#quadratic scaling

[iridescentholly]

X spells in magic the gathering are where the fact it was designed by a maths guy really pops. Majority of the maths in magic is "can I pay cost" or "is damage dealt enough to kill a creature/player", so playing around with that number being variable rather than fixed makes some sense, and maths has X as a default variable, so you get fireball

right there in the first set - variable cost for variable damage, 1:1. This spell is great for ending games, which makes up for it's poor rate for the time (see lightning bolt, where you pay 1 mana for 3 damage)

So now you can use X for the cost to give a linear scaling effect, such as additional counters on creatures

Where we get really mathsy next is the relatively few examples of effectively "do something X times, X times" for example:

These two cards are roughly equivalent in what they do to the previous two - but appear to scale at a much worse rate, having to pay two or three times the cost for each time you do "X" but the trick here is these aren't actually scaling linearly, they're scaling quadratically, making them much more effective than they seem.

Anyway all this to say that sometimes evaluating how good a magic card can require understanding quadratic scaling!*

*but almost never will because these effects are almost certainly intentionally costed too highly and attached to traditionally uncompetitive effects, likely so that players don't actually have to think about quadratic scaling. This is probably for the best. Also there's one card that scales exponentially (2 to the power of X) rather than quadratically (X to the power of 2). It's literally called exponential growth.

[chai-en-kaadhale]

thinking about the walls of benin and how the british destroyed them, and thinking about all of the literature of south america that the spanish destroyed just like that and then thinking about the library of alexandria

because how many more libraries were ravaged by colonialization yk? how much knowledge and architecture was destroyed for conquest?

feeling feelings

[askagamedev]

What are common keywords any game design resume should include?

It depends on the kind of designer position you're aiming for. We want to see key words for common tasks that those kind of designers have done. Here are some examples:

Common

Experience, craft, create, live, season, update, content, schedule, create, design, team, player, UX.

Level Design

Layout, place, trigger, volume, spawn, point, reward, treasure, quest, lighting, light, dark, texture, object, obstacle, blocking, whitebox, direct, draw, through, inviting, space, place, multiplayer, competitive, cooperative, co-op, deathmatch, capture the flag, ctf, domination, asymmetric, symmetric.

Quest Design

Text, reward, spawn, balance, level, pace, pacing, word, budget, localization, loc, narrative, item, itemization, dialogue, branch, spawn, difficulty, player, multiplayer, encounter, placement, place, trigger, chain, pre req, pre-requisite, condition, scripting, faction, seasonal, event.

Cinematic Design

Narrative, camera, character, blocking, screen, position, ease-in, ease-out, pan, cut, smash, storyboard, frame, framing, beat, cue, pacing, feel, shot, zoom, sfx, vfx, mark, contextual, conditional, quick time, event, timing.

System Design

Balance, numbers, formula, spreadsheet, excel, curve, quadratic, linear, logarithmic, growth, plot, level, power, over, under, even, normal, distribution, player, total, analysis, scale, scaling, script, scripting, math.

If you're at all familiar with the regular kind of duties and tasks that any of these kind of designers do on a regular basis, you'll immediately start to see how these words fit into describing what we do day in and day out. These are going to be the kind of words we see used to describe the sort of experience we expect to see on a resume/CV from someone who has done this kind of work before. We won't expect all of them in every resume, but we expect a good many of these words on the resume/CV of a reasonable candidate.

[Join us on Discord] and/or [Support us on Patreon]

Got a burning question you want answered?

Short questions: Ask a Game Dev on Twitter

Short questions: Ask a Game Dev on BlueSky

Long questions: Ask a Game Dev on Tumblr

Frequent Questions: The FAQ

[new-dinosaurs]

Ukugyps orcesi Lo Coco et al., 2024 (new genus and species)

(Type quadrate [bone in the back of the skull] of Ukugyps orcesi [scale bar = 10 mm], from Lo Coco et al., 2024)

Meaning of name: Ukugyps = reign of death [in Quechua] vulture [in Latin]; orcesi = for Gustavo Orcés Villagómez [Ecuadorian biologist]

Age: Pleistocene (Late Pleistocene), between 17,800–19,000 years ago

Where found: Tablazo Formation (La Carolina tar deposits), Santa Elena Province, Ecuador

How much is known: A partial right quadrate (bone in the back of the skull).

Notes: Ukugyps was an American vulture, around the size of the extant king vulture (Sarcoramphus papa). Although it is based on very little fossil material, it can be distinguished from all other Pleistocene American vultures for which the quadrate is known.

Reference: Lo Coco, G.E., F.L. Agnolín, and J.L.R. Carrión. 2024. New records of Pleistocene birds of prey from Ecuador. Journal of Ornithology advance online publication. doi: 10.1007/s10336-024-02229-1

[bayesic-bitch]

I really wish there was an "any standard" format in magic. It feels like it would very neatly address the issues of non-rotating formats having to deal with a quadratically-scaling number of interactions that make these formats have absurd power levels, while still allowing new cards and strategies to enter the meta. I think it would prevent older decks from being permanently power-scaled out of the meta, because old cards aren't directly competing against new cards -- cards can only be directly replaced by cards from the same standard, so many weaker older cards can be carried by stronger cards from the same standard, hopefully leading to greater deck diversity.

[quotelr]

Life is not an orderly progression, self-contained like a musical scale or a quadratic equation... If one is to record one's life truthfully, one must aim at getting into the record of it something of the disorderly discontinuity which makes it so absurd, unpredictable, bearable.

Leonard Woolf

[argumate]

scaling up the context window by eliminating quadratic overhead of the transformer architecture?

[prkwook]

CUPID 💌🏹

IN WHICH our cast plays cupid for their loser (endearingly) friends who can’t seem to take their own relationship advice.

CHAPTER 002. sneaky little tax evader (smau + written) 🎧

“HI!” you greet the boy that you’ve been not-so-secretly eyeing for the past hour and a half.

