granola

granola

noun

• a mixture typically of rolled oats and various added ingredients (such as brown sugar, raisins, coconut, and nuts) that is eaten especially for breakfast or as a snack

granola bar

Granola

almond

almond

noun

Almond

burrito

burrito

noun

• a flour tortilla rolled or folded around a filling (as of meat, beans, and cheese)

Burrito

casserole

casserole

noun

I made tuna casserole for dinner.

meat and noodles served in a casserole

Casserole

artichoke

artichoke

noun

Artichoke

slab

slab

noun

a slab of stone

Concrete slab

Stone slab

sanction

sanction

noun

Countries around the world are imposing fresh sanctions against Russia over its invasion of Ukraine.

What sanctions are being imposed on Russia?

crisp

crisp

adjective

• notably sharp, clean-cut, and clear

a crisp illustration

• concise and to the point

a crisp reply

crumple

crumple

verb

She crumpled the piece of paper into a ball and tossed it into the garbage can.

The car’s fender was crumpled in the accident.

taco

taco

noun

• a corn tortilla usually folded and fried and filled with a spicy mixture (as of ground meat and cheese)

Taco

albatross

albatross

noun

Albatross

kiosk

kiosk

noun

a museum with interactive kiosks

You can pick up your plane tickets at one of the airport’s kiosks.

Interactive kiosk

compass

compass

noun

Compass

Compass (drawing tool)

prance

prance

verb

horses prancing

She pranced around in her new dress.

blaze

blaze

noun

a blaze of color

the blaze of the sun

paycheck

paycheck

noun

I went to the office to pick up my paycheck.

Your weekly paycheck will be almost $600 after taxes.

Paycheck

As inflation heats up, 64% of Americans are now living paycheck to paycheck

straw

straw

noun

• a tube (as of paper, plastic, or glass) for sucking up a beverage
• a slender tube for sucking up a beverage

She drank the juice through a straw.

Drinking straw

clipboard

clipboard

noun

• a small writing board with a clip at the top for holding papers

Clipboard

• a section of computer memory that temporarily stores data (such as text or a graphics image) especially to facilitate its movement or duplication

Copy the sentence to the clipboard and paste it in a new document.

Clipboard (computing)

denim

denim

noun

He’s wearing faded denims and cowboy boots.

Denim

equinox

equinox

noun

March equinox

Spring equinox

agenda

agenda

noun

• a list of things to be done or talked about

There are several items on the agenda for tonight’s meeting.

What’s the first item on the agenda?

Agenda (meeting)

by Keith Devlin

November 2002

In late October, my new book The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time went on sale across the country, and this month sees me doing the usual round of public lectures, bookstore talks, and magazine, radio and TV interviews that these days accompany the publication of any new book the publisher thinks has even the ghost of a chance of becoming the next popular science bestseller.

Of all the books I have written for a general audience, this latest one presented by far the greatest challenge in trying to make it as accessible as possible to non-mathematicians. The seven unsolved problems I discuss – the Clay Millennium Problems – were chosen by a small, stellar, international committee of leading mathematicians appointed by the Clay Mathematics Institute, which offers a cash prize of $1 million to the first person to solve any one of the problems. The committee’s mission was to select the most difficult and most significant unsolved problems at the end of the second millennium, problems that had for many years resisted the efforts of some of the world’s greatest mathematicians to find a solution.

No one who is at all familiar with modern mathematics will be surprised to find that none of the seven problems chosen is likely to be solved by elementary methods, and even the statement of most of the problems cannot be fully understood by anyone who has not completed a mathematics major at a university.

In writing the book, I had to ignore the oft-repeated assertion that every mathematical formula you put in a book decreases the sales by 50%. (Personally, I don’t think this is literally true, but I do believe that having pages of formulas does put off a lot of potential readers.) Although my book is mostly prose, there are formulas, and some chapters have technical appendices that are little else but formulas.

Now, as I gear up for the promotional campaign, I face the same challenge again. With the book, I think I found a way to present the story of the Millennium Problems in 250 pages of text. But what can I say about the book’s contents in a twenty minutes talk in a bookstore or a ten minute interview on a radio talk show? Thinking about this made me reflect once more about the nature of modern mathematics. Put simply: Why are the Millennium Problems so hard to understand?

Imagine for a moment that Landon Clay – the wealthy mutual fund magnate who founded the Clay Institute and provided the $7 million of prize money for the seven problems – had chosen to establish his prize competition not for mathematics but for some other science, say physics, or chemistry, or biology. It surely would not have taken an entire book to explain to an interested lay audience the seven major problems in one of those disciplines. A three or four page expository article in Scientific American or 1,500 words in New Scientist would probably suffice. Indeed, when the Nobel Prizes are awarded each year, newspapers and magazines frequently manage to convey the gist of the prize-winning research in a few paragraphs. In general you can’t do that with mathematics. Mathematics is different. But how?

Part of the answer can be found in an observation first made (I believe) by the American mathematician Ronald Graham, who for most of his career was the head of mathematical research at AT&T Bell Laboratories. According to Graham, a mathematician is the only scientist who can legitimately claim: “I lie down on the couch, close my eyes, and work.”

