Δευτέρα, 31 Οκτωβρίου 2016

Μία Ανισότητα

Η Επιμονή και η υπομονή στα Μαθηματικά μετρά περισσότερο από στιγμιαίες επιτυχίες, Μ.Παπαδημητράκης

Μιχάλης Παπαδημητράκης, Καθηγητής Μαθηματικών Πανεπιστημίου Κρήτης, 1ο βραβείο ΕΜΕ 1977. 

Από ανάρτησή του στο φβ.

 



Michael Papadimitrakis Τώρα
μάλιστα! Εννοείται ότι ευχαριστώ τον Μπάμπη για τα όμορφα λόγια του.
Αλλά δεν θα ήθελα να φανεί το πράγμα με διαστάσεις πέρα από τις
αρμόζουσες. Γνωρίζω πάρα πολύ καλά το τί σημαίνει ένα βραβείο, ειδικά σε
μικρές ηλικίες. Πολλά παιδιά παίρνουν μέρ
ος
σε τέτοιους διαγωνισμούς, άλλα πρωτεύουν, άλλα έρχονται πιο κάτω κι
άλλα αποτυγχάνουν. Μερικές φορές παίζει ρόλο κι ο παράγων τύχη ή η
συγκυρία. Θα μπορούσε να ήταν κάποιο άλλο παιδί σ' εκείνη τη θέση.
Σχεδόν πάντοτε μετράει περισσότερο η διάρκεια από ότι ένα "στιγμιαίο"
βραβείο.
Και μπορεί κάποιο παιδί που ήρθε δεύτερο ή τρίτο ή που απέτυχε
να κέρδισε αργότερα στη διάρκεια. Πιστέψτε με, έχω γνωρίσει κάποιους
πάρα-πάρα πολύ μεγάλους μαθηματικούς και αυτά τα "νεανικά κατορθώματα"
έχω μάθει να τα μετράω όσο τους πρέπει. Ακριβώς γι αυτόν τον λόγο, το
συγκεκριμένο δεν το έχω αναφέρει παρά μόνο σε ελάχιστα πολύ κοντινά μου
πρόσωπα. Έχει μόνο συναισθηματική αξία για εμένα, είναι κάτι που
συμβολίζει και οριοθετεί μια εποχή αγνότητας. Δεν το αναφέρω καν στο
βιογραφικό μου, αρνούμενος να το εκθέσω στο "επαγγελματικό βλέμμα". Και
πάλι ευχαριστώ, αλλά είναι κάτι που μάλλον δεν μπορώ να διαχειριστώ
εύκολα.

Τρίτη, 25 Οκτωβρίου 2016

Χρήση των Μαθηματικών στη δημιουργία βιντεοπαιγνιδιών.

(7) Matthew S. Stenquist's answer to How is math used in video game designing and programming? - Quora






Matthew S. Stenquist
Matthew S. Stenquist, Maker of games.