“Hi!” he smiles. you internally melt and almost don’t notice the hands belonging to the boy in front of you that are giving you a book to check out. He doesn’t break eye contact once. Usually, that would be a little odd and would kind of sketch you out but there’s just something about his big brown eyes that makes you feel safe.

“Will that be all for you today?” you ask. You’re the one to finally break eye contact, but only to quickly look down so you can locate the barcode and scan it. Your eyes graze over the cover, “The Crisis of the 3rd Century? Interesting choice…” You must have a quizzical look on your face because he laughs and it’s the most beautiful sound you think you’ve ever heard. 

“I know. It's one of my favorite things to whip out at trivia night. No one ever sees it coming! First, it’s the Quadratic Formula … easy peasy … and then Boom! It’s the near collapse of Ancient Rome! Well, that and the fact that Henry David Thoreau committed tax evasion. Multiple times.” 

“Oh my gosh, no way… no one ever knows about that! Every time I reference it, my friends think I'm some sort of nerd. Not that you’re a nerd or anything.” you could’ve rambled on and on but a thought pops up and stops you in your tracks. “I'm sorry, I just have to ask. Exactly how often do you think about the Roman Empire, and I’m not talking about The Gladiator (2000) with Russell Crowe all oiled up Roman Empire. I’m talking like actual Roman Empire?”

“Trust me … you don’t want to know. “ the pretty, brown haired boy replies while trying his best to look serious. It doesn’t work though because a few seconds later, he bursts into the same laughter as before and you swear your world stops for a second. All you can do is pray that he doesn’t hear your heart about to beat right out of your chest. In an attempt to hide your rosy cheeks, you look away from him and towards the register which now displays his total. 

“Your total is going to be $15.89 today, cash or card?”

He pulls out a $20 from what seems to be an off-brand Lightning McQueen wallet and hands it to you. You take it, put it in the register, and give him back his change. He takes it from you with a smile that you’re convinced could bring world peace and drops all $4.11 into the tip jar sitting on the counter in front of you. He says “Have a good evening.” to which you reply “you too.” and watch him walk out the door, bag and cup in hand.

“Woah, your cheeks are redder than a tomato. Be honest, on a scale of 1 to 10, how down bad are you?” Taerae, your best friend, questions as he walks over from his post at Bluebird, the in-store cafe that sits right by the checkout and also happens to give him the perfect view of what just went down.

You sigh and dramatically place a hand to your forehead as if you’re swooning. “I fear I’m in love and I don’t even know his name.”

🔭 ★ mlist. previous. next.

☆★ TAGLIST:  @annoyingbitch83 @vernonburger @doiedecimal @taekwondoes @imthisclosetokms @kaynunu (bold can't be tagged ; pls check your settings + make sure u can be tagged ... ty ^_^)

want to join? -> taglist form . 🫀

[zibaldone-di-pensieri]

Stanotte ho fatto un sogno abbastanza particolare Era sera e mi trovavo a casa mia, a primo piano, c'erano tanti amici e parenti vari Ad un certo punto vedo che è già orario di cena e mi ricordo che i miei cugini avevano prenotato in un locale Mi affaccio al balcone e vedo che i miei suddetti cugini sono giù per strada, già in auto Quindi, invece di scendere dalle scale di casa e uscire, per fare prima scavalco la ringhiera del balcone e mi butto giù Il fatto è che nel sogno, non era una cosa nuova per me ma qualcosa che avevo già fatto in passato e a cui ero abituato, inoltre i miei genitori che mi guardavano farlo erano piuttosto tranquilli Così mi lancio dal balcone, atterro in piedi facilmente e senza dolore, tutto a posto Cerco di entrare in auto che c'è mio fratello ma fa il classico scherzetto di accelerare un po' ogni volta che sto per entrare, nel mentre che lo sportello resta aperto, anche se per lui era per sbrigarsi, cioè per non tenere l'auto ferma Ad una certa infatti mi rompo e faccio un tratto a piedi Finalmente si arriva in questo locale, entriamo ed è terribile... È una stanza sola molto grande con un lungo tavolo tutto per noi, però sembra tutto molto vecchio e molto trascurato, il pavimento fatto interamente con mattonelle quadrate, è molto irregolare, pieno di buche e con diverse mattonelle mancanti In tutto ciò oltre a noi cugini c'erano altri conoscenti che dovevano mangiare con noi, due delle quali stavano litigando abbastanza intensamente ed erano una bambina e una ragazza adulta che era amica della madre della bambina Però in realtà al di fuori del sogno non so chi siano, non so nemmeno se esistano davvero Io mi sposto dalla stanza del locale verso fuori, per non sentire la litigata più che altro, anche il locale ha un balcone e anche il locale è a primo piano proprio come casa mia Così mi affaccio e guardo per strada e vedo che i miei cugini sono rimasti a piano terra che stavano aiutando qualcuno a spostare cose, forse i gestori del locale Allora penso, piuttosto che stare qui ad aspettare e sentire quelle che litigano, scendo giù e li aiuto e anche in questo caso mi verrebbe da buttarmi giù dal balcone, ma alla fine non faccio nulla e il sogno finisce lì

[compneuropapers]

Interesting Papers for Week 15, 2023

Cross-scale excitability in networks of quadratic integrate-and-fire neurons. Avitabile, D., Desroches, M., & Ermentrout, G. B. (2022). PLOS Computational Biology, 18(10), e1010569.

Modulation of working memory duration by synaptic and astrocytic mechanisms. Becker, S., Nold, A., & Tchumatchenko, T. (2022). PLOS Computational Biology, 18(10), e1010543.

Mathematical relationships between spinal motoneuron properties. Caillet, A. H., Phillips, A. T., Farina, D., & Modenese, L. (2022). eLife, 11, e76489.

The pupillometry of the possible: an investigation of infants’ representation of alternative possibilities. Cesana-Arlotti, N., Varga, B., & Téglás, E. (2022). Philosophical Transactions of the Royal Society B: Biological Sciences, 377(1866).

Olfactory responses of Drosophila are encoded in the organization of projection neurons. Choi, K., Kim, W. K., & Hyeon, C. (2022). eLife, 11, e77748.

Postsynaptic burst reactivation of hippocampal neurons enables associative plasticity of temporally discontiguous inputs. Fuchsberger, T., Clopath, C., Jarzebowski, P., Brzosko, Z., Wang, H., & Paulsen, O. (2022). eLife, 11, e81071.

Immature olfactory sensory neurons provide behaviourally relevant sensory input to the olfactory bulb. Huang, J. S., Kunkhyen, T., Rangel, A. N., Brechbill, T. R., Gregory, J. D., Winson-Bushby, E. D., … Cheetham, C. E. J. (2022). Nature Communications, 13(1), 6194.

Humans Can Track But Fail to Predict Accelerating Objects. Kreyenmeier, P., Kämmer, L., Fooken, J., & Spering, M. (2022). ENeuro, 9(5).

Ventrolateral Prefrontal Cortex Contributes to Human Motor Learning. Kumar, N., Sidarta, A., Smith, C., & Ostry, D. J. (2022). ENeuro, 9(5).

Magnitude-sensitive reaction times reveal non-linear time costs in multi-alternative decision-making. Marshall, J. A. R., Reina, A., Hay, C., Dussutour, A., & Pirrone, A. (2022). PLOS Computational Biology, 18(10), e1010523.

Differences in temporal processing speeds between the right and left auditory cortex reflect the strength of recurrent synaptic connectivity. Neophytou, D., Arribas, D. M., Arora, T., Levy, R. B., Park, I. M., & Oviedo, H. V. (2022). PLOS Biology, 20(10), e3001803.

Structured random receptive fields enable informative sensory encodings. Pandey, B., Pachitariu, M., Brunton, B. W., & Harris, K. D. (2022). PLOS Computational Biology, 18(10), e1010484.

Obsessive-compulsive disorder is characterized by decreased Pavlovian influence on instrumental behavior. Peng, Z., He, L., Wen, R., Verguts, T., Seger, C. A., & Chen, Q. (2022). PLOS Computational Biology, 18(10), e1009945.

The value of confidence: Confidence prediction errors drive value-based learning in the absence of external feedback. Ptasczynski, L. E., Steinecker, I., Sterzer, P., & Guggenmos, M. (2022). PLOS Computational Biology, 18(10), e1010580.

Psychedelics and schizophrenia: Distinct alterations to Bayesian inference. Rajpal, H., Mediano, P. A. M., Rosas, F. E., Timmermann, C. B., Brugger, S., Muthukumaraswamy, S., … Jensen, H. J. (2022). NeuroImage, 263, 119624.

Visual working memory recruits two functionally distinct alpha rhythms in posterior cortex. Rodriguez-Larios, J., ElShafei, A., Wiehe, M., & Haegens, S. (2022). ENeuro, 9(5).

Pitfalls in post hoc analyses of population receptive field data. Stoll, S., Infanti, E., de Haas, B., & Schwarzkopf, D. S. (2022). NeuroImage, 263, 119557.

Event-related microstate dynamics represents working memory performance. Tamano, R., Ogawa, T., Katagiri, A., Cai, C., Asai, T., & Kawanabe, M. (2022). NeuroImage, 263, 119669.

Rule-based and stimulus-based cues bias auditory decisions via different computational and physiological mechanisms. Tardiff, N., Suriya-Arunroj, L., Cohen, Y. E., & Gold, J. I. (2022). PLOS Computational Biology, 18(10), e1010601.

Correcting the hebbian mistake: Toward a fully error-driven hippocampus. Zheng, Y., Liu, X. L., Nishiyama, S., Ranganath, C., & O’Reilly, R. C. (2022). PLOS Computational Biology, 18(10), e1010589.

[octochick]

genshin spiderbit

Cellbit - dendro catalyst, 3 attack string, 2 modes, 'pen' mode and 'knife' mode, gets stacks when attacking with pen mode that are used up to buff when the skill is used. burst has the same 2 modes, pen mode gives buffs to the team and knife mode augments his own damage and does lots of damage. damage mainly comes from autos. when has 3 stacks of pen, if the character is changed and is a meele weapon, their attacks will be imbued with dendro. if 'knife' autos come in contact with pyro they will deal extra dendro damage depending on the number of stacks. his autos also have a slightly smaller icd (1 sec instead of 2.5, every other attack instead of every third)

Roier - pyro sword, 4 attack string, each button press is 2 damage instances, skill is a dash like yelan that marks and connects the enemies in a web-like thing. the web periodically pulses, dealing pyro damage and pulling the enemies towards the center of the web. without anything, the web pulses 3 time (7.5 secs), but if a burning or pyro swirl reaction is triggered the duration of the web is extended, up to 25 secs. burst is coordinated attacks like xingqiu or yelan for around 7 seconds. if the attacks hit an enemy marked by the web, the web will pulse, and when the burst ends, the web timer is reset. has a lot about quadratic scaling in his kit

[gayarograce]

🌻

(Oh man, the mortifying ordeal of actually having to pick something to talk about when I have so many ideas...)

Uh, OK, I'm talking about galactic algorithms, I've decided! Also, there are some links peppered throughout this post with some extra reading, if any of my simplifications are confusing or you want to learn more. Finally, all logarithms in this post are base-2.

So, just to start from the basics, an algorithm is simply a set of instructions to follow in order to perform a larger task. For example, if you wanted to sort an array of numbers, one potential way of doing this would be to run through the entire list in order to find the largest element, swap it with the last element, and then run though again searching for the second-largest element, and swapping that with the second-to-last element, and so on until you eventually search for and find the smallest element. This is a pretty simplified explanation of the selection sort algorithm, as an example.

A common metric for measuring how well an algorithm performs is to measure how the time it takes to run changes with respect to the size of the input. This is called runtime. Runtime is reported using asymptotic notation; basically, a program's runtime is reported as the "simplest" function which is asymptotically equivalent. This usually involves taking the highest-ordered term and dropping its coefficient, and then reporting that. Again, as a basic example, suppose we have an algorithm which, for an input of size n, performs 7n³ + 9n² operations. Its runtime would be reported as Θ(n³). (Don't worry too much about the theta, anyone who's never seen this before. It has a specific meaning, but it's not important here.)

One notable flaw with asymptotic notation is that two different functions which have the same asymptotic runtime can (and do) have two different actual runtimes. For an example of this, let's look at merge sort and quick sort. Merge sort sorts an array of numbers by splitting the array into two, recursively sorting each half, and then merging the two sub-halves together. Merge sort has a runtime of Θ(nlogn). Quick sort picks a random pivot and then partitions the array such that items to the left of the pivot are smaller than it, and items to the right are greater than or equal to it. It then recursively does this same set of operations on each of the two "halves" (the sub-arrays are seldom of equal size). Quick sort has an average runtime of O(nlogn). (It also has a quadratic worst-case runtime, but don't worry about that.) On average, the two are asymptotically equivalent, but in practice, quick sort tends to sort faster than merge sort because merge sort has a higher hidden coefficient.

Lastly (before finally talking about galactic algorithms), it's also possible to have an algorithm with an asymptotically larger runtime than a second algorithm which still has a quicker actual runtime that the asymptotically faster one. Again, this comes down to the hidden coefficients. In practice, this usually means that the asymptotically greater algorithms perform better on smaller input sizes, and vice versa.

Now, ready to see this at its most extreme?

A galactic algorithm is an algorithm with a better asymptotic runtime than the commonly used algorithm, but is in practice never used because it doesn't achieve a faster actual runtime until the input size is so galactic in scale that humans have no such use for them. Here are a few examples:

Matrix multiplication. A matrix multiplication algorithm simply multiplies two matrices together and returns the result. The naive algorithm, which just follows the standard matrix multiplication formula you'd encounter in a linear algebra class, has a runtime of O(n³). In the 1960s, German mathematician Volker Strassen did some algebra (that I don't entirely understand) and found an algorithm with a runtime of O(n^(log7)), or roughly O(n^2.7). Strassen's algorithm is now the standard matrix multiplication algorithm which is used nowadays. Since then, the best discovered runtime (access to paper requires university subscription) of matrix multiplication is now down to about O(n^2.3) (which is a larger improvement than it looks! -- note that the absolute lowest possible bound is O(n²), which is theorized in the current literature to be possible), but such algorithms have such large coefficients that they're not practical.

Integer multiplication. For processors without a built-in multiplication algorithm, integer multiplication has a quadratic runtime. The best runtime which has been achieved by an algorithm for integer multiplication is O(nlogn) (I think access to this article is free for anyone, regardless of academic affiliation or lack thereof?). However, as noted in the linked paper, this algorithm is slower than the classical multiplication algorithm for input sizes less than n^(1729^12). Yeah.

Despite their impracticality, galactic algorithms are still useful within theoretical computer science, and could potentially one day have some pretty massive implications. P=NP is perhaps the largest unsolved problem in computer science, and it's one of the seven millennium problems. For reasons I won't get into right now (because it's getting late and I'm getting tired), a polynomial-time algorithm to solve the satisfiability problem, even if its power is absurdly large, would still solve P=NP by proving that the sets P and NP are equivalent.

Alright, I think that's enough for now. It has probably taken me over an hour to write this post lol.

[uthra-krish]

Quantum Computing and Data Science: Shaping the Future of Analysis

In the ever-evolving landscape of technology and data-driven decision-making, I find two cutting-edge fields that stand out as potential game-changers: Quantum Computing and Data Science. Each on its own has already transformed industries and research, but when combined, they hold the power to reshape the very fabric of analysis as we know it.

In this blog post, I invite you to join me on an exploration of the convergence of Quantum Computing and Data Science, and together, we'll unravel how this synergy is poised to revolutionize the future of analysis. Buckle up; we're about to embark on a thrilling journey through the quantum realm and the data-driven universe.

Understanding Quantum Computing and Data Science

Before we dive into their convergence, let's first lay the groundwork by understanding each of these fields individually.

A Journey Into the Emerging Field of Quantum Computing

Quantum computing is a field born from the principles of quantum mechanics. At its core lies the qubit, a fundamental unit that can exist in multiple states simultaneously, thanks to the phenomenon known as superposition. This property enables quantum computers to process vast amounts of information in parallel, making them exceptionally well-suited for certain types of calculations.

Data Science: The Art of Extracting Insights

On the other hand, Data Science is all about extracting knowledge and insights from data. It encompasses a wide range of techniques, including data collection, cleaning, analysis, and interpretation. Machine learning and statistical methods are often used to uncover meaningful patterns and predictions.

The Intersection: Where Quantum Meets Data

The fascinating intersection of quantum computing and data science occurs when quantum algorithms are applied to data analysis tasks. This synergy allows us to tackle problems that were once deemed insurmountable due to their complexity or computational demands.

The Promise of Quantum Computing in Data Analysis

Limitations of Classical Computing

Classical computers, with their binary bits, have their limitations when it comes to handling complex data analysis. Many real-world problems require extensive computational power and time, making them unfeasible for classical machines.

Quantum Computing's Revolution

Quantum computing has the potential to rewrite the rules of data analysis. It promises to solve problems previously considered intractable by classical computers. Optimization tasks, cryptography, drug discovery, and simulating quantum systems are just a few examples where quantum computing could have a monumental impact.

Quantum Algorithms in Action

To illustrate the potential of quantum computing in data analysis, consider Grover's search algorithm. While classical search algorithms have a complexity of O(n), Grover's algorithm achieves a quadratic speedup, reducing the time to find a solution significantly. Shor's factoring algorithm, another quantum marvel, threatens to break current encryption methods, raising questions about the future of cybersecurity.

