TobVII

hmmmm..

[MindBranches @MindBranches]

How to articulate yourself intelligently

article summary in 4k light & dark

[Mathematica @mathemetica]

Right thumb along an axis, fingers curl the positive spin. Blue marks +x, red +y, green +z. Matching rings give +θx +θy +θz in that same curl. Dashed lines run the opposites. x × y lands on z.

Flight controllers code these exact signs into a quadrotor’s attitude loop so a +θy pitch command always lifts the nose and never drives the craft inverted.

[Portal do Medo @portalmedoo]

14 anos de SINISTER

Eleito o filme de terror mais assustador de todos os tempos, segundo a ciência.

[MindBranches @MindBranches]

“If you have multiple interests, do not waste the next 2-3 years”

article summary in 4k light & dark

[Bilim Dünyası @dunyasalbilim]

Bir kürenin yüzey alanı işte bu şekilde hesaplanıyor.

[Mathematica @mathemetica]

Data structures scale by adding dimensions. A scalar is a single number. A vector is a one-dimensional array that can be a row of shape 1×3 or a column of shape 3×1. A matrix is a two-dimensional array of numbers arranged in rows and columns. A tensor is a multi-dimensional array with three or more dimensions.

Neural network training processes batches of multi-channel images as 4D tensors in convolutional models for object recognition.

[Math Hub @mathhub_vn]

Taylor Series - The Taylor Series is a way to approximate any smooth function using a polynomial, which is a simple equation made of powers of x. It works by taking a function f(x) and expanding it around a specific point 'a' as an infinite sum: f(x) = f(a) + f'(a)(x-a) + [f''(a)/2!](x-a)² + [f'''(a)/3!](x-a)³ + ... Here, f(a) is the starting value, f'(a) is the slope, f''(a) is the curvature, and so on. Each new term adds more detail to better match the original curve near point 'a'. The more terms you add, the more accurate the approximation becomes near 'a'. This useful concept is used in calculators, physics, and engineering to easily compute complex functions like sin(x), e^x, or cos(x). When the center point a = 0, it is called the Maclaurin Series....

[Mathematica @mathemetica]

Otto Hesse developed the matrix of second partial derivatives that describes the curvature of functions f: R^n → R.

Its entries are defined by [H(f)]_ij = ∂²f / ∂x_i ∂x_j . In the two-variable case, the Hessian is [[f_xx, f_xy], [f_yx, f_yy]]. Sample surfaces in two dimensions illustrate functions with critical points whose nature this matrix helps determine.

In neural network training, the Hessian of the loss function enables second-order methods to refine parameters more efficiently than gradient descent alone.

[Mathematica @mathemetica]

The Gaussian integral I = ∫_{-∞}^∞ e^{-x²} dx is evaluated by squaring it to obtain the double integral ∬ e^{-(x²+y²)} dx dy.

In polar coordinates the Jacobian r and substitution u = r² reduce it to I² = π, hence I = √π. The derivation is accompanied by a 3D plot of the symmetric Gaussian surface e^{-(x²+y²)} with its cross sections e^{-x²} and e^{-y²}.

This exact result normalizes the standard normal distribution, which is used to model and calculate probabilities for real data like experimental measurement errors in physics.

[Mustafa @oprydai]

study integration;

1. integration isn’t “area under the curve”, it’s inversion of change. differentiation fractures motion into local slope; integration rebuilds the global structure from infinitesimal pieces. If you don’t internalize this duality, and every calculus technique feels like memorization.

2. studying integration means studying accumulation rules. Substitution rewrites the coordinate system. parts decomposes the integrand into a derivative - antiderivative pair. partial fractions exposes hidden linear structure. every method is just a different way of forcing the function to reveal its generator.

3. real mastery comes when you see integrals as operators, not exercises. they map functions into new functions, preserve some properties, distort others, and encode invariants in physics, probability, geometry. once you think at the operator level, the symbolic grind vanishes and the subject becomes a language for reconstructing continuous systems.

[SpaceWeatherNews @SunWeatherMan]

THE GEOID

[Unique Wallpaper @just_unique0]

New wallpaper

[Pulaski Wallpaper @Pulaxki]

Mobile Version

[Pulaski Wallpaper @Pulaxki]

New Drop!

[Pulaski Wallpaper @Pulaxki]

Mobile Version

[Pulaski Wallpaper @Pulaxki]

Wallpaper

[طراد الأسمري @alasmari]

القطط تمتلك حاسة سادسة تشريحية تقع في سقف حلقها تُدعى "عضو جاكوبسون" (Jacobson's Organ / Vomeronasal Organ)، فعندما تجد قطك يشم شيئاً ما، ثم يفتح فمه فجأة ويثبت على وضعية "الابتسامة الغريبة" مع رفع شفته العليا وتجميد عينيه، فهذه ليست بلاهة، ��ل هي استجابة فليمن (Flehmen Response). ، لأن القط في هذه اللحظة يغلق مجرى التنفس العادي ويق��م بـ "حبس الهواء وتذوق الرائحة كيميائياً"، حيث يسحب الجزيئات الفيرمونية الطيارة عبر شق خلف أسنانه الأمامية ويوجهها مباشرة إلى عضو جاكوبسون المرتبط عصبياً بمراكز تحليل السلوك الجنسي والإقليمي في الدماغ، مما يتيح له قراءة البيانات البيولوجية الدقيقة للبيئة بدقة لا يمكن للأنف المجرد رصدها.
🐾 #باستيت

[𝓟𝓻𝓲𝓷𝓬𝓮𝓼𝓪 🐸 @princesalamwane]

Le niveau de bien-être que procure une séance de caresses à un chat est tellement sous-estimé purée

[Fabien @Fabien_Mikol]

Le mathématicien @nasqret a tenté pendant 3 ans de prouver une conjecture... qui fut finalement réfutée par ChatGPT-5.6. "This whole experience made me realize how important it has become in my scientific life to use high quality models"
A l'opposé, Villani revendique s'en passer