[{"content":"I\u0026rsquo;m ZHOU Zheng (周正). I\u0026rsquo;m curious about how things work, and if I can capitalize on that, even better.\n","date":"2 August 2026","externalUrl":null,"permalink":"/","section":"","summary":"","title":"","type":"page"},{"content":"","date":"2 August 2026","externalUrl":null,"permalink":"/musings/","section":"","summary":"","title":"","type":"musings"},{"content":"","date":"2 August 2026","externalUrl":null,"permalink":"/tags/commodities/","section":"Tags","summary":"","title":"Commodities","type":"tags"},{"content":"","date":"2 August 2026","externalUrl":null,"permalink":"/tags/futures/","section":"Tags","summary":"","title":"Futures","type":"tags"},{"content":"","date":"2 August 2026","externalUrl":null,"permalink":"/tags/","section":"Tags","summary":"","title":"Tags","type":"tags"},{"content":"","date":"2 August 2026","externalUrl":null,"permalink":"/tags/trading/","section":"Tags","summary":"","title":"Trading","type":"tags"},{"content":" Recap # Last week C2609 reached a seven-month low of 2256, then rebounded on the same day. Had I followed my plan last week, I would have been washed out.\nObservations # This week the commodity markets in China saw a broad sell-off, except for petrochemicals.\nGrains and Animal Products # Starting with the corn markets, last week C2609 seemed to have stabilized, but C2701 didn\u0026rsquo;t even have an up day. It was said that spring corn has started to enter the market, and the market expects a bumper harvest for spring corn.\nFig 1. C2609 Daily Chart with C2701 The term structure of corn futures also illustrates expectations for lower prices for spring corn.\nFig 2. Corn Futures Basis Pig farming is the biggest downstream sector for corn. Live hog futures plummeted last week, as did egg futures. This implies lower demand for corn.\nFig 3. LH2609 Daily Chart with JD2609 Infrastructure and Real Estate # Rebar futures broke the round-number level of 3000, reaching the lowest level since its first trading day, and iron ore futures also broke the round-number level of 700, also reaching the lowest level since its first trading day. It\u0026rsquo;s hard to tell which drove which — it\u0026rsquo;s possible that they drove each other.\nFig 4. RB2610 Daily Chart with I2609 From the term structure of rebar futures, we can confirm that supply far exceeds demand in the market.\nFig 5. Rebar Futures Basis Also, glass and soda ash futures also plummeted without rebounding, both reaching the lowest level since their first trading day.\nFig 6. FG609 Daily Chart with SA609 Policy Signal # On 30th July, the Politburo meeting released some interesting policy signals. Below are some excerpts:\n……实施好更加积极的财政政策和适度宽松的货币政策，充分发挥各项存量政策效能，及时谋划出台务实管用的增量政策，加大逆周期调节力度，加力扩大内需、优化供给……宏观政策要发力提效，加快财政支出和债券资金使用进度，……综合运用并适时调整货币政策工具，优化实施财政金融协同促内需政策……要有效扩大国内需求……要营造公平有序的市场竞争环境，制定实施全国统一大市场建设条例，继续综合整治“内卷式”竞争……稳定生猪等农畜产品生产和价格，促进农民稳定增收。做好农资保供稳价，抓好农业生产，夺取秋粮和全年粮食丰收……稳定房地产市场，实施好一揽子化债方案，扎实推进地方中小金融机构改革化险、减量提质。深化资本市场投融资综合改革，提升资本市场韧性和信心……\nOn the same day, treasury bond futures, especially the 30-year treasury bond futures, surged.\nFig 7. TL2609 Intraday Chart The meeting also announced that the fifth plenary session of the 20th CPC Central Committee would be held in October. Let\u0026rsquo;s wait and see if there would be any unexpected policy releases then.\nPlan # I planned to force myself to plan a trade every week. However, I don\u0026rsquo;t think it\u0026rsquo;s a good idea to trade next week, given the broad sell-off last week. Even if the prices stop plummeting, it\u0026rsquo;s hard to imagine they would rebound hard. So next week I think it\u0026rsquo;s better not to trade at all.\n","date":"2 August 2026","externalUrl":null,"permalink":"/musings/weekly-playbook-20260802/","section":"","summary":"","title":"Weekly Playbook (W/C 2026-08-02)","type":"musings"},{"content":"","date":"2 August 2026","externalUrl":null,"permalink":"/tags/weekly-playbook/","section":"Tags","summary":"","title":"Weekly-Playbook","type":"tags"},{"content":" Recap # It was clearly a big mistake to jump into a commodity only because its chart \u0026ldquo;looked attractive\u0026rdquo;. If I had followed the plan, I would have been stopped out in less than ten minutes. Nevertheless, it was a good opportunity to learn about corn, and last week turned out to be a good window to observe fear in the markets.\nFig 1. C2609 Daily Chart Observations # Why was corn so weak? # From what I could find, the fall in corn prices could be attributed to the following factors:\nTyphoon Bavi\u0026rsquo;s impact on the corn fields in Northeast China turned out to be minimal, so bullish expectations failed to materialize. The demand for corn is at a low: the largest downstream sector for corn, pig farming, is experiencing historic losses. Thus, the pig farming industry is undergoing capacity reduction, leading to lower demand for corn; the second largest downstream sector, the corn deep processing industry, was said to be undergoing maintenance shutdowns; the substitution wheat\u0026rsquo;s price was also quite low, and Sinograin released more grain into the market than it procured. Why was corn outside of China so strong? # This is mainly because China\u0026rsquo;s corn market is isolated.\nActually there is another piece of news this week saying that Argentina\u0026rsquo;s corn entered the Chinese market for the first time in over ten years. However, the amount only accounts for a tiny fraction of China\u0026rsquo;s total corn supply. Therefore, the news had little impact on the market.\nCorn prices outside China were strong because of extreme weather.