Donor-Advised Fund Tax Benefits (DAF)
Giving through a donor-advised fund can be tax-efficient. Using a donor-advised fund can reduce your tax liability while increasing your charitable impact.
Grow Your Donations Tax-Free.
Your DAF's charitable dollars can be invested before being distributed. Your DAF balance can grow with the market. This increases grantmaking funds. The assets of the DAF belong to the charitable sponsor, so you will not be taxed on any growth.
Avoid a Windfall Tax Year.
DAFs can help reduce tax burdens after a windfall like an inheritance, business sale, or strong market returns. Contributions to your DAF are immediately tax deductible, lowering your taxable income. With DAFs, you can effectively pre-fund years of giving with assets from a single high-income event.
Make a contribution to reduce or eliminate capital gains.
One of the most common ways to fund a DAF is by gifting publicly traded securities. Securities held for more than a year can be donated at fair market value and are not subject to capital gains tax. If a donor liquidates assets and then donates the proceeds to their DAF, capital gains tax reduces the amount available for philanthropy. Gifts of appreciated securities, mutual funds, real estate, and other assets are immediately tax deductible up to 30% of Adjusted gross income (AGI), with a five-year carry-forward for gifts that exceed AGI limits.
Using Appreciated Stock as a Gift
Donating appreciated stock directly to a DAF rather than liquidating it and donating the proceeds reduces philanthropists' tax liability by eliminating capital gains tax and lowering marginal income tax.
In the example below, a donor has $100,000 in long-term appreciated stock with a cost basis of $10,000:
Using a DAF would allow this donor to give more to charity while paying less taxes. This strategy often allows donors to give more than 20% more to their favorite causes.
For illustration purposes, this hypothetical example assumes a 35% income tax rate. All realized gains are subject to the federal long-term capital gains tax of 20% and the 3.8% Medicare surtax. No other state taxes are considered.
The information provided here is general and educational in nature. It is not intended to be, nor should it be construed as, legal or tax advice. NPT does not provide legal or tax advice. Furthermore, the content provided here is related to taxation at the federal level only. NPT strongly encourages you to consult with your tax advisor or attorney before making charitable contributions.
More on Economics & Investing

Cory Doctorow
2 years ago
The current inflation is unique.
New Stiglitz just dropped.
Here's the inflation story everyone believes (warning: it's false): America gave the poor too much money during the recession, and now the economy is awash with free money, which made them so rich they're refusing to work, meaning the economy isn't making anything. Prices are soaring due to increased cash and missing labor.
Lawrence Summers says there's only one answer. We must impoverish the poor: raise interest rates, cause a recession, and eliminate millions of jobs, until the poor are stripped of their underserved fortunes and return to work.
https://pluralistic.net/2021/11/20/quiet-part-out-loud/#profiteering
This is nonsense. Countries around the world suffered inflation during and after lockdowns, whether they gave out humanitarian money to keep people from starvation. America has slightly greater inflation than other OECD countries, but it's not due to big relief packages.
The Causes of and Responses to Today's Inflation, a Roosevelt Institute report by Nobel-winning economist Joseph Stiglitz and macroeconomist Regmi Ira, debunks this bogus inflation story and offers a more credible explanation for inflation.
https://rooseveltinstitute.org/wp-content/uploads/2022/12/RI CausesofandResponsestoTodaysInflation Report 202212.pdf
Sharp interest rate hikes exacerbate the slump and increase inflation, the authors argue. They compare monetary policy inflation cures to medieval bloodletting, where doctors repeated the same treatment until the patient recovered (for which they received credit) or died (which was more likely).
Let's discuss bloodletting. Inflation hawks warn of the wage price spiral, when inflation rises and powerful workers bargain for higher pay, driving up expenses, prices, and wages. This is the fairy-tale narrative of the 1970s, and it's true except that OPEC's embargo drove up oil prices, which produced inflation. Oh well.
Let's be generous to seventies-haunted inflation hawks and say we're worried about a wage-price spiral. Fantastic! No. Real wages are 2.3% lower than they were in Oct 2021 after peaking in June at 4.8%.
