Published on: 2025-06-16 10:15:45
The Hidden Subgroup Problem Introduction Two of the most famous quantum algorithms are Shor’s Algorithms for integer factorization and discrete logarithms. The significance of these algorithms is that efficient integer factorization breaks the RSA cryptosystem, and efficient discrete logarithm computation breaks the Diffie-Hellman key exchange protocol. It is natural to wonder what other types of problems that are classically considered to be hard can be efficiently solved with a quantum comp
Keywords: chi mathbb mathbf qft rangle
Find related items on AmazonPublished on: 2025-06-21 01:07:17
Methodology Our rendering pipeline uses 3D triangles as primitives, each defined by three learnable 3D vertices, color, opacity, and a smoothness parameter \( \sigma \). The triangles are projected onto the image plane using a standard pinhole camera model with known intrinsics and extrinsics. Instead of binary masks, we introduce a smooth window function that softly modulates the triangle's influence across pixels. This function is derived from the 2D signed distance field (SDF) of the triang
Keywords: function mathbf phi sigma triangle
Find related items on AmazonPublished on: 2025-06-30 17:56:12
The Transwedge Product Introductory texts on geometric algebra often begin by showing how the geometric product is a combination of the wedge product and the dot product, giving us the formula[1] \(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \cdot \mathbf b\).(1) However, the above formula holds only for vectors \(\mathbf a\) and \(\mathbf b\). When \(\mathbf a\) and \(\mathbf b\) are allowed to assume values of higher grade, the geometric product ge
Keywords: geometric mathbf mathbin product unicode
Find related items on AmazonPublished on: 2025-07-25 21:18:20
Autoregressive vs. Diffusion Language Models In the language modeling task, we have a sequence of \( L \) tokens \( \mathbf{x} = (\mathbf{x}^1, \dots, \mathbf{x}^L ) \) drawn from the data distribution \( q(\mathbf{x}) \). We aim to fit a model \( p_\theta(\mathbf{x}) \) of \( q \). Autoregressive models define a factorized distribution of the form: \[ \log p_\theta(\mathbf{x}) = \sum_{\ell=1}^L \log p_\theta(\mathbf{x}^\ell \mid \mathbf{x}^{\lt \ell}) \] However, the sequential dependencies
Keywords: _t ell mathbf p_ theta
Find related items on AmazonPublished on: 2025-08-09 05:18:52
In numerous applications of machine learning, Hessians and Jacobians exhibit sparsity, a property that can be leveraged to vastly accelerate their computation. While the usage of automatic differentiation in machine learning is ubiquitous, automatic sparse differentiation (ASD) remains largely unknown. This post introduces ASD, explaining its key components and their roles in the computation of both sparse Jacobians and Hessians. We conclude with a practical demonstration showcasing the performa
Keywords: ad jacobian mathbf matrix sparsity
Find related items on AmazonPublished on: 2025-08-30 04:30:47
Differentiable Programming from Scratch Differentiable programming has become a hot research topic, and not only due to the popularity of machine learning frameworks like TensorFlow, PyTorch, and JAX. Many fields apart from machine learning are finding differentiable programming to be a useful tool for solving optimization problems. In computer graphics, differentiable rendering, differentiable physics, and neural representations are all gaining popularity. This article received an honorable m
Keywords: frac mathbf node partial prime
Find related items on AmazonPublished on: 2025-09-01 18:01:41
Let's remove Quaternions from every 3D Engine (An Interactive Introduction to Rotors from Geometric Algebra) The clearest explanation of 3D geometric algebra within 15 minutes that I've seen so far —BrokenSymmetry I am sold. While I can understand quaternions to an extent, this way of thinking is a much more intuitive and elegant approach. —Jack Rasksilver This sets a high standard for educational material, and is a shining example of how we can improve education with today's technologies. —Se
Keywords: mathbf product vector vectors wedge
Find related items on AmazonPublished on: 2025-09-04 22:54:51
Monte Carlo Crash Course Sampling In the previous chapter, we assumed that we can uniformly randomly sample our domain. However, it’s not obvious how to actually do so—in fact, how can a deterministic computer even generate random numbers? Pseudo-Random Numbers Fortunately, Monte Carlo methods don’t need truly random numbers. Instead, we can use a pseudo-random number generator (PRNG). A PRNG produces a deterministic stream of numbers that look uniformly random: By “look uniformly random,”
Keywords: f_ frac mathbf mathcal omega
Find related items on AmazonPublished on: 2025-10-07 16:26:05
authored by Premmi and Beguène Previous Topic: An Axiomatic Study of Numbers Introduction Thinking of numbers intuitively brings to mind the simplest and most fundamental set of numbers, namely the set of natural numbers. These numbers are used to count objects like cars, books, pens, etc. If we associate natural numbers such as 1, 2, 3, etc. with counting, then with what corresponding concepts do we relate numbers like -4, \sqrt{3} \text{ and } \frac{22}{7}? To reason about all kinds of num
Keywords: axioms mathbb natural numbers set
Find related items on AmazonPublished on: 2025-10-08 17:26:05
authored by Premmi and Beguène Previous Topic: An Axiomatic Study of Numbers Introduction Thinking of numbers intuitively brings to mind the simplest and most fundamental set of numbers, namely the set of natural numbers. These numbers are used to count objects like cars, books, pens, etc. If we associate natural numbers such as 1, 2, 3, etc. with counting, then with what corresponding concepts do we relate numbers like -4, \sqrt{3} \text{ and } \frac{22}{7}? To reason about all kinds of num
Keywords: axioms mathbb natural numbers set
Find related items on AmazonPublished on: 2025-10-21 00:39:31
A hypernetwork estimates parameters $\{\mathbf{b}_1, \mathbf{W}_2\}^{(i,j)}$ of pixel-wise, local neural heat fields. The phase shifts $\mathbf{b}_1$ operate on globally learned components, before thermal activations scale each component depending on their frequency and the desired scaling factor. The components are then linearly combined using coefficients $\mathbf{W}_2$, resulting in an appropriately-blurred, continuous local neural field. This field is then rasterized at the appropriate sampl
Keywords: _1 components mathbf neural resolution
Find related items on AmazonPublished on: 2025-11-17 12:20:40
MathB.in Is Shutting Down By Susam Pal on 23 Feb 2025 Thirteen years ago, on a quiet Saturday night, I sat down and began developing MathB.in. After coding all through the night, as the sun rose on Sunday, 25 March 2012, the website was ready. I registered a new domain name and shared it with a few friends who loved mathematics. Back then, we spent hours discussing fascinating mathematics problems, and this website became a simple way for us to share snippets with each other. Word spread quick
Keywords: content mathb project server service
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