\nWhy was soybean meal so strong? # It\u0026rsquo;s natural to wonder why soybean meal was so strong, given that both corn and soybean meal are animal feeds.\nIt\u0026rsquo;s important to note that soybean meal and corn are generally not substitutes: soybean meal provides protein, while corn provides energy.\nMore importantly, China\u0026rsquo;s soybean supply depends heavily on the international market. Due to weather concerns, CBOT soybean meal futures have been rising very rapidly, which directly pushed up domestic soybean meal futures.\nFear # From what I could find, a recurring theme last week was fear. Many corn merchants were said to have bought corn at high prices. Now, with the hot and wet weather causing corn to spoil, combined with the concentrated maturity of third-party funding, the fall in corn prices triggered fear among the merchants, leading to a free fall in corn prices.\nPlan # This section is private\nThe rest of the post is public. Enter the password to read this part. Password Remember me on this device Unlock ","date":"27 July 2026","externalUrl":null,"permalink":"/musings/weekly-playbook-20260727/","section":"","summary":"","title":"Weekly Playbook (W/C 2026-07-27)","type":"musings"},{"content":" Plan your trade and trade your plan. CapitaLog is a planning, review and analytics tool for discretionary trading. It supports the full loop: write a plan before entering a position, record the trades that follow, and review the results by strategy and by conviction.\nOpen the live demo The demo is fully interactive and filled with sample data. It is reset once a day, so you can change anything. The dashboard: account performance and per-plan analytics in one place. Motivation # For a discretionary trader, keeping good records is itself an edge. A written plan makes an idea reviewable, and an honest review is what turns a set of trades into feedback. Most journals record trades after the fact. CapitaLog records the plan first, then attaches the trades to it, so that when a position closes the outcome can be compared against the conviction and the risk that were set beforehand.\nFeatures # Strategy templates # Different strategies need to record different things. A breakout setup has an entry, a stop and a catalyst; an iron condor has an underlying range and four legs with strikes. CapitaLog lets you define a template per strategy with its own fields, including nested and repeating ones, rather than fitting every strategy into one fixed form.\nEach template is tagged with a methodology (technical, fundamental, macro/catalyst, quantitative) and a style (momentum, reversion, relative value, volatility), which is what makes performance comparable across strategies later. Templates are versioned: changing the fields creates a new version, and existing plans keep the version they were created with. Individual fields can be marked read-only, so parts of a thesis cannot be edited after the trade.\nA template holds the fields one strategy needs, and is versioned so existing plans are unaffected. Trade plans # A plan holds a strategy template, a conviction level, a capital allocation, and a trade idea document written before the position is opened. Plans move across a board with four states: draft, not started, in progress, and finished or abandoned.\nA plan becomes \u0026ldquo;in progress\u0026rdquo; when a trade is linked to it, not by a manual action, so its state reflects the actual position. Finishing a plan requires all positions to be closed and a written review to be attached before the profit and loss is recorded. Abandoning a plan records a reason from a fixed list, which keeps dropped ideas in the record instead of leaving a gap.\nPlans move across the board as trades are linked to them. Risk management # Position sizing is set within an evaluation cycle: a period with a fixed starting capital and a risk framework defined at the start.\nFor each conviction level, you set the share of initial capital and the share of realized profit that a single plan may risk. A plan\u0026rsquo;s absolute risk follows from those percentages, and the total across all open plans is checked against the cycle\u0026rsquo;s budget on submission, so the framework cannot be exceeded by accident. When a position is in profit, its risk can be released to free capacity for new plans while keeping the risk the plan was originally assigned, which keeps the R-multiple figures consistent.\nConviction and risk parameters are fixed once a plan is submitted, so the analytics that group results by conviction reflect the decision made before the outcome was known.\nThe risk framework is set when a cycle opens and checked on every plan. Broker statement import # Daily data entry is where most journals are abandoned, so CapitaLog reads broker files directly. Uploading a settlement statement produces the day\u0026rsquo;s fund movements, executions and end-of-day positions, which can be reviewed and corrected before saving. Executions are then linked to plans in bulk rather than one at a time.\nThe day\u0026rsquo;s data is checked for consistency: deposits and withdrawals across accounts must net to zero, each account\u0026rsquo;s closing equity must follow from the previous day plus the day\u0026rsquo;s activity, and a position that has no recorded origin is rejected rather than saved. Before anything is written, a preview shows the account snapshots the entry will produce, along with the plans that are about to move to in progress. Data entry is final once submitted, so the preview is the point at which mistakes are caught.