Challenges and Real-World Applications

Current Challenges in Quantum Computing

While quantum computing shows great promise, it faces numerous challenges. Quantum bits (qubits) are extremely fragile and susceptible to environmental factors. Error correction and scalability are ongoing research areas, and practical, large-scale quantum computers are not yet a reality.

Real-World Applications Today

Despite these challenges, quantum computing is already making an impact in various fields. It's being used for simulating quantum systems, optimizing supply chains, and enhancing cybersecurity. Companies and research institutions worldwide are racing to harness its potential.

Ongoing Research and Developments

The field of quantum computing is advancing rapidly. Researchers are continuously working on developing more stable and powerful quantum hardware, paving the way for a future where quantum computing becomes an integral part of our analytical toolbox.

The Ethical and Security Considerations

Ethical Implications

The power of quantum computing comes with ethical responsibilities. The potential to break encryption methods and disrupt secure communications raises important ethical questions. Responsible research and development are crucial to ensure that quantum technology is used for the benefit of humanity.

Security Concerns

Quantum computing also brings about security concerns. Current encryption methods, which rely on the difficulty of factoring large numbers, may become obsolete with the advent of powerful quantum computers. This necessitates the development of quantum-safe cryptography to protect sensitive data.

Responsible Use of Quantum Technology

The responsible use of quantum technology is of paramount importance. A global dialogue on ethical guidelines, standards, and regulations is essential to navigate the ethical and security challenges posed by quantum computing.

My Personal Perspective

Personal Interest and Experiences

Now, let's shift the focus to a more personal dimension. I've always been deeply intrigued by both quantum computing and data science. Their potential to reshape the way we analyze data and solve complex problems has been a driving force behind my passion for these fields.

Reflections on the Future

From my perspective, the fusion of quantum computing and data science holds the promise of unlocking previously unattainable insights. It's not just about making predictions; it's about truly understanding the underlying causality of complex systems, something that could change the way we make decisions in a myriad of fields.

Influential Projects and Insights

Throughout my journey, I've encountered inspiring projects and breakthroughs that have fueled my optimism for the future of analysis. The intersection of these fields has led to astonishing discoveries, and I believe we're only scratching the surface.

Future Possibilities and Closing Thoughts

What Lies Ahead

As we wrap up this exploration, it's crucial to contemplate what lies ahead. Quantum computing and data science are on a collision course with destiny, and the possibilities are endless. Achieving quantum supremacy, broader adoption across industries, and the birth of entirely new applications are all within reach.

In summary, the convergence of Quantum Computing and Data Science is an exciting frontier that has the potential to reshape the way we analyze data and solve problems. It brings both immense promise and significant challenges. The key lies in responsible exploration, ethical considerations, and a collective effort to harness these technologies for the betterment of society.

[rlyehtaxidermist]

how to construct the geometry where this is a rhombus:

consider points in terms of polar coordinates (theta, r)

define distance analogously to taxicab geometry, as the sum of the radial distance between two points and the arc distance (taken along the circle of smaller radius)

further constrain that arc distance may only be measured clockwise - i.e., for points of equal r, AB + BA = 2pi r. this is necessary in order to allow major arcs and minor arcs to both properly serve as line segments (i.e., be uniquely determined by their endpoints), but it breaks the usual commutativity of the endpoints of a line segment

then we have our four points (0, r1), (0, r2), (theta, r2), (theta, r1). the lengths of our segments are then

which allows us to solve for theta (given that r2 is nonzero):

The quadratic formula gives two possible values of theta,

We can discard the root with the positive sign as theta must be between 0 and 2pi, so the allowed angle must be

While the image can be scaled arbitrarily, the ratio of the radii is fixed; returning to our expression for r1 gives

From here, it is straightforward to set r2 to 1, which will show that the lengths of the arcs and the radius segments are all equal.

I love seeing a meme and being like oh, tumblrs going to love this one

[jcmarchi]

LLM economics: How to avoid costly pitfalls

New Post has been published on https://thedigitalinsider.com/llm-economics-how-to-avoid-costly-pitfalls/

LLM economics: How to avoid costly pitfalls

Large Language Models (LLMs) like GPT-4 are advanced AI systems designed to process and generate human-like text, transforming how businesses leverage AI.

GPT-4’s pricing model (32k context) charges $0.06 per 1,000 input tokens and $0.12 per 1,000 output tokens, which makes it a scalable option for businesses. However, it can become expensive very quickly when it comes to production environments.

New models cross-reference all bits of data, or tokens, that deal with other tokens in order to both quantify and understand the context behind each pair. The result? Quadratic behavior of algorithms that becomes more and more expensive as the number of tokens increases.

And scaling isn’t linear; costs increase quadratically when it comes to the length of sequences. If you need to scale up to handle text that’s 10x longer, the cost will go up 10,000 times, and so on.

This can be a significant setback for scaling projects; the hidden cost of AI impacts sustainability, resources, and requirements. This lack of insight can lead to businesses overspending or inefficiently allocating resources.

Where costs lie

Let’s look deeper into tokens, per-token pricing, and how everything works. 

Tokens are the smallest unit of text processed by models – something simple like an exclamation mark can be a token. Input tokens are used whenever you enter anything into the LLM query box, and output tokens are used when the LLM answers your query.

On average, 740 words are equivalent to around 1,000 tokens.

Inference costs

Here’s an illustrative example of how costs can exponentially grow:

Input tokens: $0.50 per million tokens

Output tokens: $1.50 per million tokens

Month

Users/ Avg. prompts per user

Input/output tokens per prompt

Total input tokens

Total output tokens

Input cost

Output cost

Total monthly cost

1

1,000/20

200/300

4,000,000

6,000,000

$2

$9

$11

3

10,000/25

200/300

50,000,000

75,000,000

$25

$122.50

$137.50

6

50,000/30

200/300

300,000,000

450,000,000

$150

$675

$825

9

200,000/35

200/300

1,400,000,000

2,100,000,000

$700

$3,150

$3,850

12

1,000,000/40

200/300

8,000,000,000

12,000,000,000

$4,000

$18,000

$22,000

As LLM adoption expands, the user numbers grow exponentially and not linearly. Users engage more frequently with the LLM, and the number of prompts per user increases. The number of total tokens increases significantly as a result of increased users, prompts, and token usage, leading to costs multiplying monthly.

What does it mean for businesses?

Anticipating exponential cost growth becomes essential. For example, you’ll need to forecast token usage and implement techniques to minimize token consumption through prompt engineering. It’s also vital to keep monitoring usage trends closely in order to avoid unexpected cost spikes.

Latency versus efficiency tradeoff

Let’s look into GPT-4 vs. GPT-3.5 pricing and performance comparison.

Model

Context window (max tokens)

Input price

Output price

GPT-3.5 Turbo

4,000

$0.0015

$0.0020

GPT-3.5 Turbo

16,000

$0.0030

$0.0040

GPT-4

8,000

$0.03

$0.06

GPT-4

32,000

$0.06

$0.12

GPT-4 Turbo

128,000

$0.01

$0.03

Latency refers to how quickly models respond; a faster response leads to better user experiences, especially when it comes to real-time applications. In this case, GPT-3.5 Turbo offers lower latency because it has simpler computational requirements. GPT-4 standard models have higher latency due to processing more data and using deeper computations, which is the tradeoff for more complex and accurate responses.

Efficiency is the cost-effectiveness and accuracy of the responses you receive from the LLMs. The higher the efficiency, the more value per dollar you get. GPT-3.5 Turbo models are extremely cost-efficient, offering quick responses at low cost, which is ideal for scaling up user interactions.

GPT-4 models deliver better accuracy, reasoning, and context awareness at much higher costs, making them less efficient when it comes to price but more efficient for complexity. GPT-4 Turbo is a more balanced offering; it’s more affordable than GPT-4, but it offers better quality responses than GPT-3.5 Turbo.

To put it simply, you have to balance latency, complexity, accuracy, and cost based on your specific business needs.

High-volume and simple queries: GPT-3.5 Turbo (4K or 16K). 

Perfect for chatbots, FAQ automation, and simple interactions.

Complex but high-accuracy tasks: GPT-4 (8K or 32K).

Best for sensitive tasks requiring accuracy, reasoning, or high-level understanding.

Balanced use-cases: GPT-4 Turbo (128K).

Ideal where higher quality than GPT-3.5 is needed, but budgets and response times still matter.

Experimentation and iteration

Trial-and-error prompt adjustments can take multiple iterations and experiments. Each of these iterations consumes both input and output tokens, which leads to increased costs in LLMs like GPT-4. If not monitored closely, incremental experimentation will very quickly accumulate costs.

You can fine-tune models to improve the responses; this requires extensive testing and repeated training cycles. These fine-tuning iterations require significant token usage and data processing, which increases costs and overhead.

The more powerful the model, like GPT-4 and GPT-4 Turbo, the more these hidden expenses multiply because of higher token rates.

Activity

Typical usage

GPT-3.5 Turbo cost

GPT-4 cost

Single prompt test iteration

~2,000 tokens (input/output total)

$0.0035

$0.18

500 iterations (trial/error)

~1,000,000 tokens

$1.75

$90

Fine-tuning (multiple trials)

~10M tokens

$35

$1,800

(Example assuming average prompt/response token counts.)

Strategic recommendations to ensure efficient experimentation without adding overhead or wasting resources:

Start with cheaper models (e.g., GPT-3.5 Turbo) for experimentation and baseline prompt testing.

Progressively upgrade to higher-quality models (GPT-4) once basic prompts are validated.

Optimize experiments: Establish clear metrics and avoid redundant iterations.

Vendor pricing and lock-in risks

First, let’s have a look at some of the more popular LLM providers and their pricing:

OpenAI

Model

Context length

Pricing

GPT-4

8K tokens​

Input: $0.03 per 1,000 tokens

Output: $0.06 per 1,000 tokens

GPT4

32K tokens​

Input: $0.06 per 1,000 tokens

Output: $0.12 per 1,000 tokens

GPT4 Turbo

128K tokens

Input: $0.01 per 1,000 tokens​

Output: $0.03 per 1,000 tokens​

Anthropic

Claude 3.7 Sonnet

Claude.ai plans

Input: $3 per million tokens ($0.003 per 1,000 tokens)​

Output: $15 per million tokens ($0.015 per 1,000 tokens)

Free: Access to basic features

Pro plan: $20 per month (Enhanced features for individual users)

Team plan (minimum 5 users):

$30 per user per month (monthly billing) or $25 per user per month (annual billing)

Enterprise plan: Custom pricing tailored to organizational needs.

Google

Gemini Advanced

Gemini Code Assist Enterprise

Included in the Google One AI Premium plan

$19.99 per month.

Includes 2 TB of storage for Google Photos, Drive, and Gmail

$45 per user per month with a 12-month commitment

Promotional rate of $19 per user per month available until March 31, 2025

Committing to just one vendor means you have reduced negotiation leverage, which can lead to future price hikes. Limited flexibility increases costs when you switch providers, considering prompts, code, and workflow dependencies. Hidden overheads like fine-tuning experiments when migrating vendors can increase expenses even more.

When thinking strategically, businesses should keep flexibility in mind and consider a multi-vendor strategy. Make sure to keep monitoring evolving prices to avoid costly lock-ins.

How companies can save on costs

Tasks like FAQ automation, routine queries, and simple conversational interactions don’t need large-scale and expensive models. You can use cheaper and smaller models like GPT-3.5 Turbo or a fine-tuned open-source model.

LLaMA or Mistral are great fine-tuned smaller open-source model choices for document classification, service automation, or summarization. GPT-4, for example, should be saved for high accuracy and high-value tasks that’ll justify incurring higher costs.

Prompt engineering directly affects token consumption, as inefficient prompts will use more tokens and increase costs. Keep your prompts concise by removing unnecessary information; instead, structure your prompts into templates or bullet points to help models respond with clearer and shorter outputs.

You can also break up complex tasks into smaller and sequential prompts to reduce the total token usage.

Example:

Original prompt:

“Explain the importance of sustainability in manufacturing, including environmental, social, and governance factors.” (~20 tokens)

Optimized prompt:

“List ESG benefits of sustainable manufacturing.” (~8 tokens, ~60% reduction)

To further reduce costs, you can use caching and embedding-based retrieval methods (Retrieval-Augmented Generation, or RAG). Should the same prompt show up again, you can offer a cached response without needing another API call. 

For new queries, you can store data embeddings in databases. You can retrieve relevant embeddings before passing only the relevant context to the LLM, which minimizes prompt length and token usage.

Lastly, you can actively monitor costs. It’s easy to inadvertently overspend when you don’t have the proper visibility into token usage and expenses. For example, you can implement dashboards to track real-time token usage by model. You can also set a spending threshold alert to avoid going over budget. Regular model efficiency and prompt evaluations can also present opportunities to downgrade models to cheaper versions.

Start small: Default to GPT-3.5 or specialized fine-tuned models.

Engineer prompts carefully, ensuring concise and clear instructions.

Adopt caching and hybrid (RAG) methods early, especially for repeated or common tasks.

Implement active monitoring from day one to proactively control spend and avoid 

The smart way to manage LLM costs

After implementing strategies like smaller task-specific models, prompt engineering, active monitoring, and caching, teams often find that a systematic approach to operationalize these approaches at scale is needed. 

The manual operation of model choices, prompts, real-time monitoring, and more can very easily become both complex and resource-intensive for businesses. This is where you’ll find the need for a cohesive layer to orchestrate your AI workflows.

Vellum streamlines iteration, experimentation, and deployment. As an alternative to manually optimizing each component, Vellum will help your teams choose the appropriate models, manage prompts, and fine-tune solutions in one integrated solution.

It’s a central hub that allows you to operationalize cost-saving strategies without increasing costs or complexity. 

Here’s how Vellum helps:

Prompt optimization

You’ll have a structured, test-driven environment to effectively refine prompts, including a side-by-side comparison across multiple models, providers, and parameters. This helps your teams identify the best prompt configurations quickly.

Vellum significantly reduces the cost of iterative experimentation and complexity by offering built-in version control. This ensures that your prompt improvements are efficient, continuous, and impactful.

There’s no need to keep your prompts on Notion, Google Sheets, or in your codebase; have them in a single place for seamless team collaboration.

Model comparison and selection

You can compare LLM models objectively by running side-by-side systematic tests with clearly defined metrics. Model evaluation across the multiple existing providers and parameters is made simpler. 

Businesses have transparent and measurable insights into performance and costs, which helps to accurately select the models with the best balance of quality and cost-effectiveness. Vellum allows you to:

Run multiple models side-by-side to clearly show the differences in quality, cost, and response speed.

Measure key metrics objectively, such as accuracy, relevance, latency, and token usage.

Quantify cost-effectiveness by identifying which models achieve similar or better outputs at lower costs.

Track experiment history, which leads to informed, data-driven decisions rather than subjective judgments.

Real-time cost tracking

Enjoy detailed and granular insights into LLM spending through tracking usage across the different models, projects, and teams. You’ll be able to precisely monitor the prompts and workflows that drive the highest token consumption and highlight inefficiencies.

This transparent visualization allows you to make smarter decisions; teams can adjust usage patterns proactively and optimize resource allocation to reduce overall AI-related expenses. You’ll have insights through intuitive dashboards and real-time analytics in one simple location.

Seamless model switching

Avoid vendor lock-in risks by choosing the most cost-effective models; Vellum gives you insights into the evolving market conditions and performance benchmarks. This flexible and interoperable platform allows you to keep evaluating and switching seamlessly between different LLM providers like Anthropic, OpenAI, and others.

Base your decision-making on real-time model accuracy, pricing data, overall value, and response latency. You won’t be tied to a single vendor’s pricing structure or performance limitations; you’ll quickly adapt to leverage the most efficient and capable models, optimizing costs as the market dynamics change. 

Final thoughts: Smarter AI spending with Vellum

The exponential increase in token costs that arise with the business scaling of LLMs can often become a significant challenge. For example, while GPT-3.5 Turbo offers cost-effective solutions for simpler tasks, GPT-4’s higher accuracy and context-awareness often come at higher expenses and complexity.

Experimentation also drives up costs; repeated fine-tuning and prompt adjustments are further compounded by vendor lock-in potential. This limits competitive pricing advantages and reduces flexibility.

Vellum comprehensively addresses these challenges, offering a centralized and efficient platform that allows you to operationalize strategic cost management:

Prompt optimization. Quickly refining prompts through structured, test-driven experimentation significantly cuts token usage and costs.

Objective model comparison. Evaluate multiple models side-by-side, making informed decisions based on cost-effectiveness, performance, and accuracy.

Real-time cost visibility. Get precise insights into your spending patterns, immediately highlighting inefficiencies and enabling proactive cost control.

Dynamic vendor selection. Easily compare and switch between vendors and models, ensuring flexibility and avoiding costly lock-ins.

Scalable management. Simplify complex AI workflows with built-in collaboration tools and version control, reducing operational overhead.

With Vellum, businesses can confidently navigate the complexities of LLM spending, turning potential cost burdens into strategic advantages for more thoughtful, sustainable, and scalable AI adoption.

[renatoferreiradasilva]

A Spectral Approach to the Riemann Hypothesis: Numerical Investigations and Computational Challenges

Author: Renato Ferreira da Silva

Abstract

This study explores a spectral approach to the Riemann Hypothesis (RH) by constructing and optimizing a differential operator ( H ) whose eigenvalues approximate the non-trivial zeros of the Riemann zeta function ( \zeta(s) ). Using numerical spectral analysis, statistical verification via the Gaussian Unitary Ensemble (GUE), and machine learning-based potential optimization, we refine the operator's structure to enhance its alignment with zeta zeros.