Why did America's powerful workers take a paycut rather than demand inflation-based pay? Weak unions, globalization, economic developments.
Workers don't expect inflation to rise, so they're not requesting inflationary hikes. Inflationary expectations have remained moderate, consistent with our data interpretation.
https://www.newyorkfed.org/microeconomics/sce#/
Neither are workers. Working people see surplus savings as wealth and spend it gradually over their lives, despite rising demand. People may have saved money by staying in during the lockdown, but they don't eat out every night to make up for it. Instead, they keep those savings as precautionary balances. This is why the economy is lagging.
People don't buy non-traded goods with pandemic savings (basically, imports). Imports don't multiply like domestic purchases. If you buy a loaf of bread from the corner baker for $1 and they spend it at the tavern across the street, that dollar generates $3 in economic activity. Spending a dollar on foreign goods leaves the country and any multiplier effect happens there, not in the US.
Only marginally higher wages. The ECI is up 1.6% from 2019. Almost all gains went to the 25% lowest-paid Americans. Contrary to the inflation worry about too much savings, these workers don't make enough to save, even post-pandemic.
Recreation and transit spending are at or below pre-pandemic levels. Higher food and hotel prices (which doesn’t mean we’re buying more food than we were in 2019, just that it costs more).
What causes inflation if not greedy workers, free money, and high demand? The most expensive domestic goods produce the biggest revenues for their manufacturers. They charge you more without paying their workers or suppliers more.
The largest price-gougers are funneling their earnings to rich people who store it offshore through stock buybacks and dividends. A $1 billion stock buyback doesn't buy $1 billion in bread.
Five factors influence US inflation today:
I. Price rises for energy and food
II. shifts in consumer tastes
III. supply interruptions (mainly autos);
IV. increased rents (due to telecommuting);
V. monopoly (AKA price-gouging).
None can be remedied by raising interest rates or laying off workers.
Russia's invasion of Ukraine, omicron, and China's Zero Covid policy all disrupted the flow of food, energy, and production inputs. The price went higher because we made less.
After Russia invaded Ukraine, oil prices spiked, and sanctions made it worse. But that was February. By October, oil prices had returned to pre-pandemic, 2015 levels attributable to global economic adjustments, including a shift to renewables. Every new renewable installation reduces oil consumption and affects oil prices.
High food prices have a simple solution. The US and EU have bribed farmers not to produce for 50 years. If the war continues, this program may end, and food prices may decline.
Demand changes. We want different things than in 2019, not more. During the lockdown, people substituted goods. Half of the US toilet-paper supply in 2019 was on commercial-sized rolls. This is created from different mills and stock than our toilet paper.
Lockdown pushed toilet paper demand to residential rolls, causing shortages (the TP hoarding story was just another pandemic urban legend). Because supermarket stores don't have accounts with commercial paper distributors, ordering from languishing stores was difficult. Kleenex and paper towel substitutions caused greater shortages.
All that drove increased costs in numerous product categories, and there were more cases. These increases are transient, caused by supply chain inefficiencies that are resolving.
Demand for frontline staff saw a one-time repricing of pay, which is being recouped as we speak.
Illnesses. Brittle, hollowed-out global supply chains aggravated this. The constant pursuit of cheap labor and minimal regulation by monopolies that dominate most sectors means things are manufactured in far-flung locations. Financialization means any surplus capital assets were sold off years ago, leaving firms with little production slack. After the epidemic, several of these systems took years to restart.
Automobiles are to blame. Financialization and monopolization consolidated microchip and auto production in Taiwan and China. When the lockdowns came, these worldwide corporations cancelled their chip orders, and when they placed fresh orders, they were at the back of the line.
That drove up car prices, which is why the US has slightly higher inflation than other wealthy countries: the economy is car-centric. Automobile prices account for 9% of the CPI. France: 3.6%
Rent shocks and telecommuting. After the epidemic, many professionals moved to exurbs, small towns, and the countryside to work from home. As commercial properties were vacated, it was impractical to adapt them for residential use due to planning restrictions. Addressing these restrictions will cut rent prices more than raising inflation rates, which halts housing construction.