\nCash equity, exchange-traded stock options, and futures with futures options are supported together, each valued according to how its market settles.\nSelect a statement type, upload, review, then preview the exact snapshots before submitting.\nPlan timeline # Each plan keeps a full history: the snapshot as it was created, the trade idea document, every field edit with its previous and new values, status changes, the executions linked to it, the review document, and the abandon reason where applicable.\nThe timeline also charts the plan\u0026rsquo;s profit and loss by trading day over its whole life, so a result is not reduced to a single closing number. This shows, for example, whether a profitable plan spent time underwater along the way.\nEach plan keeps its full history and its day-by-day profit and loss. Dashboard and analytics # The dashboard covers both account performance and the review process.\nFor the account, it shows an equity curve as return with drawdown on a shared axis, change rates over standard windows, and a performance overview with total net value and the profit and loss from trades made outside any plan. The overview links to a page where each account\u0026rsquo;s snapshot on any past date can be checked against the broker.\nFor the process, it shows cumulative profit and loss by strategy, monthly results, and four analytics for reviewing discretionary decisions:\nR-multiple distribution: results in units of the risk taken, which allows a large position and a small one to be compared. Conviction versus reality: outcomes grouped by the conviction assigned beforehand, with win rate and profit factor per group. Holding period versus efficiency: one point per finished plan, relating time held to result. Rolling expectancy: expectancy in R over a moving window, by conviction level. Past cycles remain available in a read-only view, so a previous period\u0026rsquo;s dashboard, plans and risk framework stay as they were.\nFour views of results, each in units of the risk taken. Status # CapitaLog is a personal tool built to product standards, and its data model, validation and analytics are designed to support multiple users without change. It is in daily use against real broker statements. The interface is available in Chinese and English.\nTry it ","date":"23 July 2026","externalUrl":null,"permalink":"/projects/capitalog/","section":"Projects","summary":"A web app for discretionary traders to plan trades, record what happens, and review performance by strategy and conviction.","title":"CapitaLog","type":"projects"},{"content":"","date":"23 July 2026","externalUrl":null,"permalink":"/tags/project/","section":"Tags","summary":"","title":"Project","type":"tags"},{"content":"","date":"23 July 2026","externalUrl":null,"permalink":"/projects/","section":"Projects","summary":"","title":"Projects","type":"projects"},{"content":"","date":"23 July 2026","externalUrl":null,"permalink":"/tags/web-app/","section":"Tags","summary":"","title":"Web App","type":"tags"},{"content":" Recap # RB2610 reached a historic low (for this contract) at 3052 on Monday, nearly touched the stop-loss I had set. Fortunately, it reached 3110 on Wednesday, so I could have made 2% on this trade had I followed the plan.\nFig 1. RB2610 Daily Chart Observations # On Rebar # I would not go long on rebar recently, because from the price of glass futures, which has been free-falling recently, I think the demand for real estate is not promising.\nFig 2. FG609 Daily Chart Then What # This section is private\nThe rest of the post is public. Enter the password to read this part. Password Remember me on this device Unlock Plan # This section is private\nUnlocks with the section above. ","date":"19 July 2026","externalUrl":null,"permalink":"/musings/weekly-playbook-20260719/","section":"","summary":"","title":"Weekly Playbook (W/C 2026-07-19)","type":"musings"},{"content":" Recap # RB2610 gapped up this Monday and touched 3070, my target buy price near the close. Had I submitted the order at the open, it would not have been unlikely to get filled. There was a surge in price on Wednesday, which was most likely due to the US-Iran conflict.\nFig 1. RB2610 Daily Chart Observations # Basis # There has been a significant recovery in basis, which is a good sign.\nFig 2. RB Basis Profit Margin and Inventory # There isn\u0026rsquo;t much change in the profit margin and inventory of of either coke or rebar this week.\nFig 3. Profit Margin \u0026amp; Inventory Further Comments # The most active contract for the coke futures J2609 reached a 6-week low on Thursday, and it doesn\u0026rsquo;t seem that it will stop there. This is a risk for rebar.\nPlan # This section is private\nThe rest of the post is public. Enter the password to read this part. Password Remember me on this device Unlock ","date":"12 July 2026","externalUrl":null,"permalink":"/musings/weekly-playbook-20260712/","section":"","summary":"","title":"Weekly Playbook (W/C 2026-07-12)","type":"musings"},{"content":"","date":"6 July 2026","externalUrl":null,"permalink":"/notes/","section":"","summary":"","title":"","type":"notes"},{"content":"","date":"6 July 2026","externalUrl":null,"permalink":"/tags/calculus/","section":"Tags","summary":"","title":"Calculus","type":"tags"},{"content":"","date":"6 July 2026","externalUrl":null,"permalink":"/series/core-topics-of-university-math/","section":"Series","summary":"","title":"Core Topics of University Math","type":"series"},{"content":" Background # There is no need to emphasize the importance of math. However, for most people learning math, efforts are put into calculations rather than theory. In some circumstances, a lack of theoretical knowledge leads to the misuse of theorems, yielding wrong results.\nThis series includes notes based on Calculus, a two-volume textbook series by Tom M. Apostol. The series not only covers calculus (including single-variable calculus and multivariable calculus) on a relatively strong theoretical foundation, but also covers linear algebra and probability. These topics1 are the main topics taught in university for a variety of programs.\nThe notes follow the same order in which I read the chapters of the books, rather than the order given by the books, because, as stated by the author in the preface of Volume 2, the chapters can be read in different orders. Chapters I deemed unimportant are also skipped.\nIntroduction to Calculus # What is Calculus? # Calculus originates from two fundamental geometric problems formulated from concepts in various fields. Consider a curve \\(C\\) lying above a horizontal base line intersected by any vertical line at most once. The two problems are:\nThe Area Problem: To measure the area bounded by \\(C\\), the base line, and two vertical segments. The Tangent Problem: To measure the steepness of a line tangent to \\(C\\). Calculus provides the precise formulation and solution for these problems, enabling us to define and calculate area and tangents. Integral calculus deals with the area problem, and differential calculus deals with the tangent problem.\nSome History Integral calculus traces back to ancient Greece, where the method of exhaustion was used to compute areas. For instance, Archimedes showed that a parabolic segment with base length \\(b\\) has area \\(b^3/3\\)2, which was accepted as a mathematical theorem until mathematicians realized that it could not be so considered until a satisfactory definition of area was given first.3\nNevertheless, Archimedes\u0026rsquo; work suggests a reasonable way to define area for arbitrary figures, and to define a much more general concept known as the integral.\nA Set of Axioms for the Real-Number System # A thorough and complete treatment of calculus depends on a careful study of the real number system. There are many ways to introduce the real-number system.4 The approach taken in the book is to take the real numbers as the undefined objects satisfying a number of properties used as axioms.\nBasic Concepts of the Theory of Sets In discussing any branch of math, it\u0026rsquo;s helpful to use the notation and terminology of set theory5. Before continuing, please make sure you understand the basic notions:\nWhat is a set? What are elements or members of a set? What is the roster notation? What is the definition of set equality? What is the definition of a subset? What about a proper subset? If \\(A \\subseteq B\\) and \\(B \\subseteq A\\), what can we say about \\(A = B\\)? What about the converse? What is the universal set? What is the empty set or the void set? What is the relationship between the empty set and any other set? What are Venn diagrams? What are set operations? What operations are commutative or associative? What is a class? What does \\(\\bigcup_{A \\in \\mathcal{F}} A\\) or \\(\\bigcap_{A \\in \\mathcal{F}} A\\) mean? The axioms fall into three groups: the field axioms, the order axioms, and the least-upper-bound axiom (also called the completeness axiom).\nThe field axioms # Along with the set \\(\\mathbb{R}\\) of real numbers, we assume the existence of two operations called addition and multiplication, such that for every pair of real numbers \\(x\\) and \\(y\\) we can form other real numbers:\nthe sum of \\(x\\) and \\(y\\), denoted by \\(x + y\\) the product of \\(x\\) and \\(y\\), denoted by \\(x \\cdot y\\) It is assumed that \\(x + y\\) and \\(x \\cdot y\\) are uniquely determined by \\(x\\) and \\(y\\).\nAxiom 1. Commutative Laws. \\(x+y=y+x,\\ xy=yx.\\)\nAxiom 2. Associative Laws. \\(x+(y+z)=(x+y)+z,\\ x(yz)=(xy)z.\\)\nAxiom 3. Distributive Law. \\(x(y+z)=xy+xz.\\)\nAxiom 4. Existence of Identity Element. There exist two distinct real numbers, denoted by \\(0\\) and \\(1\\), such that for every real \\(x\\) we have \\(x+0=x\\) and \\(1\\cdot x=x\\).\nAxiom 5. Existence of Negatives. For every real number \\(x\\) there is a real number \\(y\\) such that \\(x+y=0\\).\nAxiom 6. Existence of Reciprocals. For every real number \\(x\\neq0\\) there is a real number \\(y\\) such that \\(xy=1\\).\nNote: The numbers \\(0\\) and \\(1\\) in Axioms 5 and 6 are those of Axiom 4.\nFrom the above axioms we can deduce all the usual laws of elementary algebra. Here are some of them6:\nTheorem I.1. Cancellation Law for Addition. If \\(a+b=a+c\\), then \\(b=c\\). (In particular, this shows that the number \\(0\\) of Axiom 4 is unique.)\nTheorem I.2. Possibility of Subtraction. Given \\(a\\) and \\(b\\), there is exactly one \\(x\\) such that \\(a+x=b\\). This \\(x\\) is denoted by \\(b-a\\). In particular, \\(0-a\\) is written simply \\(-a\\) and is called the negative of \\(a\\).\nExample: Obtaining Theorem I.1. as a Consequence of the Axioms Proof: Given \\(a+b=a+c\\). By Axiom 5, there is a number \\(y\\) such that \\(y+a=0\\). Since sums are uniquely determined, we have \\(y+(a+b)=y+(a+c)\\). Using the associative law, we obtain \\((y+a)+b=(y+a)+c\\) or \\(0+b=0+c\\). But by Axiom 4 we have \\(0+b=b\\) and \\(0+c=c\\), so that \\(b=c\\).\nThe order axioms # This group of axioms allow us to establish an ordering among the real numbers. We introduce the order properties as a set of axioms about an undefined concept called positiveness, and define terms like less than and greater than in terms of positiveness.\nIt is assumed that there exists a subset \\(\\mathbb{R^+}\\subset\\mathbb{R}\\), called the set of positive numbers, which satisfies the three order axioms:\nAxiom 7. If \\(x\\) and \\(y\\) are in \\(\\mathbb{R^+}\\), so are \\(x+y\\) and \\(xy\\).\nAxiom 8. For every real \\(x\\neq0\\), either \\(x\\in\\mathbb{R^+}\\) or \\(-x\\in\\mathbb{R^+}\\), but not both.\nAxiom 9. \\(0\\notin\\mathbb{R^+}\\).\nThen we can define the symbols \\(\\lt, \\gt, \\leq, \\geq\\), called, respectively, less than, greater than, less than or equal to, greater than or equal to, as follows:\n\\(x\\lt y\\) means that \\(y-x\\) is positive \\(y\\gt x\\) means that \\(x\\lt y\\) \\(x\\leq y\\) means that either \\(x\\lt y\\) or \\(x=y\\) \\(y\\geq x\\) means that \\(x\\leq y\\) From the order axioms we can derive all the usual rules for calculating with inequalities. Here are some of them7:\nTheorem I.16. Trichotomy Law. For arbitrary real numbers \\(a\\) and \\(b\\), exactly one of the three relations \\(a\\lt b\\), \\(b\\lt a\\), \\(a=b\\) holds.\nTheorem I.17. Transitive Law. If \\(a\\lt b\\) and \\(b\\lt c\\), then \\(a\\lt c\\).\nIntegers, Rational Numbers, and Irrational Numbers The set of integers \\(\\mathbb{Z}\\) consists of the positive integers8, their negatives, and 0. The set of rational numbers \\(\\mathbb{Q}\\) consists of all quotients \\(a/b\\) where \\(a, b \\in \\mathbb{Z}\\) and \\(b \\neq 0\\). \\(\\mathbb{Z}\\) and \\(\\mathbb{Q}\\) are special subsets of \\(\\mathbb{R}\\) in that they satisfy all the field axioms and the order axioms. 9\nReal numbers not in \\(\\mathbb{Q}\\) are called irrational numbers. They arise when we solve certain equations like \\(x^2=2\\). From the nine axioms above, we cannot prove that such an \\(x\\) exists in \\(\\mathbb{R}\\), because these nine axioms are also satisfied by \\(\\mathbb{Q}\\), and there is no rational number \\(x\\) whose square is 2.\nThe least-upper bound axiom (completeness axiom) # To establish the existence of irrational numbers, we need another axiom. It allows us to introduce irrational numbers in the real-number system, giving it a property of continuity that is a keystone in the logical structure of calculus.\nBefore introducing Axiom 10, we define some basic terminology. Suppose \\(S\\) is a nonempty set of real numbers and there is a number \\(B\\) such that \\(x\\leq B\\) for every \\(x\\) in \\(S\\), then \\(S\\) is bounded above by \\(B\\), and \\(B\\) is an upper bound for \\(S\\). If an upper bound \\(B\\) is also a member of \\(S\\), then \\(B\\) is the largest member or the maximum element of \\(S\\). A set with no upper bound is unbounded above.\nFor sets that are bounded above but have no maximum element (e.g. \\(\\{x\\mid x\\in\\mathbb{R},\\ 0\\leq x\\lt 1\\}\\)), the least upper bound takes the place of the maximum element.\nDefinition of Least Upper Bound. A number \\(B\\) is called a least upper bound of a nonempty set \\(S\\) if:\n(a) \\(B\\) is an upper bound for S;\n(b) no number less than \\(B\\) is an upper bound for \\(S\\).\nTheorem I.26. Two different numbers cannot be least upper bounds for the same set.\\\nHint for Proof Suppose \\(B\\) and \\(C\\) are two least upper bounds for a set \\(S\\). Use property (b) and the Trichotomy Law.\nThis theorem tells us that if there is a least upper bound for a set \\(S\\), there is only one, and we may speak of the least upper bound, or the more concise term supremum, abbreviated sup.\nAXIOM 10. Every nonempty set \\(S\\) of real numbers which is bounded above has a supremum; that is, there is a real number \\(B\\) such that \\(B = \\sup S\\).\nBy symmetry, definitions of lower bound, bounded below, smallest member (or minimum element), greatest lower bound (or infimum) can be similarly formulated.\nTheorem I.27. Every nonempty set \\(S\\) that is bounded below has a greatest lower bound; that is, there is a real number \\(L\\) such that \\(L = \\inf S\\).\nHint for Proof Let \\(-S\\) denote the set of negatives of numbers in \\(S\\) and apply Axiom 10.\nExistence of Square Roots of Nonnegative Real Numbers The existence of at least one square root10 can be deduced from an important theorem in calculus11, but it\u0026rsquo;s instructive to see how it can be proved directly from Axiom 10.\nTheorem I.35.12 Every nonnegative real number \\(a\\) has a unique nonnegative square root.\nProof Sketch. Let \\(S = \\{x \u003e 0 \\mid x^2 \u003c a\\}\\). First we show \\(S\\) is nonempty and bounded above. By Axiom 10, \\(b = \\sup S\\) exists. Then, show that both \\(b^2 \u003e a\\) and \\(b^2 \u003c a\\) lead to contradictions: for \\(b^2 \u003e a\\), construct \\(c = \\frac12(b + a/b)\\) and show that \\(c \u003c b\\) and \\(c^2 \u003e a\\); for \\(b^2 \u003c a\\), choose a positive number \\(c\\) such that \\(c \\lt b \\) and \\(c\\lt(a-b^2)/(3b)\\), and show that \\(b + c\\) is in \\(S\\). Uniqueness follows from \\(x^2 - y^2 = (x-y)(x+y) = 0\\).\nThe least-upper-bound axiom can also be used to show the existence of roots of higher order. For example, if \\(n\\) is a positive odd integer, then for each real \\(x\\) there is exactly one real \\(y\\) such that \\(y^n=x\\). This \\(y\\) is called the \\(n\\)th root of \\(x\\), denoted by \\(y=x^{1/n}\\).\nUsing these roots, we can define rational powers: for \\(r = m/n\\) where \\(m\\) and \\(n\\) are positive integers, we define \\(x^r = (x^m)^{1/n}\\) and \\(x^{-r} = 1/x^r\\).\nRepresentation of Real Numbers by Decimals Decimal expansions can be defined analytically using the least-upper-bound axiom.13 For a positive real number \\(x\\), let \\(a_0\\) be the largest integer \\(\\leq x\\). Having chosen \\(a_0, a_1, \\dots, a_{n-1}\\), let \\(a_n\\) be the largest integer such that $$ \\tag{I.17} a_0 + \\frac{a_1}{10} + \\dots + \\frac{a_n}{10^n} \\leq x. $$ Let \\(S\\) be the set of all numbers \\(a_0 + a_1/10 + \\dots + a_n/10^n\\) for \\(n = 0, 1, 2, \\dots\\). Then \\(S\\) is nonempty, bounded above, and \\(x = \\sup S\\). The decimal expansion \\(x = a_0.a_1a_2a_3\\cdots\\) is defined by this construction, where the \\(n\\)th digit \\(a_n\\) is the largest integer satisfying (I.17).\nIf we replace \\(\\leq\\) in (I.17) by \\(\u003c\\), we obtain a slightly different definition of decimal expansions. The supremum of the corresponding set is still \\(x\\), although the integers \\(a_0, a_1, a_2, \\dots\\) need not be the same as those satisfying (I.17).\nThe fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.\nFurther Properties of the Real-Number System # The field axioms, order axioms, and completeness axiom are the fundamental properties of the real-number system. There are several other important properties that are worth discussing.\nThe Archimedean Property # This section contains a number of important properties of the real-number system which are consequences of the least-upper-bound axiom.\nTheorem I.28. The set \\(\\mathbb{P}\\) of positive integers \\(1, 2, 3, \\dots\\) is unbounded above.\nHint for Proof Assume \\(\\mathbb{P}\\) is bounded above and use Axiom 10 to find a least upper bound for \\(\\mathbb{P}\\). Find a contradiction.\nAs corollaries of Theorem I.28, we obtain the following consequences:\nTheorem I.29. For every real \\(x\\) there exists a positive integer \\(n\\) such that \\(n \\gt x\\).\nHint for Proof Assume this were not so and find a contradiction with Theorem I.28.\nTheorem I.30. Archimedean Property of the Real-number System. If \\(x \\gt 0\\) and if \\(y\\) is an arbitrary real number, there exists a positive integer \\(n\\) such that \\(nx \\gt y\\).\nHint for Proof Replace \\(x\\) by \\(\\frac{y}{x}\\) in Theorem I.29.\nTheorem I.31. If three real numbers \\(a\\), \\(x\\), and \\(y\\) satisfy the inequalities \\(a \\leq x \\leq a+\\frac{y}{n}\\) for every integer \\(n \\gt 1\\), then \\(x = a\\).\nHint for Proof Assume \\(x\\gt a\\) and find a contradiction using Theorem I.30.\nFundamental Properties of the Supremum and Infimum # This section discusses three fundamental properties of the supremum and infimum that are used in our development of calculus.\nTheorem I.32. Let \\(h\\) be a given positive number and let \\(S\\) be a set of real numbers.\n(a) If \\(S\\) has a supremum, then for some \\(x\\) in \\(S\\) we have \\(x \\gt \\sup S - h\\).\n(b) If \\(S\\) has an infimum, then for some \\(x\\) in \\(S\\) we have \\(x \\lt \\inf S + h\\).\nHint for Proof For (a), assume \\(x \u003c \\sup S - h\\) for all \\(x\\) in \\(S\\) and find a contradiction using the definition of supremum.\nTheorem I.33. Additive Property. Given nonempty subsets \\(A\\) and \\(B\\) of \\(\\mathbb{R}\\), let \\(C\\) denote the set \\(C = \\{a + b \\mid a \\in A, b \\in B\\}\\).\n(a) If each of \\(A\\) and \\(B\\) has a supremum, then \\(C\\) has a supremum, and \\(\\sup C = \\sup A + \\sup B\\).\n(b) If each of \\(A\\) and \\(B\\) has an infimum, then \\(C\\) has an infimum, and \\(\\inf C = \\inf A + \\inf B\\).\nHint for Proof For (a), show that \\(\\sup C \\lt \\sup A + \\sup B\\), then use Theorem I.32. (with \\(h = 1/n\\)) and Theorem I.31.\nTheorem I.34. Given two nonempty subsets \\(S\\) and \\(T\\) of \\(R\\) such that \\(s \\leq t\\) for every \\(s\\) in \\(S\\) and every \\(t\\) in \\(T\\). Then \\(S\\) has a supremum, and \\(T\\) has an infimum, and they satisfy the inequality \\(\\sup S \\leq \\inf T\\).\nHint for Proof Each \\(t\\) in \\(T\\) is an upper bound for \\(S\\).\nMathematical Tools for Calculus # Mathematical Induction # Method of Proof by Induction. Let \\(A(n)\\) be an assertion involving an integer \\(n\\). We conclude that \\(A(n)\\) is true for every \\(n \\geq n_1\\) if we can perform the following two steps:\n(a) Prove that \\(A(n_1)\\) is true.\n(b) Let \\(k\\) be an arbitrary but fixed integer \\(\\geq n_1\\). Assume that \\(A(k)\\) is true and prove that \\(A(k + 1)\\) is also true.\nThe logical justification for this method of proof is the following theorem about real numbers.\nTheorem 1.36. Principle of Mathematical Induction. Let \\(S\\) be a set of positive integers which has the following two properties: (a) The number \\(1\\) is in the set \\(S\\); (b) If an integer \\(k\\) is in \\(S\\), then so is \\(k + 1\\). Then every positive integer is in the set \\(S\\).\nHint for Proof Properties (a) and (b) imply that \\(S\\) is an inductive set.\nThe Well-ordering Principle The well-ordering principle is another important property of the positive integers. It is equivalent to the principle of mathematical induction.\nTheorem I.37. Well-ordering Principle. Every nonempty set of positive integers contains a smallest member.\nProof Sketch. Let \\(T\\) be a nonempty collection of positive integers. Suppose \\(T\\) has no smallest member. Show that this leads to a contradiction.\nWhat\u0026rsquo;s Wrong with the Following \u0026quot;Proof\u0026quot; by Induction? Statement. Given any collection of \\(n\\) blonde girls. If at least one of the girls has blue eyes, then all \\(n\\) of them have blue eyes.\n\u0026quot;Proof\u0026quot;. The statement is obviously true when \\(n = 1\\). The step from \\(k\\) to \\(k + 1\\) can be illustrated by going from \\(n = 3\\) to \\(n = 4\\). Assume, therefore, that the statement is true when \\(n = 3\\) and let \\(G_1, G_2, G_3, G_4\\) be four blonde girls, at least one of which, say \\(G_1\\) has blue eyes. Taking \\(G_1\\), \\(G_2\\), and \\(G_3\\) together and using the fact that the statement is true when \\(n = 3\\), we find that \\(G_2\\) and \\(G_3\\) also have blue eyes. Repeating the process with \\(G_1\\), \\(G_2\\), and \\(G_4\\) we find that \\(G_4\\) has blue eyes. Thus all four have blue eyes. A similar argument allows us to make the step from \\(k\\) to \\(k + 1\\) in general.\nCorollary. All blonde girls have blue eyes.\nProof. Since there exists at least one blonde girl with blue eyes, we can apply the foregoing result to the collection consisting of all blonde girls.14\nThe Summation Notation # We denote the sum of several real numbers \\(a_1, a_2, \\dots, a_n\\) by the symbol $$\\tag{I.21} a_1 + a_2 + \\dots + a_n$$ or, using summation notation: $$\\tag{I.22}\\sum_{k=1}^n a_k.$$From a strictly logical standpoint, the symbols in (I.21) and (I.22) do not appear among the primitive symbols for the real-number system. In a more careful treatment, we could define these new symbols in terms of the primitive undefined symbols of our system using definition by induction:\n(a) We define \\(\\sum_{k=1}^1 a_k = a_1\\).\n(b) Assuming that we have defined \\(\\sum_{k=1}^n a_k\\) for a fixed \\(n\\geq1\\), we further define \\(\\sum_{k=1}^{n+1} a_k = (\\sum_{k=1}^n a_k) + a_{n+1}\\).\nNotice that definition by induction and proof by induction involve the same underlying idea. A definition by induction is also called a recursive definition.\nAbsolute Values and the Triangle Inequality # Calculations with inequalities arise frequently in calculus. They are of particular importance in dealing with the notion of absolute value.\nIf x is a real number, the absolute value of x is a nonnegative real number denoted by \\(\\lvert x \\rvert\\) and defined as \\(\\lvert x \\rvert = \\begin{cases} x \u0026 \\text{if}\\ x \\geq 0,\\\\ -x \u0026 \\text{if}\\ x \\leq 0.\\end{cases}\\)\nTheorem I.38. If \\(a \\gt 0\\), then \\(\\lvert x \\rvert \\leq a\\) if and only if \\(-a \\leq x \\leq a\\).\nHint for Proof Use the definition of absolute value.\nThe following theorem is a consequence of Theorem I.38.\nTheorem I.39. Triangle Inequality.15 For arbitrary real numbers x and y, we have \\(\\lvert x + y \\rvert \\leq \\lvert x \\rvert + \\lvert y \\rvert \\).\nHint for Proof Add the inequalities \\(-\\lvert x\\rvert \\lt x \\lt \\lvert x \\rvert\\) and \\(-\\lvert y \\rvert \\lt y \\lt \\lvert y \\rvert\\), and use Theorem I.38.\nUsing mathematical induction, we may extend the triangle inequality as follows:\nTheorem I.40. For arbitrary real numbers \\(a_1, a_2, \\dots, a_n\\), we have \\( \\lvert \\sum_{k=1}^n a_k \\rvert \\leq \\sum_{k=1}^n \\lvert a_k \\rvert \\).\nHint for Proof Use mathematical induction.\nThe next theorem describes an important inequality that we shall use later in connection with our study of vector algebra.\nTheorem I.41. The Cauchy-Schwarz Inequality. If \\(a_1, \\dots, a_n\\) and \\(b_1, \\dots, b_n\\) are arbitrary real numbers, we have \\( (\\sum_{k=1}^n a_kb_k)^2 \\leq (\\sum_{k=1}^n a_k^2)(\\sum_{k=1}^n b_k^2) \\). The equality sign holds if and only if there is a real number \\(x\\) such that \\(a_kx + b_k = 0\\) for each \\(k = 1,2,\\dots,n\\).\nHint for Proof We have \\(\\sum_{k=1}^n(a_kx+b_k)^2 \\geq 0\\) for every real \\(x\\). Write it in the form \\(Ax^2 + 2Bx + C \\geq 0\\).\nStrictly speaking statistics should be included, but Calculus by Tom M. Apostol does not cover this topic. This series can therefore be regarded as notes on Calculus by Tom M. Apostol.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nPlease check the textbook for detailed derivation.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nIn a deductive system, a number of \u0026ldquo;undefined\u0026rdquo; concepts are chosen in advance as axioms or postulates, and all other concepts in the system are defined in terms of these. Statements deduced from axioms are called theorems.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nOne popular method is constructive, taking the positive integers as undefined objects and stating some axioms concerning them, then using them to build systems consisting of positive rational numbers, positive irrational numbers, and finally negative real numbers and zero. The point of view adopted by the book is nonconstructive.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nSet theory has unified many seemingly disconnected ideas and has helped to reduce many mathematical concepts to their logical foundations in an elegant and systematic way.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nPlease check the textbook for Theorem I.3. through Theorem I.15.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nPlease check the textbook for Theorem I.18. through Theorem I.25.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nFormal definition of positive integers: A real number is called a positive integer if it belongs to every inductive set, where an inductive set is a set of real numbers such that (a) the number 1 is in the set; (b) for every \\(x\\) in the set, the number \\(x+1\\) is also in the set. Clearly, the set of all positive integers, denoted by \\(\\mathbb{P}\\), is the smallest inductive set.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nA set satisfying all the field axioms and the order axioms is called an ordered field.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nEach \\(x\\) is called a square root of \\(a\\) if \\(x^2 = a\\).\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nWhich is the intermediate-value theorem for continuous functions.