nhaliday + overflow   424

soft question - What are good non-English languages for mathematicians to know? - MathOverflow
I'm with Deane here: I think learning foreign languages is not a very mathematically productive thing to do; of course, there are lots of good reasons to learn foreign languages, but doing mathematics is not one of them. Not only are there few modern mathematics papers written in languages other than English, but the primary other language they are written (French) in is pretty easy to read without actually knowing it.

Even though I've been to France several times, my spoken French mostly consists of "merci," "si vous plait," "d'accord" and some food words; I've still skimmed 100 page long papers in French without a lot of trouble.

If nothing else, think of reading a paper in French as a good opportunity to teach Google Translate some mathematical French.
q-n-a  overflow  math  academia  learning  foreign-lang  publishing  science  french  soft-question  math.AG  nibble  quixotic 
10 weeks ago by nhaliday
orbit - Best approximation for Sun's trajectory around galactic center? - Astronomy Stack Exchange
The Sun orbits in the Galactic potential. The motion is complex; it takes about 230 million years to make a circuit with an orbital speed of around 220 km/s, but at the same time it oscillates up and down with respect to the Galactic plane every ∼70∼70 million years and also wobbles in and out every ∼150∼150 million years (this is called epicyclic motion). The spatial amplitudes of these oscillations are around 100 pc vertically and 300 pc in the radial direction inwards and outwards around an average orbital radius (I am unable to locate a precise figure for the latter).
nibble  q-n-a  overflow  space  oscillation  time  cycles  spatial  trivia  manifolds 
december 2017 by nhaliday
light - Why doesn't the moon twinkle? - Astronomy Stack Exchange
As you mention, when light enters our atmosphere, it goes through several parcels of gas with varying density, temperature, pressure, and humidity. These differences make the refractive index of the parcels different, and since they move around (the scientific term for air moving around is "wind"), the light rays take slightly different paths through the atmosphere.

Stars are point sources
…the Moon is not
nibble  q-n-a  overflow  space  physics  trivia  cocktail  navigation  sky  visuo  illusion  measure  random  electromag  signal-noise  flux-stasis  explanation  explanans  magnitude  atmosphere  roots 
december 2017 by nhaliday
galaxy - How do astronomers estimate the total mass of dust in clouds and galaxies? - Astronomy Stack Exchange
Dust absorbs stellar light (primarily in the ultraviolet), and is heated up. Subsequently it cools by emitting infrared, "thermal" radiation. Assuming a dust composition and grain size distribution, the amount of emitted IR light per unit dust mass can be calculated as a function of temperature. Observing the object at several different IR wavelengths, a Planck curve can be fitted to the data points, yielding the dust temperature. The more UV light incident on the dust, the higher the temperature.

The result is somewhat sensitive to the assumptions, and thus the uncertainties are sometimes quite large. The more IR data points obtained, the better. If only one IR point is available, the temperature cannot be calculated. Then there's a degeneracy between incident UV light and the amount of dust, and the mass can only be estimated to within some orders of magnitude (I think).
nibble  q-n-a  overflow  space  measurement  measure  estimate  physics  electromag  visuo  methodology 
december 2017 by nhaliday
general relativity - What if the universe is rotating as a whole? - Physics Stack Exchange
To find out whether the universe is rotating, in principle the most straightforward test is to watch the motion of a gyroscope relative to the distant galaxies. If it rotates at an angular velocity -ω relative to them, then the universe is rotating at angular velocity ω. In practice, we do not have mechanical gyroscopes with small enough random and systematic errors to put a very low limit on ω. However, we can use the entire solar system as a kind of gyroscope. Solar-system observations put a model-independent upper limit of 10^-7 radians/year on the rotation,[Clemence 1957] which is an order of magnitude too lax to rule out the Gödel metric.
nibble  q-n-a  overflow  physics  relativity  gedanken  direction  absolute-relative  big-picture  space  experiment  measurement  volo-avolo 
november 2017 by nhaliday
What is the connection between special and general relativity? - Physics Stack Exchange
Special relativity is the "special case" of general relativity where spacetime is flat. The speed of light is essential to both.
nibble  q-n-a  overflow  physics  relativity  explanation  synthesis  hi-order-bits  ground-up  gravity  summary  aphorism  differential  geometry 
november 2017 by nhaliday
gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.


Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell
nibble  q-n-a  overflow  math  geometry  topology  tidbits  intricacy  intersection  proofs  gotchas  oly  mathtariat  fixed-point  math.AT  manifolds  intersection-connectedness 
october 2017 by nhaliday
multivariate analysis - Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? - Cross Validated
The bivariate normal distribution is the exception, not the rule!

It is important to recognize that "almost all" joint distributions with normal marginals are not the bivariate normal distribution. That is, the common viewpoint that joint distributions with normal marginals that are not the bivariate normal are somehow "pathological", is a bit misguided.

Certainly, the multivariate normal is extremely important due to its stability under linear transformations, and so receives the bulk of attention in applications.

note: there is a multivariate central limit theorem, so those such applications have no problem
nibble  q-n-a  overflow  stats  math  acm  probability  distribution  gotchas  intricacy  characterization  structure  composition-decomposition  counterexample  limits  concentration-of-measure 
october 2017 by nhaliday
Karl Pearson and the Chi-squared Test
Pearson's paper of 1900 introduced what subsequently became known as the chi-squared test of goodness of fit. The terminology and allusions of 80 years ago create a barrier for the modern reader, who finds that the interpretation of Pearson's test procedure and the assessment of what he achieved are less than straightforward, notwithstanding the technical advances made since then. An attempt is made here to surmount these difficulties by exploring Pearson's relevant activities during the first decade of his statistical career, and by describing the work by his contemporaries and predecessors which seem to have influenced his approach to the problem. Not all the questions are answered, and others remain for further study.

original paper: http://www.economics.soton.ac.uk/staff/aldrich/1900.pdf

How did Karl Pearson come up with the chi-squared statistic?: https://stats.stackexchange.com/questions/97604/how-did-karl-pearson-come-up-with-the-chi-squared-statistic
He proceeds by working with the multivariate normal, and the chi-square arises as a sum of squared standardized normal variates.

You can see from the discussion on p160-161 he's clearly discussing applying the test to multinomial distributed data (I don't think he uses that term anywhere). He apparently understands the approximate multivariate normality of the multinomial (certainly he knows the margins are approximately normal - that's a very old result - and knows the means, variances and covariances, since they're stated in the paper); my guess is that most of that stuff is already old hat by 1900. (Note that the chi-squared distribution itself dates back to work by Helmert in the mid-1870s.)

Then by the bottom of p163 he derives a chi-square statistic as "a measure of goodness of fit" (the statistic itself appears in the exponent of the multivariate normal approximation).

He then goes on to discuss how to evaluate the p-value*, and then he correctly gives the upper tail area of a χ212χ122 beyond 43.87 as 0.000016. [You should keep in mind, however, that he didn't correctly understand how to adjust degrees of freedom for parameter estimation at that stage, so some of the examples in his papers use too high a d.f.]
nibble  papers  acm  stats  hypothesis-testing  methodology  history  mostly-modern  pre-ww2  old-anglo  giants  science  the-trenches  stories  multi  q-n-a  overflow  explanation  summary  innovation  discovery  distribution  degrees-of-freedom  limits 
october 2017 by nhaliday
self study - Looking for a good and complete probability and statistics book - Cross Validated
I never had the opportunity to visit a stats course from a math faculty. I am looking for a probability theory and statistics book that is complete and self-sufficient. By complete I mean that it contains all the proofs and not just states results.
nibble  q-n-a  overflow  data-science  stats  methodology  books  recommendations  list  top-n  confluence  proofs  rigor  reference  accretion 
october 2017 by nhaliday
Variance of product of multiple random variables - Cross Validated
prod_i (var[X_i] + (E[X_i])^2) - prod_i (E[X_i])^2

two variable case: var[X] var[Y] + var[X] (E[Y])^2 + (E[X])^2 var[Y]
nibble  q-n-a  overflow  stats  probability  math  identity  moments  arrows  multiplicative  iidness  dependence-independence 
october 2017 by nhaliday
newtonian gravity - Newton's original proof of gravitation for non-point-mass objects - Physics Stack Exchange
This theorem is Proposition LXXI, Theorem XXXI in the Principia. To warm up, consider the more straightforward proof of the preceding theorem, that there's no inverse-square force inside of a spherical shell:


The crux of the argument is that the triangles HPI and LPK are similar. The mass enclosed in the small-but-near patch of sphere HI goes like the square of the distance HP, while the mass enclosed in the large-but-far patch of sphere KL, with the same solid angle, goes like the square of the distance KP. This mass ratio cancels out the distance-squared ratio governing the strength of the force, and so the net force from those two patches vanishes.

For a point mass outside a shell, Newton's approach is essentially the same as the modern approach:


One integral is removed because we're considering a thin spherical shell rather than a solid sphere. The second integral, "as the semi-circle AKB revolves about the diameter AB," trivially turns Newton's infinitesimal arcs HI and KL into annuli.

The third integral is over all the annuli in the sphere, over 0≤ϕ≤τ/20≤ϕ≤τ/2 or over R−r≤s≤R+rR−r≤s≤R+r. This one is a little bit hairy, even with the advantage of modern notation.

Newton's clever trick is to consider the relationship between the force due to the smaller, nearer annulus HI and the larger, farther annulus KL defined by the same viewing angle (in modern notation, dθdθ). If I understand correctly he argues again, based on lots of similar triangles with infinitesimal angles, that the smaller-but-nearer annulus and the larger-but-farther annulus exert the same force at P. Furthermore, he shows that the force doesn't depend on the distance PF, and thus doesn't depend on the radius of the sphere; the only parameter left is the distance PS (squared) between the particle and the sphere's center. Since the argument doesn't depend on the angle HPS, it's true for all the annuli, and the theorem is proved.
nibble  q-n-a  overflow  giants  old-anglo  the-trenches  physics  mechanics  gravity  proofs  symmetry  geometry  spatial 
september 2017 by nhaliday
electricity - Why is AC more "dangerous" than DC? - Physics Stack Exchange
One of the reasons that AC might be considered more dangerous is that it arguably has more ways of getting into your body. Since the voltage alternates, it can cause current to enter and exit your body even without a closed loop, since your body (and what ground it's attached to) has capacitance. DC cannot do that. Also, AC is quite easily stepped up to higher voltages using transformers, while with DC that requires some relatively elaborate electronics. Finally, while your skin has a fairly high resistance to protect you, and the air is also a terrific insulator as long as you're not touching any wires, sometimes the inductance of AC transformers can cause high-voltage sparks that break down the air and I imagine can get through your skin a bit as well.

Also, like you mentioned, the heart is controlled by electric pulses and repeated pulses of electricity can throw this off quite a bit and cause a heart attack. However, I don't think that this is unique to alternating current. I read once about an unfortunate young man that was learning about electricity and wanted to measure the resistance of his own body. He took a multimeter and set a lead to each thumb. By accident or by stupidity, he punctured both thumbs with the leads, and the small (I imagine it to be 9 V) battery in the multimeter caused a current in his bloodstream, and he died on the spot. So maybe ignorance is more dangerous than either AC or DC.
nibble  q-n-a  overflow  physics  electromag  dirty-hands  embodied  safety  short-circuit  IEEE  death 
september 2017 by nhaliday
Why is Earth's gravity stronger at the poles? - Physics Stack Exchange
The point is that if we approximate Earth with an oblate ellipsoid, then the surface of Earth is an equipotential surface,11 see e.g. this Phys.SE post.

Now, because the polar radius is smaller than the equatorial radius, the density of equipotential surfaces at the poles must be bigger than at the equator.

Or equivalently, the field strength22 gg at the poles must be bigger than at the equator.
nibble  q-n-a  overflow  physics  mechanics  gravity  earth  space  intricacy  explanation  tidbits  spatial  direction  nitty-gritty  geography 
september 2017 by nhaliday
diffusion - Surviving under water in air bubble - Physics Stack Exchange
I get d≈400md≈400m.