Statistical mirages cause some rent inflation. The CPI estimates what homeowners would pay to rent their properties. When rents rise in your neighborhood, the CPI believes you're spending more on rent even if you have a 30-year fixed-rate mortgage.
Market dominance. Almost every area of the US economy is dominated by monopolies, whose CEOs disclose on investor calls that they use inflation scares to jack up prices and make record profits.
https://pluralistic.net/2022/02/02/its-the-economy-stupid/#overinflated
Long-term profit margins are rising. Markups averaged 26% from 1960-1980. 2021: 72%. Market concentration explains 81% of markup increases (e.g. monopolization). Profit margins reach a 70-year high in 2022. These elements interact. Monopolies thin out their sectors, making them brittle and sensitive to shocks.
If we're worried about a shrinking workforce, there are more humanitarian and sensible solutions than causing a recession and mass unemployment. Instead, we may boost US production capacity by easing workers' entry into the workforce.
https://pluralistic.net/2022/06/01/factories-to-condos-pipeline/#stuff-not-money
US female workforce participation ranks towards the bottom of developed countries. Many women can't afford to work due to America's lack of daycare, low earnings, and bad working conditions in female-dominated fields. If America doesn't have enough workers, childcare subsidies and minimum wages can help.
By contrast, driving the country into recession with interest-rate hikes will reduce employment, and the last recruited (women, minorities) are the first fired and the last to be rehired. Forcing America into recession won't enhance its capacity to create what its people want; it will degrade it permanently.
Nothing the Fed does can stop price hikes from international markets, lack of supply chain investment, COVID-19 disruptions, climate change, the Ukraine war, or market power. They can worsen it. When supply problems generate inflation, raising interest rates decreases investments that can remedy shortages.
Increasing interest rates won't cut rents since landlords pass on the expenses and high rates restrict investment in new dwellings where tenants could escape the costs.
Fixing the supply fixes supply-side inflation. Increase renewables investment (as the Inflation Reduction Act does). Monopolies can be busted (as the IRA does). Reshore key goods (as the CHIPS Act does). Better pay and child care attract employees.
Windfall taxes can claw back price-gouging corporations' monopoly earnings.
https://pluralistic.net/2022/03/15/sanctions-financing/#soak-the-rich
In 2008, we ruled out fiscal solutions (bailouts for debtors) and turned to monetary policy (bank bailouts). This preserved the economy but increased inequality and eroded public trust.
Monetary policy won't help. Even monetary policy enthusiasts recognize an 18-month lag between action and result. That suggests monetary tightening is unnecessary. Like the medieval bloodletter, central bankers whose interest rate hikes don't work swiftly may do more of the same, bringing the economy to its knees.
Interest rates must rise. Zero-percent interest fueled foolish speculation and financialization. Increasing rates will stop this. Increasing interest rates will destroy the economy and dampen inflation.
Then what? All recent evidence indicate to inflation decreasing on its own, as the authors argue. Supply side difficulties are finally being overcome, evidence shows. Energy and food prices are showing considerable mean reversion, which is disinflationary.
The authors don't recommend doing nothing. Best case scenario, they argue, is that the Fed won't keep raising interest rates until morale improves.

Trevor Stark
3 years ago
Economics is complete nonsense.
Mainstream economics haven't noticed.
What come to mind when I say the word "economics"?
Probably GDP, unemployment, and inflation.
If you've ever watched the news or listened to an economist, they'll use data like these to defend a political goal.
The issue is that these statistics are total bunk.
I'm being provocative, but I mean it:
The economy is not measured by GDP.
How many people are unemployed is not counted in the unemployment rate.
Inflation is not measured by the CPI.
All orthodox economists' major economic statistics are either wrong or falsified.
Government institutions create all these stats. The administration wants to reassure citizens the economy is doing well.
GDP does not reflect economic expansion.
GDP measures a country's economic size and growth. It’s calculated by the BEA, a government agency.
The US has the world's largest (self-reported) GDP, growing 2-3% annually.
If GDP rises, the economy is healthy, say economists.
