/2, 5/2)$.let $t: r^2 \rightarrow r^2$, $t(x,y) = (x+2y, 3x - 2y)$.with $b_1$, $m(t) = \begin{bmatrix} 1 & 2 \\ 3 & -2\\ \end{bmatrix}$, and $t(2,3) = m(t).v = \begin{bmatrix} 1 & 2 \\ 3 & -2\\ \end{bmatrix} .\begin{bmatrix} 2 \\ 3\\ \end{bmatrix} =\begin{bmatrix} 8\\ 0\\ \end{bmatrix}$with $b_2$...





reality check - in a spinning dyson ring, would objects not touching the surface experience gravity?

the short answer is that birds would feel your pseudo-gravity.the long answer involves making some sense of the situation with coordinate transforms.the simplest system to explore is the inertial system, where the coordinate system is fixed. in this coordinate system, the dyson ring appears to be spinning at a high rate. objects travel in a straight line unless forces are put on them. one way forces can be put on them is if they are in physical contact with the ring. they can also have forces put on them by the air. but what forces are on the air itself? that question is a little difficult to answer in inertial systems. it can be done, but it'll be easier of we change things up.consider a rotating system, which rotates at the same rate as the surface of the dyson ring. in this coordinate ...





what does it mean to "burn a zero day"?

a zero-day is a vulnerability that is unknown by the software manufacturer and for which no patch exists.when using a zero-day vulnerability against a remote server, it may give away how it works. the administrators of the application may notice they have been hacked, look in the logs and discover the vulnerability that was used to hack them. if they then fix the vulnerability, the vulnerability is no longer a zero-day vulnerability and knowledge about it is useless.for example, this flash zero-day was actively being exploited and that was how adobe learned about the vulnerability.in some cases, using a zero-day does not expose the vulnerability. for example, zero-days to root a mobile os may be used without the software manufacturer learning about the vulnerability.





dnd 5e - can i see a hidden creature that is not obscured at all?

see? yes. notice? maybe.the rules on hiding and stealth give a lot of leeway to a dm, so you are absolutely within your rights to declare that once a creature is looking directly at you, you cease to be hidden from it. but keep in mind a couple of important distinctions: 1. hiding is different than being unseenany creature that is in a heavily obscured area, or invisible, or fighting a blinded enemy is "unseen". but a creature that is hiding is not only unseen, but also unheard, and generally unperceived. they might keep very still, or stand close to similarly colored parts of their surrounding, or compact their body so their silhouette no longer seems humanoid. in short, they are doing more than being out of line of sight. as evidence of this distinction, consider the following on page ...