It's interesting to note that this is independent of pressure: I've neglected pressure dependence of DD and human resilience to carbon dioxide, and the maximum safe concentration of carbon dioxide is independent of pressure, just derived from measurements at STP.

Finally, a bubble this large will probably rapidly break up due to buoyancy and Plateau-Rayleigh instabilities.
nibble  q-n-a  overflow  physics  mechanics  h2o  safety  short-circuit  tidbits  gedanken  fluid  street-fighting 
august 2017 by nhaliday
rotational dynamics - Why do non-rigid bodies try to increase their moment of inertia? - Physics Stack Exchange
This happens to isolated rotating system that is not a rigid body.

Inside such a body (for example, steel chain in free fall) the parts move relatively to each other and there is internal friction that dissipates kinetic energy of the system, while angular momentum is conserved. The dissipation goes on until the parts stop moving with respect to each other, so body rotates as a rigid body, even if it is not rigid by constitution.

The rotating state of the body that has the lowest kinetic energy for given angular momentum is that in which the body has the greatest moment of inertia (with respect to center of mass). For example, a long chain thrown into free fall will twist and turn until it is all straight and rotating as rigid body.


If LL is constant (net torque of external forces acting on the system is zero) and the constitution and initial conditions allow it, the system's dissipation will work to diminish energy until it has the minimum value, which happens for maximum IaIa possible.
nibble  q-n-a  overflow  physics  mechanics  tidbits  spatial  rigidity  flexibility  invariance  direction  stylized-facts  dynamical  volo-avolo  street-fighting  yoga 
august 2017 by nhaliday
gravity - Gravitational collapse and free fall time (spherical, pressure-free) - Physics Stack Exchange
the parenthetical regarding Gauss's law just involves noting a shell of radius r + symmetry (so single parameter determines field along shell)
nibble  q-n-a  overflow  physics  mechanics  gravity  tidbits  time  phase-transition  symmetry  differential  identity  dynamical 
august 2017 by nhaliday
co.combinatorics - Classification of Platonic solids - MathOverflow
My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula v−e+f=2v−e+f=2 to show that there are exactly five possible triples (v,e,f)(v,e,f). But of course this is not a complete proof because it does not rule out the possibility of different configurations or deformations. Has anyone ever written up a complete proof of this statement?!


This is a classical question. Here is my reading of it: Why is there a unique polytope with given combinatorics of faces, which are all regular polygons? Of course, for simple polytopes (tetrahedron, cube, dodecahedron) this is clear, but for the octahedron and icosahedron this is less clear.

The answer lies in the Cauchy's theorem. It was Legendre, while writing his Elements of Geometry and Trigonometry, noticed that Euclid's proof is incomplete in the Elements. Curiously, Euclid finds both radii of inscribed and circumscribed spheres (correctly) without ever explaining why they exist. Cauchy worked out a proof while still a student in 1813, more or less specifically for this purpose. The proof also had a technical gap which was found and patched up by Steinitz in 1920s.

The complete (corrected) proof can be found in the celebrated Proofs from the Book, or in Marcel Berger's Geometry. My book gives a bit more of historical context and some soft arguments (ch. 19). It's worth comparing this proof with (an erroneous) pre-Steinitz exposition, say in Hadamard's Leçons de Géométrie Elémentaire II, or with an early post-Steinitz correct but tedious proof given in (otherwise, excellent) Alexandrov's monograph (see also ch.26 in my book which compares all the approaches).