Why is the GDP flawed?
GDP measures a country's yearly spending.
The government may adjust this to make the economy look good.
GDP = C + G + I + NX
C = Consumer Spending
G = Government Spending
I = Investments (Equipment, inventories, housing, etc.)
NX = Exports minus Imports
GDP is a country's annual spending.
The government can print money to boost GDP. The government has a motive to increase and manage GDP.
Because government expenditure is part of GDP, printing money and spending it on anything will raise GDP.
They've done this. Since 1950, US government spending has grown 8% annually, faster than GDP.
In 2022, government spending accounted for 44% of GDP. It's the highest since WWII. In 1790-1910, it was 3% of GDP.
Who cares?
The economy isn't only spending. Focus on citizens' purchasing power or quality of life.
Since GDP just measures spending, the government can print money to boost GDP.
Even if Americans are poorer than last year, economists can say GDP is up and everything is fine.
How many people are unemployed is not counted in the unemployment rate.
The unemployment rate measures a country's labor market. If unemployment is high, people aren't doing well economically.
The BLS estimates the (self-reported) unemployment rate as 3-4%.
Why is the unemployment rate so high?
The US government surveys 100k persons to measure unemployment. They extrapolate this data for the country.
They come into 3 categories:
Employed
People with jobs are employed … duh.
Unemployed
People who are “jobless, looking for a job, and available for work” are unemployed
Not in the labor force
The “labor force” is the employed + the unemployed.
The unemployment rate is the percentage of unemployed workers.
Problem is unemployed definition. You must actively seek work to be considered unemployed.
You're no longer unemployed if you haven't interviewed in 4 weeks.
This shit makes no goddamn sense.
Why does this matter?
You can't interview if there are no positions available. You're no longer unemployed after 4 weeks.
In 1994, the BLS redefined "unemployed" to exclude discouraged workers.
If you haven't interviewed in 4 weeks, you're no longer counted in the unemployment rate.
If unemployment were measured by total unemployed, it would be 25%.
Because the government wants to keep the unemployment rate low, they modify the definition.
If every US resident was unemployed and had no job interviews, economists would declare 0% unemployment. Excellent!
Inflation is not measured by the CPI.
The BLS measures CPI. This month was the highest since 1981.
CPI measures the cost of a basket of products across time. Food, energy, shelter, and clothes are included.
A 9.1% CPI means the basket of items is 9.1% more expensive.
What is the CPI problem?
Here's a more detailed explanation of CPI's flaws.
In summary, CPI is manipulated to be understated.
Housing costs are understated to manipulate CPI. Housing accounts for 33% of the CPI because it's the biggest expense for most people.
This signifies it's the biggest CPI weight.
Rather than using actual house prices, the Bureau of Labor Statistics essentially makes shit up. You can read more about the process here.
Surprise! It’s bullshit
The BLS stated Shelter's price rose 5.5% this month.
House prices are up 11-21%. (Source 1, Source 2, Source 3)
Rents are up 14-26%. (Source 1, Source 2)
Why is this important?
If CPI included housing prices, it would be 12-15 percent this month, not 9.1 percent.
9% inflation is nuts. Your money's value halves every 7 years at 9% inflation.
Worse is 15% inflation. Your money halves every 4 years at 15% inflation.
If everyone realized they needed to double their wage every 4-5 years to stay wealthy, there would be riots.
Inflation drains our money's value so the government can keep printing it.
The Solution
Most individuals know the existing system doesn't work, but can't explain why.
People work hard yet lag behind. The government lies about the economy's data.
In reality:
GDP has been down since 2008
25% of Americans are unemployed
Inflation is actually 15%
People might join together to vote out kleptocratic politicians if they knew the reality.
Having reliable economic data is the first step.
People can't understand the situation without sufficient information. Instead of immigrants or billionaires, people would blame liar politicians.
Here’s the vision:
A decentralized, transparent, and global dashboard that tracks economic data like GDP, unemployment, and inflation for every country on Earth.
Government incentives influence economic statistics.
ShadowStats has already started this effort, but the calculations must be transparent, decentralized, and global to be effective.
If interested, email me at trevorstark02@gmail.com.
Here are some links to further your research:

Wayne Duggan
3 years ago
What An Inverted Yield Curve Means For Investors
The yield spread between 10-year and 2-year US Treasury bonds has fallen below 0.2 percent, its lowest level since March 2020. A flattening or negative yield curve can be a bad sign for the economy.
What Is An Inverted Yield Curve?
In the yield curve, bonds of equal credit quality but different maturities are plotted. The most commonly used yield curve for US investors is a plot of 2-year and 10-year Treasury yields, which have yet to invert.
A typical yield curve has higher interest rates for future maturities. In a flat yield curve, short-term and long-term yields are similar. Inverted yield curves occur when short-term yields exceed long-term yields. Inversions of yield curves have historically occurred during recessions.
Inverted yield curves have preceded each of the past eight US recessions. The good news is they're far leading indicators, meaning a recession is likely not imminent.
Every US recession since 1955 has occurred between six and 24 months after an inversion of the two-year and 10-year Treasury yield curves, according to the San Francisco Fed. So, six months before COVID-19, the yield curve inverted in August 2019.
Looking Ahead
The spread between two-year and 10-year Treasury yields was 0.18 percent on Tuesday, the smallest since before the last US recession. If the graph above continues, a two-year/10-year yield curve inversion could occur within the next few months.
According to Bank of America analyst Stephen Suttmeier, the S&P 500 typically peaks six to seven months after the 2s-10s yield curve inverts, and the US economy enters recession six to seven months later.
Investors appear unconcerned about the flattening yield curve. This is in contrast to the iShares 20+ Year Treasury Bond ETF TLT +2.19% which was down 1% on Tuesday.
Inversion of the yield curve and rising interest rates have historically harmed stocks. Recessions in the US have historically coincided with or followed the end of a Federal Reserve rate hike cycle, not the start.
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Julie Plavnik
3 years ago
Why the Creator Economy needs a Web3 upgrade
Looking back into the past can help you understand what's happening today and why.
"Creator economy" conjures up images of originality, sincerity, and passion. Where do Michelangelos and da Vincis push advancement with their gifts without battling for bread and proving themselves posthumously?
Creativity has been as long as humanity, but it's just recently become a new economic paradigm. We even talk about Web3 now.
Let's examine the creative economy's history to better comprehend it. What brought us here? Looking back can help you understand what's happening now.
No yawning, I promise 😉.
Creator Economy's history
Long, uneven transition to creator economy. Let's examine the economic and societal changes that led us there.
1. Agriculture to industry
Mid-18th-century Industrial Revolution led to shift from agriculture to manufacturing. The industrial economy lasted until World War II.
The industrial economy's principal goal was to provide more affordable, accessible commodities.
Unlike today, products were scarce and inaccessible.
To fulfill its goals, industrialization triggered enormous economic changes, moving power from agrarians to manufacturers. Industrialization brought hard work, rivalry, and new ideas connected to production and automation. Creative thinkers focused on that then.
It doesn't mean music, poetry, or painting had no place back then. They weren't top priority. Artists were independent. The creative field wasn't considered a different economic subdivision.
2. The consumer economy
Manufacturers produced more things than consumers desired after World War II. Stuff was no longer scarce.
The economy must make customers want to buy what the market offers.
The consumer economic paradigm supplanted the industrial one. Customers (or consumers) replaced producers as the new economic center.
Salesmen, marketing, and journalists also played key roles (TV, radio, newspapers, etc.). Mass media greatly boosted demand for goods, defined trends, and changed views regarding nearly everything.
Mass media also gave rise to pop culture, which focuses on mass-market creative products. Design, printing, publishing, multi-media, audio-visual, cinematographic productions, etc. supported pop culture.
The consumer paradigm generated creative occupations and activities, unlike the industrial economy. Creativity was limited by the need for wide appeal.
Most creators were corporate employees.
Creating a following and making a living from it were difficult.
Paul Saffo said that only journalists and TV workers were known. Creators who wished to be known relied on producers, publishers, and other gatekeepers. To win their favor was crucial. Luck was the best tactic.
3. The creative economy
Consumer economy was digitized in the 1990s. IT solutions transformed several economic segments. This new digital economy demanded innovative, digital creativity.
Later, states declared innovation a "valuable asset that creates money and jobs." They also introduced the "creative industries" and the "creative economy" (not creator!) and tasked themselves with supporting them. Australia and the UK were early adopters.
Individual skill, innovation, and intellectual property fueled the creative economy. Its span covered design, writing, audio, video material, etc. The creative economy required IT-powered activity.
The new challenge was to introduce innovations to most economic segments and meet demand for digital products and services.
Despite what the title "creative economy" may imply, it was primarily oriented at meeting consumer needs. It didn't provide inventors any new options to become entrepreneurs. Instead of encouraging innovators to flourish on their own, the creative economy emphasized "employment-based creativity."
4. The creator economy
Next, huge IT platforms like Google, Facebook, YouTube, and others competed with traditional mainstream media.
During the 2008 global financial crisis, these mediums surpassed traditional media. People relied on them for information, knowledge, and networking. That was a digital media revolution. The creator economy started there.
The new economic paradigm aimed to engage and convert clients. The creator economy allowed customers to engage, interact, and provide value, unlike the consumer economy. It gave them instruments to promote themselves as "products" and make money.
Writers, singers, painters, and other creators have a great way to reach fans. Instead of appeasing old-fashioned gatekeepers (producers, casting managers, publishers, etc.), they can use the platforms to express their talent and gain admirers. Barriers fell.
It's not only for pros. Everyone with a laptop and internet can now create.
2022 creator economy:
Since there is no academic description for the current creator economy, we can freestyle.
The current (or Web2) creator economy is fueled by interactive digital platforms, marketplaces, and tools that allow users to access, produce, and monetize content.
No entry hurdles or casting in the creative economy. Sign up and follow platforms' rules. Trick: A platform's algorithm aggregates your data and tracks you. This is the payment for participation.
The platforms offer content creation, design, and ad distribution options. This is platforms' main revenue source.
The creator economy opens many avenues for creators to monetize their work. Artists can now earn money through advertising, tipping, brand sponsorship, affiliate links, streaming, and other digital marketing activities.
Even if your content isn't digital, you can utilize platforms to promote it, interact and convert your audience, and more. No limits. However, some of your income always goes to a platform (well, a huge one).
The creator economy aims to empower online entrepreneurship by offering digital marketing tools and reducing impediments.
Barriers remain. They are just different. Next articles will examine these.
Why update the creator economy for Web3?
I could address this question by listing the present creator economy's difficulties that led us to contemplate a Web3 upgrade.
I don't think these difficulties are the main cause. The mentality shift made us see these challenges and understand there was a better reality without them.
Crypto drove this thinking shift. It promoted disintermediation, independence from third-party service providers, 100% data ownership, and self-sovereignty. Crypto has changed the way we view everyday things.
Crypto's disruptive mission has migrated to other economic segments. It's now called Web3. Web3's creator economy is unique.
Here's the essence of the Web3 economy:
Eliminating middlemen between creators and fans.
100% of creators' data, brand, and effort.
Business and money-making transparency.
Authentic originality above ad-driven content.
In the next several articles, I'll explain. We'll also discuss the creator economy and Web3's remedies.
Final thoughts
The creator economy is the organic developmental stage we've reached after all these social and economic transformations.
The Web3 paradigm of the creator economy intends to allow creators to construct their own independent "open economy" and directly monetize it without a third party.
If this approach succeeds, we may enter a new era of wealth creation where producers aren't only the products. New economies will emerge.
This article is a summary. To read the full post, click here.

Vitalik
3 years ago
An approximate introduction to how zk-SNARKs are possible (part 2)
If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.
Polynomials
Polynomials are a special class of algebraic expressions of the form:
- x+5
- x^4
- x^3+3x^2+3x+1
- 628x^{271}+318x^{270}+530x^{269}+…+69x+381
i.e. they are a sum of any (finite!) number of terms of the form cx^k
There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.
Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:
- A(0)+B(0)=C(0)
- A(1)+B(1)=C(1)
- A(2)+B(2)=C(2)
- A(3)+B(3)=C(3)
And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:
- 12+1=13
- 10+8=18
- 15+8=23
- 15+13=28
You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:
- A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
- B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
- C(x)=5x+13
Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.
Comparing a polynomial to itself
You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)
As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.
Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).
Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!
Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.
see part 3

Will Lockett
3 years ago
Russia's nukes may be useless
Russia's nuclear threat may be nullified by physics.
Putin seems nostalgic and wants to relive the Cold War. He's started a deadly war to reclaim the old Soviet state of Ukraine and is threatening the West with nuclear war. NATO can't risk starting a global nuclear war that could wipe out humanity to support Ukraine's independence as much as they want to. Fortunately, nuclear physics may have rendered Putin's nuclear weapons useless. However? How will Ukraine and NATO react?
To understand why Russia's nuclear weapons may be ineffective, we must first know what kind they are.
Russia has the world's largest nuclear arsenal, with 4,447 strategic and 1,912 tactical weapons (all of which are ready to be rolled out quickly). The difference between these two weapons is small, but it affects their use and logistics. Strategic nuclear weapons are ICBMs designed to destroy a city across the globe. Russia's ICBMs have many designs and a yield of 300–800 kilotonnes. 300 kilotonnes can destroy Washington. Tactical nuclear weapons are smaller and can be fired from artillery guns or small truck-mounted missile launchers, giving them a 1,500 km range. Instead of destroying a distant city, they are designed to eliminate specific positions, bases, or military infrastructure. They produce 1–50 kilotonnes.
These two nuclear weapons use different nuclear reactions. Pure fission bombs are compact enough to fit in a shell or small missile. All early nuclear weapons used this design for their fission bombs. This technology is inefficient for bombs over 50 kilotonnes. Larger bombs are thermonuclear. Thermonuclear weapons use a small fission bomb to compress and heat a hydrogen capsule, which undergoes fusion and releases far more energy than ignition fission reactions, allowing for effective giant bombs.
Here's Russia's issue.
A thermonuclear bomb needs deuterium (hydrogen with one neutron) and tritium (hydrogen with two neutrons). Because these two isotopes fuse at lower energies than others, the bomb works. One problem. Tritium is highly radioactive, with a half-life of only 12.5 years, and must be artificially made.
Tritium is made by irradiating lithium in nuclear reactors and extracting the gas. Tritium is one of the most expensive materials ever made, at $30,000 per gram.
Why does this affect Putin's nukes?
Thermonuclear weapons need tritium. Tritium decays quickly, so they must be regularly refilled at great cost, which Russia may struggle to do.
Russia has a smaller economy than New York, yet they are running an invasion, fending off international sanctions, and refining tritium for 4,447 thermonuclear weapons.
The Russian military is underfunded. Because the state can't afford it, Russian troops must buy their own body armor. Arguably, Putin cares more about the Ukraine conflict than maintaining his nuclear deterrent. Putin will likely lose power if he loses the Ukraine war.
It's possible that Putin halted tritium production and refueling to save money for Ukraine. His threats of nuclear attacks and escalating nuclear war may be a bluff.
This doesn't help Ukraine, sadly. Russia's tactical nuclear weapons don't need expensive refueling and will help with the invasion. So Ukraine still risks a nuclear attack. The bomb that destroyed Hiroshima was 15 kilotonnes, and Russia's tactical Iskander-K nuclear missile has a 50-kiloton yield. Even "little" bombs are deadly.
We can't guarantee it's happening in Russia. Putin may prioritize tritium. He knows the power of nuclear deterrence. Russia may have enough tritium for this conflict. Stockpiling a material with a short shelf life is unlikely, though.
This means that Russia's most powerful weapons may be nearly useless, but they may still be deadly. If true, this could allow NATO to offer full support to Ukraine and push the Russian tyrant back where he belongs. If Putin withholds funds from his crumbling military to maintain his nuclear deterrent, he may be willing to sink the ship with him. Let's hope the former.