MediMath Science

 In this blog, I provide a brief introduction to the famous Black–Scholes partial differential equation and examine it from Charlie Munger's critical perspective. While the equation revolutionized option pricing and became a cornerstone of modern quantitative finance, its assumptions may limit its usefulness when applied to long-term stock investing. Before proceeding, I should note that I am not an expert in this field. The Black–Scholes framework begins with the assumption that a stock price follows a stochastic process, $dS=\mu Sdt+\sigma S dW$, where $S$ denotes the stock price, $\mu$ is the expected growth rate, $\sigma$ represents volatility, and $dW$ is Brownian motion. In essence, stock prices are modeled as evolving continuously through a combination of deterministic growth and random uncertainty. Using stochastic calculus, one arrives at $\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0,$ where $V$ i...

I have watched a few recent UFC fights, and the bout between Ilia Topuria and Justin Gaethje in mid-June 2026 was particularly memorable. Most people predicted a victory for Ilia, but the outcome was different from expectations. Both fighters displayed exceptional stamina and endurance, suggesting that they had invested a tremendous amount of time in training. However, some analysts believe that although Ilia preferred high-intensity training camps, he may have absorbed fewer realistic punches in sparring because his training partners were reluctant to engage him at full intensity due to the risk of being knocked out by his formidable punching power. Many critics dismiss mixed martial arts as little more than a modern version of Roman gladiatorial combat.  (The comparison, however, is imperfect. Gladiators were often forced into combat, whereas UFC fighters voluntarily dedicate years of their lives to earning a place among the sport's elite. Countless athletes train relentlessly fo...

This blog examines the AI-related stock market, which has delivered exceptional returns over the past several years and whose long-term prospects remain promising. Despite this favorable outlook, AI-related stocks have recently experienced significant volatility. Much of this reflects the tension between exceptionally high market expectations and uncertainty surrounding AI monetization, compounded by elevated valuations, interest-rate sensitivity, geopolitical risks, and export controls. The objective of this blog is to develop a framework for understanding the long-term evolution of the AI investment landscape. Before proceeding, I should note that I am not a professional financial analyst and have been actively investing for only about one year. My professional background lies instead in mathematics, artificial intelligence, engineering, medical imaging, and related quantitative disciplines. These perspectives, together with discussions with friends and extensive use of ChatGPT as a ...

Before beginning this blog, I should acknowledge that I do not have the ability to speak with complete confidence about the current confrontation between the U.S. and Iran (March 2026). Those who are not direct participants in the conflict cannot fully understand the deeper motives, internal calculations, and hidden tensions between the parties. For that reason, as an outside observer, I am not in a position to make definitive judgments, since much of the true structure of the conflict is neither visible nor easily knowable. Most media reports portray the current confrontation between the U.S. and Iran as a struggle between good and evil, or as the product of the greed of a few actors. While this framing attracts attention, it is analytically weak. Consequently, individual investors influenced by such narratives may misinterpret the prospects for war and make costly financial decisions. A recurring reality is that individual investors who simply follow sensational news headlines of...

The performance of supervised learning in digital medical imaging modalities such as ultrasound and low-dose CBCT depends critically on the availability of paired datasets. These datasets must capture variability across patients, anatomical structures, and disease presentations, while providing accurate and consistent labels aligned with the measured images. Diagnostic tasks—including segmentation and detection—are particularly dependent on such paired data, requiring reliable annotations such as lesion localization, bounding regions, and clinically meaningful diagnostic labels. Consequently, robust model training requires large-scale datasets with high-quality annotations spanning diverse patient populations. However, in real clinical settings, such high-quality paired datasets are often unavailable due to the limited representation of abnormal cases, the absence of ground truth, inter-observer variability in annotations, patient-specific image heterogeneity, and the inherent variabil...

This blog examines a recurring pattern that has persisted in mathematics since the mid-20th century. A substantial fraction of highly capable researchers devote their efforts to extending or resolving longstanding theoretical problems inherited from earlier generations. This, in itself, is not surprising—mathematics is inherently cumulative, and deep problems often require decades of sustained attention. What is striking, however, is the scale of this concentration. Today, the global population of mathematicians exceeds, by a wide margin, the total number that existed prior to the mid-20th century. At the same time, the set of mathematically grounded problems emerging from modern society—ranging from medical imaging and data-driven modeling to complex systems and engineering constraints—has expanded dramatically. Yet a significant portion of mathematical effort remains focused on classical, internally defined questions rather than on these rapidly growing external demands. At first gla...