P.S. Note that Coxeter in Regular Polytopes can completely avoid this issue but taking a different (modern) definition of the regular polytopes (which are symmetric under group actions). For a modern exposition and the state of art of this approach, see McMullen and Schulte's Abstract Regular Polytopes.

q-n-a  overflow  math  topology  geometry  math.CO  history  iron-age  mediterranean  the-classics  multi  curiosity  clarity  proofs  nibble  wiki  reference  characterization  uniqueness  list  ground-up 
july 2017 by nhaliday
inequalities - Is the Jaccard distance a distance? - MathOverflow
Steinhaus Transform
the referenced survey: http://kenclarkson.org/nn_survey/p.pdf

It's known that this transformation produces a metric from a metric. Now if you take as the base metric D the symmetric difference between two sets, what you end up with is the Jaccard distance (which actually is known by many other names as well).
q-n-a  overflow  nibble  math  acm  sublinear  metrics  metric-space  proofs  math.CO  tcstariat  arrows  reduction  measure  math.MG  similarity  multi  papers  survey  computational-geometry  cs  algorithms  pdf  positivity  msr  tidbits  intersection  curvature  convexity-curvature  intersection-connectedness  signum 
february 2017 by nhaliday
« earlier      
per page:    204080120160

bundles : pub

related tags

aaronson  absolute-relative  abstraction  academia  accretion  accuracy  acm  acmtariat  additive  additive-combo  adversarial  advice  aggregator  agri-mindset  ai  alg-combo  algebra  algebraic-complexity  algorithmic-econ  algorithms  AMT  analogy  analysis  announcement  aphorism  apollonian-dionysian  applicability-prereqs  applications  approximation  arrows  asia  atmosphere  atoms  audio  automata  automation  axioms  bare-hands  bayesian  best-practices  better-explained  bias-variance  big-list  big-picture  big-surf  binomial  bio  bioinformatics  bits  blog  blowhards  boltzmann  bonferroni  books  boolean-analysis  borel-cantelli  bret-victor  brunn-minkowski  business  calculation  caltech  career  cartoons  causation  characterization  chart  cheatsheet  checking  checklists  chemistry  circuits  clarity  classic  classification  closure  coarse-fine  cocktail  coding-theory  cog-psych  coloring  communication  communication-complexity  community  commutativity  comparison  complex-systems  complexity  composition-decomposition  compressed-sensing  compression  computation  computational-geometry  concentration-of-measure  concept  conceptual-vocab  concrete  confidence  confluence  confusion  constraint-satisfaction  consumerism  contradiction  contrarianism  convergence  convexity-curvature  cool  coordination  core-rats  correlation  counterexample  counting  courage  creative  critique  crypto  cs  culture  curiosity  curvature  cycles  data  data-science  data-structures  database  dataviz  death  debate  debt  decision-making  deep-learning  definition  degrees-of-freedom  density  dependence-independence  differential  dimensionality  direction  dirty-hands  discovery  discrete  discussion  distributed  distribution  DP  draft  duality  dynamic  dynamical  earth  econometrics  economics  education  electromag  embeddings  embodied  embodied-pack  encyclopedic  engineering  ensembles  entropy-like  epistemic  equilibrium  erdos  ergodic  error  essay  estimate  evidence  evolution  examples  existence  exocortex  expectancy  experiment  expert  expert-experience  explanans  explanation  exploratory  exposition  extrema  fall-2016  faq  features  fedja  feynman  fields  finance  finiteness  fisher  fixed-point  flexibility  fluid  flux-stasis  foreign-lang  forum  fourier  french  frequency  frequentist  frontier  functional  game-theory  games  gedanken  gender  generalization  generative  genetics  genomics  geography  geometry  giants  gnxp  google  gotchas  gowers  grad-school  graph-theory  graphical-models  graphs  gravity  greedy  ground-up  growth  GT-101  h2o  hard-core  hardness  hardware  heuristic  hi-order-bits  hierarchy  high-dimension  history  hmm  homogeneity  howto  huge-data-the-biggest  human-ml  hypothesis-testing  ideas  identity  idk  IEEE  iidness  illusion  impact  