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nYou may wonder about the numbering of theorems. I use the same numbers as the textbook, but I don\u0026rsquo;t present them in the same order as the book.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nPlease check the book for more information about decimal representations, including approximating an arbitrary real number to any desired degree of accuracy, and sequences of nested intervals which is a concept that is sometimes used as a basis for constructing the irrational numbers from the rational numbers.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nThis example is from G. Pólya, who suggests that the reader may want to test the validity of the statement by experiment.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\nIt is so called because when it is generalized to vectors it states that the length of any side of a triangle is less than or equal to the sum of the lengths of the other two sides.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\n","date":"6 July 2026","externalUrl":null,"permalink":"/notes/foundations-of-calculus/","section":"","summary":"","title":"Foundations of Calculus","type":"notes"},{"content":"","date":"6 July 2026","externalUrl":null,"permalink":"/tags/math/","section":"Tags","summary":"","title":"Math","type":"tags"},{"content":"","date":"6 July 2026","externalUrl":null,"permalink":"/series/","section":"Series","summary":"","title":"Series","type":"series"},{"content":" Recap # This week RB2610 fell through 3080 and reached a four-month low at 3057, so if I had traded as planned, I would have lost 1% of my capital. To be honest, I could not explain the price decline in terms of the fundamentals. It was probably due to the price decline of coke and weak expectations.\nFig 1. RB2610 Daily Chart Observations # Compared to last week, the price of rebar looks more attractive, and I didn\u0026rsquo;t see the fundamentals turn significantly worse.\nBasis # On Friday, the basis fell even further compared with last week. This provides a better margin of safety.\nFig 2. RB Basis Profit Margin and Inventory # Last week, there was a moderate fall in the profit margin of coking, while there was no significant change in the profit margin of rebar. There was no significant change in the inventory of coke and rebar either.\nFig 3. Profit Margin \u0026amp; Inventory Further Comments # East China has entered the plum rain season recently based on my observation in Shanghai, so construction activities have probably slowed down. The demand for rebar may be affected. This is one of the risks I must consider.\nPlan # This section is private\nThe rest of the post is public. Enter the password to read this part. Password Remember me on this device Unlock ","date":"5 July 2026","externalUrl":null,"permalink":"/musings/weekly-playbook-20260705/","section":"","summary":"","title":"Weekly Playbook (W/C 2026-07-05)","type":"musings"},{"content":" Background # The most active contract for the rebar futures RB2610 fell to as low as 3080 last week, and it\u0026rsquo;s starting to look attractive to me. Since the last low is 3083, and in recent months RB2610 never fell below 3050, longing RB2610 seems to have a good risk-reward profile. Before I commit to the trade, I need to take a closer look at the fundamentals and technicals.\nFig 1. RB2610 Daily Chart Observations # Technicals # There is a bullish divergence of MACD histogram (default parameters), which is a bullish signal. Also, the RSI touches the oversold region, so there might be a pullback.\nBasis # The following figure demonstrates the differences of the closing prices of listed contracts relative to the front month contract over the past several days. We can see that in the last week, the differences are all well below 0, providing a margin of safety to take a long position.\nFig 2. RB Basis Profit Margin and Inventory # The upper subplot of the following figure shows the futures-implied profit margin of coking and rebar (fixed constants excluded; only the absolute levels are meaningful). We can see that the profit margin of rebar is at a very low level, which means it\u0026rsquo;s likely that the manufacturer will slow down production, leading to tighter supply. However, the profit margin of coking is still high, so there\u0026rsquo;s still a chance that the price of rebar will go lower, which is one of the risks.\nThe lower subplot of the figure shows the inventory of coke and rebar. The high inventory of coke and relatively high inventory of rebar implies that there\u0026rsquo;s still space for further decline, another risk.\nFig 3. Profit Margin \u0026amp; Inventory Further Comments # The discussions above only cover a tiny fraction of the fundamentals, and the positioning and sentiment were not covered. I would further improve my analysis in the future.\nPlan # This section is private\nThe rest of the post is public. Enter the password to read this part. Password Remember me on this device Unlock ","date":"28 June 2026","externalUrl":null,"permalink":"/musings/weekly-playbook-20260628/","section":"","summary":"","title":"Weekly Playbook (W/C 2026-06-28)","type":"musings"},{"content":"","externalUrl":null,"permalink":"/authors/","section":"Authors","summary":"","title":"Authors","type":"authors"},{"content":"","externalUrl":null,"permalink":"/categories/","section":"Categories","summary":"","title":"Categories","type":"categories"},{"content":"","externalUrl":null,"permalink":"/gallery/","section":"","summary":"","title":"Gallery","type":"page"},{"content":"","externalUrl":null,"permalink":"/resume/","section":"","summary":"","title":"Resume","type":"page"}]