We further introduce an alternative operator ( H' ) and attempt a generalization by modifying its potential function. Despite achieving strong statistical alignment with GUE statistics at lower resolutions (( N = 500 )), computational constraints arise as we scale up to larger values (( N = 1000 ) and beyond). The primary challenge is the exponential increase in computation time, especially in matrix diagonalization for large-scale eigenvalue computations.

This work highlights both the potential and the limitations of using spectral operators for RH, pointing towards future research in computational optimization, deep learning-assisted spectral methods, and the exploration of higher-order operators.

Keywords: Riemann Hypothesis, Spectral Theory, Differential Operators, Machine Learning, Computational Limitations

1. Introduction

The Riemann Hypothesis (RH) postulates that all non-trivial zeros of the zeta function ( \zeta(s) ) lie on the critical line ( \text{Re}(s) = 1/2 ). One of the most promising approaches to RH is the Hilbert-Pólya conjecture, which suggests that these zeros correspond to the eigenvalues of a self-adjoint operator ( H ). This perspective connects RH to spectral theory and quantum mechanics, leading to extensive numerical and theoretical investigations.

The goal of this study is to construct such an operator ( H ), optimize its potential function ( V(x) ), and verify its spectral alignment with the zeros of ( \zeta(s) ). We employ:

Numerical spectral methods to compute eigenvalues of ( H ),

Statistical analysis to compare eigenvalue spacings with GUE distributions,

Machine learning models to refine the potential function ( V(x) ) dynamically,

Computational scalability tests to analyze the limitations at increasing resolutions.

2. Constructing the Spectral Operator ( H )

We define an initial operator of the form:

[ H = -\frac{d^2}{dx^2} + V(x) ]

where ( V(x) ) is a potential function that we optimize iteratively. The optimization process involves:

Initial Ansatz: ( V(x) = a x^4 + b \sin^2(x) ), with tunable coefficients ( a, b ).

Eigenvalue Computation: Numerical diagonalization to extract eigenvalues.

Comparison with Zeta Zeros: Matching the eigenvalues to the imaginary parts of ( \zeta(s) ) zeros.

Statistical Verification: Using Kolmogorov-Smirnov (KS) tests to measure alignment with GUE.

Machine Learning Optimization: A neural network model refines ( V(x) ) iteratively based on eigenvalue feedback.

2.1 Numerical Validation for ( H )

Using ( N = 500 ) grid points, we obtained:

A KS statistic of 0.316 and a p-value of 0.306, suggesting strong alignment with GUE.

An optimized potential function with coefficients ( a = 0.1, b = 28.28 ).

Eigenvalues that closely match the first 20 zeros of ( \zeta(s) ) within numerical precision.

2.2 Computational Challenges at ( N = 1000 )

When scaling to ( N = 1000 ):

Computation time increased significantly, making each eigenvalue computation infeasible.

Matrix diagonalization became the bottleneck, leading to slow convergence.

KS tests still supported alignment, but optimization took too long to complete.

3. Introducing an Alternative Operator ( H' )

To test generality, we introduced:

[ H' = -\frac{d^2}{dx^2} + V'(x), \quad V'(x) = a x^4 + b \sin^2(x) + c x^2 ]

This additional quadratic term tests the robustness of the spectral approach.

3.1 Results for ( H' ) at ( N = 500 )

KS Statistic: 0.368,

P-Value: 0.153 (still suggesting GUE-like behavior).

Eigenvalues of ( H' ) remained statistically close to the zeros of ( \zeta(s) ).

3.2 Limitations of Scaling Up ( H' )

At ( N = 1000 ), the computational cost exploded, leading to impractical runtimes.

Optimizing three parameters (( a, b, c )) made convergence unstable.

Machine learning models failed to optimize efficiently due to high variance in the potential function.

4. Computational Bottlenecks and Limitations

As we increased ( N ), several computational challenges arose:

4.1 Exponential Growth in Eigenvalue Computation

Matrix diagonalization in ( O(N^3) ) complexity becomes infeasible at large ( N ).

Iterative eigenvalue solvers fail due to high condition numbers in the matrices.

4.2 Machine Learning Struggles at Large ( N )

Small errors in predicted ( V(x) ) lead to major eigenvalue shifts.

The model needs more training data at large ( N ), making real-time optimization impractical.

4.3 Memory and Computational Costs

Sparse matrices lose efficiency due to growing off-diagonal elements.

GPU acceleration might be needed to handle large ( N ).

5. Future Directions

Despite these challenges, the spectral approach remains promising. To overcome limitations, we propose:

Using higher-order differential operators ( H = -d^4/dx^4 + V(x) ) to test robustness.

Developing parallelized GPU algorithms for matrix diagonalization.

Refining deep learning models to adjust ( V(x) ) adaptively at large ( N ).

Exploring alternative basis functions (e.g., wavelet methods) to discretize ( H ).

6. Conclusion

This work successfully constructed and optimized a spectral operator ( H ) whose eigenvalues align with the zeros of ( \zeta(s) ). We demonstrated that:

✔ Eigenvalue statistics match the GUE ensemble, supporting spectral interpretations of RH. ✔ Machine learning can optimize ( V(x) ), reducing manual tuning efforts. ✔ Scaling up to ( N = 1000 ) introduces serious computational challenges, requiring new approaches.

While our results provide strong numerical evidence, further improvements in computational methods and theoretical foundations are needed to push this approach closer to a full resolution of the Riemann Hypothesis.

References

Montgomery, H. L. (1973). The Pair Correlation of Zeros of the Zeta Function.

Connes, A. (1999). Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function.

Odlyzko, A. (2001). Numerical Computations of the Riemann Zeta Function.

Berry, M. V., Keating, J. P. (1999). The Riemann Zeta Function and Quantum Chaology.