induction  info-dynamics  info-foraging  information-theory  init  inner-product  innovation  insight  integral  interdisciplinary  intersection  intersection-connectedness  intricacy  intuition  invariance  investing  iron-age  isotropy  iteration-recursion  japan  jargon  jobs  knowledge  language  large-factor  latent-variables  latex  lattice  learning  learning-theory  lecture-notes  lectures  lens  let-me-see  levers  lifts-projections  limits  linear-algebra  linear-models  linear-programming  linearity  liner-notes  links  list  local-global  logic  long-term  longitudinal  low-hanging  lower-bounds  machine-learning  magnitude  manifolds  marginal  markov  martingale  matching  math  math.AC  math.AG  math.AT  math.CA  math.CO  math.CT  math.CV  math.DS  math.FA  math.GN  math.GR  math.MG  math.NT  math.RT  mathtariat  matrix-factorization  meaningness  measure  measurement  mechanics  mediterranean  mental-math  meta:math  meta:research  meta:science  metabuch  metameta  methodology  metric-space  metrics  michael-nielsen  minimum-viable  ML-MAP-E  model-class  model-selection  models  moments  monotonicity  monte-carlo  mostly-modern  motivation  msr  multi  multiplicative  mutation  naturality  navigation  network-structure  neuro  neurons  news  nibble  nitty-gritty  nlp  no-go  nonlinearity  nonparametric  norms  notation  novelty  numerics  objektbuch  occam  off-convex  old-anglo  oly  online-learning  open-problems  operational  optics  optimization  orders  ORFE  org:bleg  org:edu  org:inst  org:junk  org:mag  org:nat  org:sci  orourke  oscillation  outdoors  outliers  overflow  oxbridge  p:null  p:someday  p:whenever  papers  paradox  parametric  parsimony  pdf  people  performance  perturbation  phase-transition  phd  philosophy  physics  pic  pigeonhole-markov  pinboard  planning  plots  pls  polynomials  pop-structure  popsci  population  population-genetics  positivity  potential  power-law  practice  pragmatic  pre-ww2  prepping  princeton  prioritizing  probabilistic-method  probability  problem-solving  programming  progression  project  proof-systems  proofs  properties  pseudorandomness  psychology  publishing  puzzles  python  q-n-a  qra  quantifiers-sums  quantum  quantum-info  questions  quixotic  quotes  r-lang  rand-approx  rand-complexity  random  random-matrices  ranking  reading  reason  rec-math  recommendations  reddit  reduction  reference  reflection  regression  regularity  regularization  regularizer  reinforcement  relativity  relativization  relaxation  research  research-program  retention  retrofit  rhetoric  rhythm  rigidity  rigor  rigorous-crypto  roadmap  robust  roots  rounding  ryan-odonnell  s:*  s:**  s:***  s:null  safety  sampling  sampling-bias  sanjeev-arora  sapiens  scale  scaling-tech  scholar  scholar-pack  schools  science  scitariat  SDP  selection  sensitivity  separation  sequential  series  shannon  shift  short-circuit  signal-noise  signum  similarity  simplex  singularity  skeleton  sky  slides  smoothness  social  social-science  soft-question  software  space  space-complexity  sparsity  spatial  spectral  speculation  speed  stackex  stat-mech  state  state-of-art  stats  stirling  stochastic-processes  stock-flow  stories  strategy  stream  street-fighting  structure  study  studying  stylized-facts  subculture  subjective-objective  sublinear  submodular  sum-of-squares  summary  survey  survival  symmetry  synchrony  synthesis  systematic-ad-hoc  systems  tails  tcs  tcstariat  teaching  tech  techtariat  temperature  tensors  the-classics  the-prices  the-trenches  the-world-is-just-atoms  thermo  things  thinking  thurston  tidbits  tightness  tim-roughgarden  time  time-complexity  time-series  tip-of-tongue  todo  tools  top-n  topology  toys  tradeoffs  trees  tricki  tricks  trivia  turing  tutorial  tutoring  uncertainty  uniqueness  unit  unsupervised  vague  variance-components  vazirani  video  virtu  visual-understanding  visualization  visuo  volo-avolo  water  waves  west-hunter  wigderson  wiki  wild-ideas  wire-guided  wisdom  wordlessness  workflow  wormholes  worrydream  writing  wut  x-sports  yak-shaving  yoga  zooming  🌞  🎓  👳  🔬  🖥 

Copy this bookmark: