More on Cooking
Alexandra Walker-Jones
3 years ago
These are the 15 foods you should eat daily and why.
Research on preventing disease, extending life, and caring for your body from the inside out
Grapefruit and pomegranates aren't on the list, so ignore that. Mostly, I enjoyed the visual, but those fruits are healthful, too.
15 (or 17 if you consider the photo) different foods a day sounds like a lot. If you're not used to it — it is.
These lists don't aim for perfection. Instead, use this article and the science below to eat more of these foods. If you can eat 5 foods one day and 5 the next, you're doing well. This list should be customized to your requirements and preferences.
“Every time you eat or drink, you are either feeding disease or fighting it” -Heather Morgan.
The 15 Foods That You Should Consume Daily and Why:
1. Dark/Red Berries
(blueberries, blackberries, acai, goji, cherries, strawberries, raspberries)
The 2010 Global Burden of Disease Study is the greatest definitive analysis of death and disease risk factors in history. They found the primary cause of both death, disability, and disease inside the United States was diet.
Not eating enough fruit, and specifically berries, was one of the best predictors of disease (1).
What's special about berries? It's their color! Berries have the most antioxidants of any fruit, second only to spices. The American Cancer Society found that those who ate the most berries were less likely to die of cardiovascular disease.
2. Beans
Soybeans, black beans, kidney beans, lentils, split peas, chickpeas.
Beans are one of the most important predictors of survival in older people, according to global research (2).
For every 20 grams (2 tablespoons) of beans consumed daily, the risk of death is reduced by 8%.
Soybeans and soy foods are high in phytoestrogen, which reduces breast and prostate cancer risks. Phytoestrogen blocks the receptors' access to true estrogen, mitigating the effects of weight gain, dairy (high in estrogen), and hormonal fluctuations (3).
3. Nuts
(almonds, walnuts, pecans, pistachios, Brazil nuts, cashews, hazelnuts, macadamia nuts)
Eating a handful of nuts every day reduces the risk of chronic diseases like heart disease and diabetes. Nuts also reduce oxidation, blood sugar, and LDL (bad) cholesterol, improving arterial function (4).
Despite their high-fat content, studies have linked daily nut consumption to a slimmer waistline and a lower risk of obesity (5).
4. Flaxseed
(milled flaxseed)
2013 research found that ground flaxseed had one of the strongest anti-hypertensive effects of any food. A few tablespoons (added to a smoothie or baked goods) lowered blood pressure and stroke risk 23 times more than daily aerobic exercise (6).
Flax shouldn't replace exercise, but its nutritional punch is worth adding to your diet.
5. Other seeds
(chia seeds, hemp seeds, pumpkin seeds, sesame seeds, fennel seeds)
Seeds are high in fiber and omega-3 fats and can be added to most dishes without being noticed.
When eaten with or after a meal, chia seeds moderate blood sugar and reduce inflammatory chemicals in the blood (7). Overall, a great daily addition.
6. Dates
Dates are one of the world's highest sugar foods, with 80% sugar by weight. Pure cake frosting is 60%, maple syrup is 66%, and cotton-candy jelly beans are 70%.
Despite their high sugar content, dates have a low glycemic index, meaning they don't affect blood sugar levels dramatically. They also improve triglyceride and antioxidant stress levels (8).
Dates are a great source of energy and contain high levels of dietary fiber and polyphenols, making 3-10 dates a great way to fight disease, support gut health with prebiotics, and satisfy a sweet tooth (9).
7. Cruciferous Veggies
(broccoli, Brussel sprouts, horseradish, kale, cauliflower, cabbage, boy choy, arugula, radishes, turnip greens)
Cruciferous vegetables contain an active ingredient that makes them disease-fighting powerhouses. Sulforaphane protects our brain, eyesight, against free radicals and environmental hazards, and treats and prevents cancer (10).
Unless you eat raw cruciferous vegetables daily, you won't get enough sulforaphane (and thus, its protective nutritional benefits). Cooking destroys the enzyme needed to create this super-compound.
If you chop broccoli, cauliflower, or turnip greens and let them sit for 45 minutes before cooking them, the enzyme will have had enough time to work its sulforaphane magic, allowing the vegetables to retain the same nutritional value as if eaten raw. Crazy, right? For more on this, see What Chopping Your Vegetables Has to Do with Fighting Cancer.
8. Whole grains
(barley, brown rice, quinoa, oats, millet, popcorn, whole-wheat pasta, wild rice)
Whole-grains are one of the healthiest ways to consume your daily carbs and help maintain healthy gut flora.
This happens when fibre is broken down in the colon and starts a chain reaction, releasing beneficial substances into the bloodstream and reducing the risk of Type 2 Diabetes and inflammation (11).
9. Spices
(turmeric, cumin, cinnamon, ginger, saffron, cloves, cardamom, chili powder, nutmeg, coriander)
7% of a person's cells will have DNA damage. This damage is caused by tiny breaks in our DNA caused by factors like free-radical exposure.
Free radicals cause mutations that damage lipids, proteins, and DNA, increasing the risk of disease and cancer. Free radicals are unavoidable because they result from cellular metabolism, but they can be avoided by consuming anti-oxidant and detoxifying foods.
Including spices and herbs like rosemary or ginger in our diet may cut DNA damage by 25%. Yes, this damage can be improved through diet. Turmeric worked better at a lower dose (just a pinch, daily). For maximum free-radical fighting (and anti-inflammatory) effectiveness, use 1.5 tablespoons of similar spices (12).
10. Leafy greens
(spinach, collard greens, lettuce, other salad greens, swiss chard)
Studies show that people who eat more leafy greens perform better on cognitive tests and slow brain aging by a year or two (13).
As we age, blood flow to the brain drops due to a decrease in nitric oxide, which prevents blood vessels from dilatation. Daily consumption of nitrate-rich vegetables like spinach and swiss chard may prevent dementia and Alzheimer's.
11. Fermented foods
(sauerkraut, tempeh, kombucha, plant-based kefir)
Miso, kimchi, and sauerkraut contain probiotics that support gut microbiome.
Probiotics balance the good and bad bacteria in our bodies and offer other benefits. Fermenting fruits and vegetables increases their antioxidant and vitamin content, preventing disease in multiple ways (14).
12. Sea vegetables
(seaweed, nori, dulse flakes)
A population study found that eating one sheet of nori seaweed per day may cut breast cancer risk by more than half (15).
Seaweed and sea vegetables may help moderate estrogen levels in the metabolism, reducing cancer and disease risk.
Sea vegetables make up 30% of the world's edible plants and contain unique phytonutrients. A teaspoon of these super sea-foods on your dinner will help fight disease from the inside out.
13. Water
I'm less concerned about whether you consider water food than whether you drink enough. If this list were ranked by what single item led to the best health outcomes, water would be first.
Research shows that people who drink 5 or more glasses of water per day have a 50% lower risk of dying from heart disease than those who drink 2 or less (16).
Drinking enough water boosts energy, improves skin, mental health, and digestion, and reduces the risk of various health issues, including obesity.
14. Tea
All tea consumption is linked to a lower risk of stroke, heart disease, and early death, with green tea leading for antioxidant content and immediate health benefits.
Green tea leaves may also be able to interfere with each stage of cancer formation, from the growth of the first mutated cell to the spread and progression of cancer in the body. Green tea is a quick and easy way to support your long-term and short-term health (17).
15. Supplemental B12 vitamin
B12, or cobalamin, is a vitamin responsible for cell metabolism. Not getting enough B12 can have serious consequences.
Historically, eating vegetables from untreated soil helped humans maintain their vitamin B12 levels. Due to modern sanitization, our farming soil lacks B12.
B12 is often cited as a problem only for vegetarians and vegans (as animals we eat are given B12 supplements before slaughter), but recent studies have found that plant-based eaters have lower B12 deficiency rates than any other diet (18).
Article Sources:
Joseph Mavericks
3 years ago
Apples Top 100 Meeting: Steve Jobs's Secret Agenda's Lessons
Jobs' secret emails became public due to a litigation with Samsung.
Steve Jobs sent Phil Schiller an email at the end of 2010. Top 100 A was the codename for Apple's annual Top 100 executive meetings. The 2011 one was scheduled.
Everything about this gathering is secret, even attendance. The location is hidden, and attendees can't even drive themselves. Instead, buses transport them to a 2-3 day retreat.
Due to a litigation with Samsung, this Top 100 meeting's agenda was made public in 2014. This was a critical milestone in Apple's history, not a Top 100 meeting. Apple had many obstacles in the 2010s to remain a technological leader. Apple made more money with non-PC goods than with its best-selling Macintosh series. This was the last Top 100 gathering Steve Jobs would attend before passing, and he wanted to make sure his messages carried on before handing over his firm to Tim Cook.
In this post, we'll discuss lessons from Jobs' meeting agenda. Two sorts of entrepreneurs can use these tips:
Those who manage a team in a business and must ensure that everyone is working toward the same goals, upholding the same principles, and being inspired by the same future.
Those who are sole proprietors or independent contractors and who must maintain strict self-discipline in order to stay innovative in their industry and adhere to their own growth strategy.
Here's Steve Jobs's email outlining the annual meeting agenda. It's an 11-part summary of the company's shape and strategy.
Steve Jobs outlines Apple's 2011 strategy, 10/24/10
1. Correct your data
Business leaders must comprehend their company's metrics. Jobs either mentions critical information he already knows or demands slides showing the numbers he wants. These numbers fall under 2 categories:
Metrics for growth and strategy
As we will see, this was a crucial statistic for Apple since it signaled the beginning of the Post PC era and required them to make significant strategic changes in order to stay ahead of the curve. Post PC products now account for 66% of our revenues.
Within six months, iPad outsold Mac, another sign of the Post-PC age. As we will see, Jobs thought the iPad would be the next big thing, and item number four on the agenda is one of the most thorough references to the iPad.
Geographical analysis: Here, Jobs emphasizes China, where the corporation has a slower start than anticipated. China was dominating Apple's sales growth with 16% of revenue one year after this meeting.
Metrics for people & culture
The individuals that make up a firm are more significant to its success than its headcount or average age. That holds true regardless of size, from a 5-person startup to a Fortune 500 firm. Jobs was aware of this, which is why his suggested agenda begins by emphasizing demographic data.
Along with the senior advancements in the previous year's requested statistic, it's crucial to demonstrate that if the business is growing, the employees who make it successful must also grow.
2. Recognize the vulnerabilities and strengths of your rivals
Steve Jobs was known for attacking his competition in interviews and in his strategies and roadmaps. This agenda mentions 18 competitors, including:
Google 7 times
Android 3 times
Samsung 2 times
Jobs' agenda email was issued 6 days after Apple's Q4 results call (2010). On the call, Jobs trashed Google and Android. His 5-minute intervention included:
Google has acknowledged that the present iteration of Android is not tablet-optimized.
Future Android tablets will not work (Dead On Arrival)
While Google Play only has 90,000 apps, the Apple App Store has 300,000.
Android is extremely fragmented and is continuing to do so.
The App Store for iPad contains over 35,000 applications. The market share of the latest generation of tablets (which debuted in 2011) will be close to nil.
Jobs' aim in blasting the competition on that call was to reassure investors about the upcoming flood of new tablets. Jobs often criticized Google, Samsung, and Microsoft, but he also acknowledged when they did a better job. He was great at detecting his competitors' advantages and devising ways to catch up.
Jobs doesn't hold back when he says in bullet 1 of his agenda: "We further lock customers into our ecosystem while Google and Microsoft are further along on the technology, but haven't quite figured it out yet tie all of our goods together."
The plan outlined in bullet point 5 is immediately clear: catch up to Android where we are falling behind (notifications, tethering, and speech), and surpass them (Siri,). It's important to note that Siri frequently let users down and never quite lived up to expectations.
Regarding MobileMe, see Bullet 6 Jobs admits that when it comes to cloud services like contacts, calendars, and mail, Google is far ahead of Apple.
3. Adapt or perish
Steve Jobs was a visionary businessman. He knew personal computers were the future when he worked on the first Macintosh in the 1980s.
Jobs acknowledged the Post-PC age in his 2010 D8 interview.
Will the tablet replace the laptop, Walt Mossberg questioned Jobs? Jobs' response:
“You know, when we were an agrarian nation, all cars were trucks, because that’s what you needed on the farm. As vehicles started to be used in the urban centers and America started to move into those urban and suburban centers, cars got more popular and innovations like automatic transmission and things that you didn’t care about in a truck as much started to become paramount in cars. And now, maybe 1 out of every 25 vehicles is a truck, where it used to be 100%. PCs are going to be like trucks. They’re still going to be around, still going to have a lot of value, but they’re going to be used by one out of X people.”
Imagine how forward-thinking that was in 2010, especially for the Macintosh creator. You have to be willing to recognize that things were changing and that it was time to start over and focus on the next big thing.
Post-PC is priority number 8 in his 2010 agenda's 2011 Strategy section. Jobs says Apple is the first firm to get here and that Post PC items account about 66% of our income. The iPad outsold the Mac in 6 months, and the Post-PC age means increased mobility (smaller, thinner, lighter). Samsung had just introduced its first tablet, while Apple was working on the iPad 3. (as mentioned in bullet 4).
4. Plan ahead (and different)
Jobs' agenda warns that Apple risks clinging to outmoded paradigms. Clayton Christensen explains in The Innovators Dilemma that huge firms neglect disruptive technologies until they become profitable. Samsung's Galaxy tab, released too late, never caught up to Apple.
Apple faces a similar dilemma with the iPhone, its cash cow for over a decade. It doesn't sell as much because consumers aren't as excited about new iPhone launches and because technology is developing and cell phones may need to be upgraded.
Large companies' established consumer base typically hinders innovation. Clayton Christensen emphasizes that loyal customers from established brands anticipate better versions of current products rather than something altogether fresh and new technologies.
Apple's marketing is smart. Apple's ecosystem is trusted by customers, and its products integrate smoothly. So much so that Apple can afford to be a disruptor by doing something no one has ever done before, something the world's largest corporation shouldn't be the first to try. Apple can test the waters and produce a tremendous innovation tsunami, something few corporations can do.
In March 2011, Jobs appeared at an Apple event. During his address, Steve reminded us about Apple's brand:
“It’s in Apple’s DNA, that technology alone is not enough. That it’s technology married with liberal arts, married with the humanities that yields us the results that make our hearts sink. And nowhere is that more true that in these Post-PC devices.“
More than a decade later, Apple remains one of the most innovative and trailblazing companies in the Post-PC world (industry-disrupting products like Airpods or the Apple Watch came out after that 2011 strategy meeting), and it has reinvented how we use laptops with its M1-powered line of laptops offering unprecedented performance.
A decade after Jobs' death, Apple remains the world's largest firm, and its former CEO had a crucial part in its expansion. If you can do 1% of what Jobs did, you may be 1% as successful.
Not bad.
Scott Hickmann
3 years ago Draft
This is a draft
My wallpape
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Jari Roomer
3 years ago
Successful people have this one skill.
Without self-control, you'll waste time chasing dopamine fixes.
I found a powerful quote in Tony Robbins' Awaken The Giant Within:
“Most of the challenges that we have in our personal lives come from a short-term focus” — Tony Robbins
Most people are short-term oriented, but highly successful people are long-term oriented.
Successful people act in line with their long-term goals and values, while the rest are distracted by short-term pleasures and dopamine fixes.
Instant gratification wrecks lives
Instant pleasure is fleeting. Quickly fading effects leave you craving more stimulation.
Before you know it, you're in a cycle of quick fixes. This explains binging on food, social media, and Netflix.
These things cause a dopamine spike, which is entertaining. This dopamine spike crashes quickly, leaving you craving more stimulation.
It's fine to watch TV or play video games occasionally. Problems arise when brain impulses aren't controlled. You waste hours chasing dopamine fixes.
Instant gratification becomes problematic when it interferes with long-term goals, happiness, and life fulfillment.
Most rewarding things require delay
Life's greatest rewards require patience and delayed gratification. They must be earned through patience, consistency, and effort.
Ex:
A fit, healthy body
A deep connection with your spouse
A thriving career/business
A healthy financial situation
These are some of life's most rewarding things, but they take work and patience. They all require the ability to delay gratification.
To have a healthy bank account, you must save (and invest) a large portion of your monthly income. This means no new tech or clothes.
If you want a fit, healthy body, you must eat better and exercise three times a week. So no fast food and Netflix.
It's a battle between what you want now and what you want most.
Successful people choose what they want most over what they want now. It's a major difference.
Instant vs. delayed gratification
Most people subconsciously prefer instant rewards over future rewards, even if the future rewards are more significant.
We humans aren't logical. Emotions and instincts drive us. So we act against our goals and values.
Fortunately, instant gratification bias can be overridden. This is a modern superpower. Effective methods include:
#1: Train your brain to handle overstimulation
Training your brain to function without constant stimulation is a powerful change. Boredom can lead to long-term rewards.
Unlike impulsive shopping, saving money is boring. Having lots of cash is amazing.
Compared to video games, deep work is boring. A successful online business is rewarding.
Reading books is boring compared to scrolling through funny videos on social media. Knowledge is invaluable.
You can't do these things if your brain is overstimulated. Your impulses will control you. To reduce overstimulation addiction, try:
Daily meditation (10 minutes is enough)
Daily study/work for 90 minutes (no distractions allowed)
First hour of the day without phone, social media, and Netflix
Nature walks, journaling, reading, sports, etc.
#2: Make Important Activities Less Intimidating
Instant gratification helps us cope with stress. Starting a book or business can be intimidating. Video games and social media offer a quick escape in such situations.
Make intimidating tasks less so. Break them down into small tasks. Start a new business/side-hustle by:
Get domain name
Design website
Write out a business plan
Research competition/peers
Approach first potential client
Instead of one big mountain, divide it into smaller sub-tasks. This makes a task easier and less intimidating.
#3: Plan ahead for important activities
Distractions will invade unplanned time. Your time is dictated by your impulses, which are usually Netflix, social media, fast food, and video games. It wants quick rewards and dopamine fixes.
Plan your days and be proactive with your time. Studies show that scheduling activities makes you 3x more likely to do them.
To achieve big goals, you must plan. Don't gamble.
Want to get fit? Schedule next week's workouts. Want a side-job? Schedule your work time.
Theresa W. Carey
3 years ago
How Payment for Order Flow (PFOF) Works
What is PFOF?
PFOF is a brokerage firm's compensation for directing orders to different parties for trade execution. The brokerage firm receives fractions of a penny per share for directing the order to a market maker.
Each optionable stock could have thousands of contracts, so market makers dominate options trades. Order flow payments average less than $0.50 per option contract.
Order Flow Payments (PFOF) Explained
The proliferation of exchanges and electronic communication networks has complicated equity and options trading (ECNs) Ironically, Bernard Madoff, the Ponzi schemer, pioneered pay-for-order-flow.
In a December 2000 study on PFOF, the SEC said, "Payment for order flow is a method of transferring trading profits from market making to brokers who route customer orders to specialists for execution."
Given the complexity of trading thousands of stocks on multiple exchanges, market making has grown. Market makers are large firms that specialize in a set of stocks and options, maintaining an inventory of shares and contracts for buyers and sellers. Market makers are paid the bid-ask spread. Spreads have narrowed since 2001, when exchanges switched to decimals. A market maker's ability to play both sides of trades is key to profitability.
Benefits, requirements
A broker receives fees from a third party for order flow, sometimes without a client's knowledge. This invites conflicts of interest and criticism. Regulation NMS from 2005 requires brokers to disclose their policies and financial relationships with market makers.
Your broker must tell you if it's paid to send your orders to specific parties. This must be done at account opening and annually. The firm must disclose whether it participates in payment-for-order-flow and, upon request, every paid order. Brokerage clients can request payment data on specific transactions, but the response takes weeks.
Order flow payments save money. Smaller brokerage firms can benefit from routing orders through market makers and getting paid. This allows brokerage firms to send their orders to another firm to be executed with other orders, reducing costs. The market maker or exchange benefits from additional share volume, so it pays brokerage firms to direct traffic.
Retail investors, who lack bargaining power, may benefit from order-filling competition. Arrangements to steer the business in one direction invite wrongdoing, which can erode investor confidence in financial markets and their players.
Pay-for-order-flow criticism
It has always been controversial. Several firms offering zero-commission trades in the late 1990s routed orders to untrustworthy market makers. During the end of fractional pricing, the smallest stock spread was $0.125. Options spreads widened. Traders found that some of their "free" trades cost them a lot because they weren't getting the best price.
The SEC then studied the issue, focusing on options trades, and nearly decided to ban PFOF. The proliferation of options exchanges narrowed spreads because there was more competition for executing orders. Options market makers said their services provided liquidity. In its conclusion, the report said, "While increased multiple-listing produced immediate economic benefits to investors in the form of narrower quotes and effective spreads, these improvements have been muted with the spread of payment for order flow and internalization."
The SEC allowed payment for order flow to continue to prevent exchanges from gaining monopoly power. What would happen to trades if the practice was outlawed was also unclear. SEC requires brokers to disclose financial arrangements with market makers. Since then, the SEC has watched closely.
2020 Order Flow Payment
Rule 605 and Rule 606 show execution quality and order flow payment statistics on a broker's website. Despite being required by the SEC, these reports can be hard to find. The SEC mandated these reports in 2005, but the format and reporting requirements have changed over the years, most recently in 2018.
Brokers and market makers formed a working group with the Financial Information Forum (FIF) to standardize order execution quality reporting. Only one retail brokerage (Fidelity) and one market maker remain (Two Sigma Securities). FIF notes that the 605/606 reports "do not provide the level of information that allows a retail investor to gauge how well a broker-dealer fills a retail order compared to the NBBO (national best bid or offer’) at the time the order was received by the executing broker-dealer."
In the first quarter of 2020, Rule 606 reporting changed to require brokers to report net payments from market makers for S&P 500 and non-S&P 500 equity trades and options trades. Brokers must disclose payment rates per 100 shares by order type (market orders, marketable limit orders, non-marketable limit orders, and other orders).
Richard Repetto, Managing Director of New York-based Piper Sandler & Co., publishes a report on Rule 606 broker reports. Repetto focused on Charles Schwab, TD Ameritrade, E-TRADE, and Robinhood in Q2 2020. Repetto reported that payment for order flow was higher in the second quarter than the first due to increased trading activity, and that options paid more than equities.
Repetto says PFOF contributions rose overall. Schwab has the lowest options rates, while TD Ameritrade and Robinhood have the highest. Robinhood had the highest equity rating. Repetto assumes Robinhood's ability to charge higher PFOF reflects their order flow profitability and that they receive a fixed rate per spread (vs. a fixed rate per share by the other brokers).
Robinhood's PFOF in equities and options grew the most quarter-over-quarter of the four brokers Piper Sandler analyzed, as did their implied volumes. All four brokers saw higher PFOF rates.
TD Ameritrade took the biggest income hit when cutting trading commissions in fall 2019, and this report shows they're trying to make up the shortfall by routing orders for additional PFOF. Robinhood refuses to disclose trading statistics using the same metrics as the rest of the industry, offering only a vague explanation on their website.
Summary
Payment for order flow has become a major source of revenue as brokers offer no-commission equity (stock and ETF) orders. For retail investors, payment for order flow poses a problem because the brokerage may route orders to a market maker for its own benefit, not the investor's.
Infrequent or small-volume traders may not notice their broker's PFOF practices. Frequent traders and those who trade larger quantities should learn about their broker's order routing system to ensure they're not losing out on price improvement due to a broker prioritizing payment for order flow.
This post is a summary. Read full article here
Shalitha Suranga
3 years ago
The Top 5 Mathematical Concepts Every Programmer Needs to Know
Using math to write efficient code in any language
Programmers design, build, test, and maintain software. Employ cases and personal preferences determine the programming languages we use throughout development. Mobile app developers use JavaScript or Dart. Some programmers design performance-first software in C/C++.
A generic source code includes language-specific grammar, pre-implemented function calls, mathematical operators, and control statements. Some mathematical principles assist us enhance our programming and problem-solving skills.
We all use basic mathematical concepts like formulas and relational operators (aka comparison operators) in programming in our daily lives. Beyond these mathematical syntaxes, we'll see discrete math topics. This narrative explains key math topics programmers must know. Master these ideas to produce clean and efficient software code.
Expressions in mathematics and built-in mathematical functions
A source code can only contain a mathematical algorithm or prebuilt API functions. We develop source code between these two ends. If you create code to fetch JSON data from a RESTful service, you'll invoke an HTTP client and won't conduct any math. If you write a function to compute the circle's area, you conduct the math there.
When your source code gets more mathematical, you'll need to use mathematical functions. Every programming language has a math module and syntactical operators. Good programmers always consider code readability, so we should learn to write readable mathematical expressions.
Linux utilizes clear math expressions.
Inbuilt max and min functions can minimize verbose if statements.
How can we compute the number of pages needed to display known data? In such instances, the ceil function is often utilized.
import math as m
results = 102
items_per_page = 10
pages = m.ceil(results / items_per_page)
print(pages)Learn to write clear, concise math expressions.
Combinatorics in Algorithm Design
Combinatorics theory counts, selects, and arranges numbers or objects. First, consider these programming-related questions. Four-digit PIN security? what options exist? What if the PIN has a prefix? How to locate all decimal number pairs?
Combinatorics questions. Software engineering jobs often require counting items. Combinatorics counts elements without counting them one by one or through other verbose approaches, therefore it enables us to offer minimum and efficient solutions to real-world situations. Combinatorics helps us make reliable decision tests without missing edge cases. Write a program to see if three inputs form a triangle. This is a question I commonly ask in software engineering interviews.
Graph theory is a subfield of combinatorics. Graph theory is used in computerized road maps and social media apps.
Logarithms and Geometry Understanding
Geometry studies shapes, angles, and sizes. Cartesian geometry involves representing geometric objects in multidimensional planes. Geometry is useful for programming. Cartesian geometry is useful for vector graphics, game development, and low-level computer graphics. We can simply work with 2D and 3D arrays as plane axes.
GetWindowRect is a Windows GUI SDK geometric object.
High-level GUI SDKs and libraries use geometric notions like coordinates, dimensions, and forms, therefore knowing geometry speeds up work with computer graphics APIs.
How does exponentiation's inverse function work? Logarithm is exponentiation's inverse function. Logarithm helps programmers find efficient algorithms and solve calculations. Writing efficient code involves finding algorithms with logarithmic temporal complexity. Programmers prefer binary search (O(log n)) over linear search (O(n)). Git source specifies O(log n):
Logarithms aid with programming math. Metas Watchman uses a logarithmic utility function to find the next power of two.
Employing Mathematical Data Structures
Programmers must know data structures to develop clean, efficient code. Stack, queue, and hashmap are computer science basics. Sets and graphs are discrete arithmetic data structures. Most computer languages include a set structure to hold distinct data entries. In most computer languages, graphs can be represented using neighboring lists or objects.
Using sets as deduped lists is powerful because set implementations allow iterators. Instead of a list (or array), store WebSocket connections in a set.
Most interviewers ask graph theory questions, yet current software engineers don't practice algorithms. Graph theory challenges become obligatory in IT firm interviews.
Recognizing Applications of Recursion
A function in programming isolates input(s) and output(s) (s). Programming functions may have originated from mathematical function theories. Programming and math functions are different but similar. Both function types accept input and return value.
Recursion involves calling the same function inside another function. In its implementation, you'll call the Fibonacci sequence. Recursion solves divide-and-conquer software engineering difficulties and avoids code repetition. I recently built the following recursive Dart code to render a Flutter multi-depth expanding list UI:
Recursion is not the natural linear way to solve problems, hence thinking recursively is difficult. Everything becomes clear when a mathematical function definition includes a base case and recursive call.
Conclusion
Every codebase uses arithmetic operators, relational operators, and expressions. To build mathematical expressions, we typically employ log, ceil, floor, min, max, etc. Combinatorics, geometry, data structures, and recursion help implement algorithms. Unless you operate in a pure mathematical domain, you may not use calculus, limits, and other complex math in daily programming (i.e., a game engine). These principles are fundamental for daily programming activities.
Master the above math fundamentals to build clean, efficient code.