Math is an absolute fundamental foundation to successful game development and game design.
Math
is everything when it comes to games. From having the ability to
calculating the trajectory of an Angry Bird flying through the sky, to
ensuring that a character can jump and come back down to the ground --
without the help of mathematics, games simply wouldn't work.
A
character wouldn't be able to walk up a slope, slide down a slide, fire
a bullet from a gun, or even jump without the help of the mathematics.
The most basic of games use some form of math to the most complex of games.
Math
is essential to the production of games. It is the flour to the cake
that game developers are trying to bake. Without it, the cake wouldn't
rise.
Math is used in every aspect of game
development, including art. Maya is a math based program that plots out
the vertices and normals in mathematical form while the artist just uses
a tool that allows them to create stunning 3d graphics without worrying
about math. Simply put, you could model Godzilla in notepad and pushing
it into Maya, if you knew were to plot the points in numerical form
(which is extremely difficult).
However, a lot
of the math is computed at runtime and handled by the game engines that
render back face culling, and the other nitty gritty things that would
be too cumbersome to do without the use of using an engine to alleviate
the math calculation portion at runtime.
A lot
of math in gameplay scripting is fairly simple, but math used in game
engine architecture is far more complex and a lot more taxing mentally.
Math in game development is simulated either by the developer or handled
by the engine at runtime by running computations to calculate the
operation that is needed.
Here are some examples.
  • Ocean waves crashing nicely against your boat in ASC: Black Flag? Math.
  • Those bullets flying over your head in Call of Duty: Ghosts? Math.
  • That fancy UI animation that's procedurally generated? Math.
  • Sonic being able to run fast and Mario being able to jump? Math.
  • Drifting around that corner in Need for Speed at 80mph? Math.
  • Snowboarding down a slope in SSX? Math.
  • That rocket blasting off in Kerbal Space Program? Math.
Here's an expanded example using Mario's jumping mechanic.
  • When
    Mario jumps he isn't just going straight up and down on the Y-Axis but
    rather creating a parabola to jump up and down. It would be quite silly
    for Mario to beam up and then beam across on the X axis and then
    straight down on the Y-axis. Certainly, it would look goofy and not
    quite right.
Another example for instance
is, Kerbal Space Program; a game that relies heavily on math. The
entire game is pretty much all math when you think about it.
Newtonian
physics are heavily simulated in KSP. Without the use of math in KSP,
the game wouldn't be able to do much. Your rocket wouldn't be able to
get off the ground and out of the atmosphere into space. The simulations
in the game would be boring, and quite frankly the game wouldn't be
able to do much. A rocket wouldn't be able to get off the ground because
thrust doesn't exist, and or be able to pitch,yaw,and roll when in the
sky.
In game development we're allowed to get
away with faking math so that it is to work with the design of a game,
but a lot of the time the math in games mirror real world physics and
math principles. Math helps out with calculating everything from what a
particles velocity should be, to the spread of a shotgun blast, to using
gravity to bring a ball back down to the ground, and vice versa. These
are all very basic math principles that a lot of games use. And their is
still much more sophisticated and advanced math used in AAA games that I
haven't covered.
What are the main branches of math used in game development?
  • Algebra
  • Discrete Mathematics
  • Trigonometry
  • Calculus
  • Linear Algebra
  • Applied mathematics
  • And much much more....
Some of the most important math used in games.
  • Dot Product, Cross Product.
  • Scaling Vectors, Unit Vectors, and Vectors.
  • Reflection, Matrices.
  • Scalar Manipulation.
  • COS,SIN,TAN.
  • Delta Time.
  • Domain and Range.
Most
of these math topics are all used all together in very advanced games.
But, in simpler games the math may only require trig and algebra, to a
handful of math using only scalar multiplication.
How is math used in game programming and game scripting?
Math
in game programming can be as easy as adding x + y, to calculating and
manipulating cos, sin, tan with added variables while stringing it to
your function, all the while their is still much much more advanced ways
to use math in games.
Let's take a look at
creating a constant variable called MAX_SPEED and MIN_SPEED (vectors)
and adding that to the game update loop while adding a speed to a
spaceship so that it can move, to creating an array that runs an
algorithm that loads a random level every time the game is played, to
simply altering a child's X (vectors) in the update loop, or to
adjusting the angle of rotation once it hits a wall (reflection) on a
spin wheel when a ball hits it. Doing any sort of basic thing, such as
movement in a game will incorporate some form of math.
Math
is essential ingredient that is necessary to the production of games.
Without the use of math in games we would just have pretty art that's
semi-interactive. Math adds a whole new element to entertainment by
blending the sciences and arts into a whole.
With
that being said, mathematics allows games to do incredible things that
the real world cannot simulate by allowing new ways for the game script
and engine to handle calculations on the fly that would not be feasible
to do in the real world. So next time when you question your math
teacher if you're going to ever use "Algebra" in the real world when
you're done with your studies, you sure will especially if you pursue a
career in game development.
What can math do in game design?
  • Fluid water simulation.
  • Animation.
  • Algorithms.
  • Game engine architecture.
  • Gameplay scripting. (walking, shooting, jumping)
  • Analytics and data mining. (analyzing players interaction)
  • Timers programming (delta time, etc).
  • Physics programming (inertia, friction, etc).
  • Graphics/Shaders programming (matrices,etc).
  • Path finding/AI.
  • Procedural level design.
  • Handling Polygon rendering.
  • And much much more.
The things listed above are only but a fraction of what math is able to help out with in game development and game design.
Here are some books I recommend reading.

About the Author

Matthew S. Stenquist

Matthew S. Stenquist

I snowboard professionally, code, and design.
Frontend Developer
Live here in the Winter
Pro Snowboarder
507.5k answer views43.7k this month
Το 1509 ο DeBouvelles ισχυρίστηκε ότι ένας τουλάχιστον από τους αριθμούς α=6ν-1 ,β=6ν+1 με ν θετικό ακέραιο είναι πρώτος.


α) Να δώσετε παραδείγματα , στα οποία να φαίνεται ότι κανένας από τους δύο αριθμούς δεν είναι πάντα πρώτος .

β) Να αποδείξετε ότι υπάρχουν άπειροι αριθμοί ν , για τους οποίους και οι δύο αριθμοί α, β είναι συγχρόνως σύνθετοι.

Δευτέρα, 24 Οκτωβρίου 2016

Διασκεδαστικά Μαθηματικά: Η άσκηση της ημέρας (25 - 10 - 2016)

Διασκεδαστικά Μαθηματικά: Η άσκηση της ημέρας (25 - 10 - 2016)



 

24-10 3 β λυκειου

24-10 2 β λυκειου

24-10 1 β λυκειου

Πέμπτη, 20 Οκτωβρίου 2016

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Στη συγκεκριμένη περίπτωση η διαδικασία της παραγοντοποίησης μπορεί να γίνει και ως εξής: 

 
 
Αν τώρα θεωρήσουμε εξαρχής το τριώνυμο ως προς x: 
τότε θα έχει αυτό διακρίνουσα: 


και οι λύσεις θα είναι: 

και η παράσταση παραγοντοποιείται: 

που ήταν το ζητούμενο.







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