Stress states collapse to Mohr circles on the σₙ–σₛ plane. - Hydrostatic pressure appears as a lone point; - Uniaxial compression or tension yields a single circle rooted at the origin; - Confined axial and triaxial fields generate nested circles whose diameters equal principal differences; - Pure shear centers at zero mean stress; - Differential stress is simply σ₁ − σ₃; - Pore pressure p_f shifts every circle leftward to give effective stress. Principal matrices sit beside each plot. From the same constructions, tectonicists forecast crustal faults and mining engineers gauge tunnel stability.

THE MATH NEEDED FOR TRADING (COMPLETE ROADMAP): today I'll will break down the essential math you need for trading & this is the exact roadmap that helped me personally when i started, i thought math was for interviews, two months into live trading i realized every position i took was pure math running in production here's the complete map of what math actually fires on real trades: --------------- 1. statistics and probability every price move is signal plus randomness. statistics separates the two what you need: > mean, median, expected value = EV formula (win% × avg win) - (loss% × avg loss) is what you're actually maximizing > variance and standard deviation = foundation of every position sizing formula, becomes volatility when applied to returns > correlation from -1 to +1 = tells you if strategies are actually independent > correlation 0.9 across 3 strategies = you have one strategy dressed as three > conditional probability = the biggest edge upgrade retail misses. P(win) = 55% unconditionally, but 70% when VIX < 15 > Bayes' theorem = how you update beliefs when new information arrives. never work with static beliefs > central limit theorem = why portfolio-level statistics behave cleaner than individual trades > linear and logistic regression = building blocks for mean reversion and binary prediction --------------- 2. linear algebra the moment you hold multiple positions, you're doing linear algebra whether you know it or not what you need: > scalars, vectors, matrices = your portfolio is a weighted sum of vectors > portfolio variance = w^T Σ w. not the sum of individual variances. one matrix operation > eigenvalues and eigenvectors = reveal where risk actually lives. in a 500-stock universe, top 5 eigenvectors explain 70% of variance. the other 495 are noise > PCA and SVD = reduce 50 correlated indicators into 5 independent factors explaining 90% of variation --------------- 3. time series analysis markets have memory. today's price depends on yesterday's. volatility clusters. trends persist what you need: > stationarity = assumption most statistical tests make, but markets aren't stationary, this is why strategies decay when regime shifts > autocorrelation = positive means momentum, negative means mean reversion, zero means random walk > ARIMA = framework for forecasting returns and volatility > GARCH = formalizes what every trader knows, volatility clusters. after a big move expect more volatility > cointegration = the foundation of pairs trading. two assets can both trend but their spread stays stationary --------------- 4. risk management math edge doesn't matter if you size wrong what you need: > Value at Risk = 95% VaR of $5,000 means 95% of the time you won't lose more, but 5% of the time you might lose much more > Sharpe ratio = (return - risk-free rate) / volatility. institutional threshold is Sharpe > 1.5 before deployment > maximum drawdown = biggest peak-to-trough loss. more intuitive than volatility for most traders > Monte Carlo simulation = randomizes trade sequencing to show the range of possible outcomes > Kelly criterion = f* = (bp - q) / b. professionals use 0.25x to 0.5x fractional Kelly because your true edge is never certain --------------- 5. stochastic calculus (for options) if you trade options, every price on your screen came from a stochastic differential equation what you need: > Black-Scholes = dS = μS dt + σS dW. the underlying follows geometric Brownian motion > Ito's Lemma = why the σ² term exists. this is why gamma exists > Heston stochastic volatility = dv = κ(θ - v)dt + ξ√v dW. captures the volatility smile that Black-Scholes misses > delta hedging = stochastic calculus running in production. every rehedge is dictated by the SDE governing the underlying --------------- MINIMUM TO START you don't need everything above to start for your first backtest: > mean, median, standard deviation > correlation > basic probability > Sharpe ratio and max drawdown start with statistics, that alone separates you from 95% of retail traders --------------- every real trade is math executing in production: > entry = conditional probability > validation = statistics > portfolio = linear algebra > sizing = Kelly optimization > risk = VaR, Sharpe, max drawdown > options = stochastic calculus the traders who make consistent money see markets as continuous equations, everyone else guesses if you're a complete beginner shoot me a DM and I'll share the resources with you MATH IS EVERYTHING <3

Why $WALL3 is still under the radar & imo an extremely undervalued token ; When they launched there was no info on how the token would accrue value & no real timeline on when the treasury would open Robinhood pushes for RWA & agentic trading. @Wall3_RL does both. And they literally require the onchain stock liquidity/infrastructure to improve in order to get their fund onchain. Uniquely aligned goals with @vladtenev @RobinhoodCrypto There are so many external treasuries farming yield/sitting in stablecoins or more recently, idle in onchain stocks. Isn’t it more interesting to direct funds to the first hedge fund onchain? Or just anyone. Run it non custodially in your own wallet. Because that’s what it is. The first onchain hedge fund, doxxed team (founders sold company before & worked at multi billion asset management firms), do your research into them From their disclaimers you can see they are being serious about legal compliance, and from their timing/what they’ve presented so far you can assume they’ve been working on this quite some time. Literally frontrunning everyone. A couple days ago, they released a paper on the utility of their token & here’s a TLDR on it 👇 — How much of the fees go to buybacks The lever is the "support share" (σ): •100% at launch. Every dollar of fee cash left after direct costs goes to buying $WALL3 on the open market and burning it. •Steps down to a floor of 50% over three years — but only once FDV passes $100M or TVL passes $50M, and never before October 2030. On the paper's own modelled path the step-down starts October 2030. •Fees are always charged in USDC, never in the token. Anything a customer chooses to pay in $WALL3 (10% discount) is burned directly, on top of the buyback. •Execution cap: purchases are limited to 25% of daily volume. Whatever the cap can't absorb goes into the pool as protocol-owned liquidity until depth hits 5% of FDV, then it also goes to buybacks. •Governance: holders of the access deposit are "intended" to govern share, floor, trigger and cadence so the parameters can change. The pitch: five fee-generating businesses (own treasury, partner wallets, model portfolios, research seats, a trade-time intelligence API, plus a pooled access vault), all charging traditional hedge-fund style fees ; 1.5% + 10–15% above a high-water mark, or 0.75% + 10% over the S&P for the long-only portfolios. All fees in stablecoins, all routed through one rule into buying and burning the token. The token's job is access: hold 2% of your tracked capital in $WALL3, or pay a 0.5%/yr pass instead. The headline model: $200k of seed treasury in October, modelled TVL of $944M by 2031 and $6.4B by 2036, with $11M and $70.9M of buybacks in those years. —- Not saying they will reach these numbers but even if its a fraction of it, the current price makes $WALL3 seem extremely undervalued to me 2.5M FDV, under 20% of that is circulating I will personally (high likelyhood) park 50-100k in the fund myself

Approximating the Heaviside step function & its derivative with sigmoids Top: Smooth approximations to H(x) = 0 (x < 0), 1 (x > 0) using sigmoid(x; σ) = 1 / (1 + e^{-σ x}) for σ = 1 (orange), 2 (green), 4 (red), 8 (purple). Larger σ → sharper step. Bottom: Their derivatives d/dx [sigmoid] = σ ⋅ sigmoid(x; σ) ⋅ (1 − sigmoid(x; σ)) which converge to the Dirac delta δ(x) as σ → ∞. This Enables gradient-based optimization in neural networks and physics simulations where the true step function is non-differentiable.

Numbers rarely sit exactly at their average - the standard deviation formula shows the typical distance they stray from it. Population: σ = √[∑(xᵢ − μ)² / N] xᵢ = elements in population μ = population mean N = population size Sample: s = √[∑(xᵢ − x̄)² / (n − 1)] xᵢ = elements in sample x̄ = sample mean n = sample size The sample version is applied by bank risk analysts to measure day-to-day swings in currency exchange rates.

bitcoin:native just hit $80K — and the way it got there matters. The +24.8% seven-day run is one of the most extreme weekly moves since 2020: a +2.5σ event, in the top 1% of all weeks. It's happened 7 times before. Two months later: ➡️ 6 of 7 were higher ➡️ +18.3% average return ➡️ shallow pullbacks along the way (median dip just −1.9%) Rare momentum — but to notice it's a small sample. History rhymes, it doesn't repeat.

Backpropagation in a simple neural network. This diagram walks through the full process: > Forward pass with sigmoid activations σ > Mean squared error loss E > Backward pass computing gradients ∂E/∂z, ∂E/∂a, ∂E/∂w, and ∂E/∂b Clearly shows how error flows backward to update weights and biases using the chain rule.

What's the latest with Felix lending? Over the past week, deposits across Felix Vanilla + CDP have grown from $573M to $645M, up by ~12.6%, driven by continued HYPE growth. Stablecoin deposits have remained steady around $95-100M. Borrow demand remains healthy across the platform, with USDC Flagship at 4.32% APY and USDH Flagship at 5.11% APY as of today. Across Felix Vanilla, borrowing has continued to increase alongside that growth. Total borrows now stand at roughly $122M, with 6.7K active positions across Felix lending markets. Most borrow positions remain well-collateralized, with 4,241 positions above 1.35 HF. Only 104 positions currently sit in the 1.00-1.20 HF buffer range, with collateral at risk of liquidation at roughly $2.9M, or about 0.8% of total collateral. As far as DaR + CaR today, we monitor debt at risk and collateral at risk through real time scenario testing across the vault suite. Under a 3σ collateral drawdown, modeled DaR is about $1.58M and CaR about $2.15M. Even a 5σ shock remains fully liquidatable. Our DEX + orderbook monitoring makes sure ≤ 3σ swings do not threaten solvency. Supply utilization across Felix Vanilla sits now over 91% ($122m assets borrowed of $134m assets lent out). If interested in lending to the borrower base on Felix and looking for support, feel free to reach out. More information to come on spot equities borrow/lend.