Mathematics is almost entirely cerebral – the actual work is done not in a laboratory or an office or a factory, but in the head. Of course, that head is attached to a body which might well be in an office – or on a couch – but the mathematics itself goes on in the brain, without any direct connection to something in the physical world. This is not to imply that other scientists don’t do mental work. But in physics or chemistry or biology, the object of the scientist’s thought is generally some phenomenon in the physical world. Although you and I cannot get inside the scientist’s mind and experience her thoughts, we do live in the same world, and that provides the key connection, an initial basis for the scientist to explain her thoughts to us. Even in the case of physicists trying to understand quarks or biologists grappling with DNA, although we have no everyday experience of those objects, even a nonscientifically trained mind has no trouble thinking about them. In a deep sense, the typical artist’s renderings of quarks as clusters of colored billiard balls and DNA as a spiral staircase might well be (in fact are) “wrong,” but as mental pictures that enable us to visualize the science they work just fine.

Mathematics does not have this. Even when it is possible to draw a picture, more often than not the illustration is likely to mislead as much as it helps, which leaves the expositor having to make up with words what is lacking or misleading in the picture. But how can the nonmathematical reader understand those words, when they in turn don’t link to anything in everyday experience?

Even for the committed spectator of mathematics, this task is getting harder as the subject grows more and more abstract and the objects the mathematician discusses become further and further removed from the everyday world. Indeed, for some contemporary problems, such as the Hodge Conjecture – one of the seven Millennium Problems – we may have already reached the point where the outsider simply can’t make the connection. It’s not that the human mind requires time to come to terms with new levels of abstraction. That’s always been the case. Rather, the degree and the pace of abstraction may have finally reached a stage where only the expert can keep up.

Two and a half thousand years ago, a young follower of Pythagoras proved that the square root of 2 is not a rational number, that is, cannot be expressed as a fraction. This meant that what they took to be the numbers (the whole numbers and the fractions) were not adequate to measure the length of the hypotenuse of a right triangle with width and height both equal to 1 unit (which Pythagoras’ theorem says will have length the square root of 2). This discovery came as such a shock to the Pythagoreans that their progress in mathematics came to a virtual halt. Eventually, mathematicians found a way out of the dilemma, by changing their conception of what a number is to what we nowadays call the real numbers.

To the Greeks, numbers began with counting (the natural numbers) and in order to measure lengths you extended them to a richer system (the rational numbers) by declaring that the result of dividing one natural number by another was itself a number. The discovery that the rational numbers were not in fact adequate for measuring lengths led later mathematicians to abandon this picture, and instead declare that numbers simply are the points on a line! This was a major change, and it took two thousand years for all the details to be worked out. Only toward the end of the nineteenth century did mathematicians finally work out a rigorous theory of the real numbers. Even today, despite the simple picture of the real numbers as the points on a line, university students of mathematics always have trouble grasping the formal (and highly abstract) development of the real numbers.

Numbers less than zero presented another struggle. These days we think of negative numbers as simply the points on the number line that lie to the left of 0, but mathematicians resisted their introduction until the end of the seventeenth century. Similarly, most people have difficulty coming to terms with complex numbers – numbers that involve the square root of negative quantities – even though there is a simple intuitive picture of the complex numbers as the points in a two-dimensional plane.

These days, even many nonmathematicians feel comfortable using real numbers, complex numbers, and negative numbers. That is despite the fact that these are highly abstract concepts that bear little relationship with counting, the process with which numbers began some ten thousand years ago, and even though, in our everyday lives, we never encounter a concrete example of an irrational real number or a number involving the square root of -1.

Similarly in geometry, the discovery in the eighteenth century that there were other geometries besides the one that Euclid had described in his famous book Elements caused both the experts and the nonmathematicians enormous conceptual problems. Only during the nineteenth century did the idea of “non-Euclidean geometries“ gain widespread acceptance. That acceptance came even though the world of our immediate, everyday experience is entirely Euclidean.

With each new conceptual leap, even mathematicians need time to come to terms with the new ideas, to accept them as part of the overall background against which they do their work. Until recently, the pace of progress in mathematics was such that, by and large, the interested observer could catch up with one new advance before the next one came along. But it has been getting steadily harder. To understand what the Riemann Hypothesis says, the first problem on the Millennium list, you need to have understood, and feel comfortable with, not only complex numbers (and their arithmetic) but also advanced calculus, and what it means to add together infinitely many (complex) numbers and to multiply together infinitely many (complex) numbers.

Now that kind of knowledge is restricted almost entirely to people who have majored in mathematics at university. Only they are in a position to see the Riemann Hypothesis as a simple statement, not significantly different from the way an average person views Pythagoras’ theorem. My task in writing my book, then, was not only to explain what the Riemann Hypothesis says but to provide all of the preliminary material as well. Clearly, I cannot do that in a ten minute radio interview!

The root of the problem is that, in most cases, the preparatory material cannot be explained in terms of everyday phenomena, the way that physicists, for example, can explain the latest, deepest, cutting-edge theory of the universe – Superstring Theory – in terms of the intuitively simple picture of tiny, vibrating loops of energy (the “strings” of the theory).

Most mathematical concepts are built up not from everyday phenomena but from earlier mathematical concepts. That means that the only route to getting even a superficial understanding of those concepts is to follow the entire chain of abstractions that leads to them. My readers will decide how well I succeed in the book. But that avenue is not available to me in a short talk.

Perhaps, then, instead of trying to describe the Millennium Problems themselves, I’ll tell my audiences why they are so hard to understand. I’ll explain that the concepts involved in the Millennium Problems are not so much inherently difficult – for they are not – as they are very, very unfamiliar. Much as the idea of complex numbers or non-Euclidean geometries would have seemed incomprehensibly strange to the ancient Greeks. Today, having grown familiar with these ideas, we can see how they grow naturally out of concepts the Greeks knew as commonplace mathematics.

Perhaps the best way to approach the Millennium Problems, I will say, is to think of the seven problems as the commonplace mathematics of the 25th century.

And maybe that will turn out to be the case.

• The inaccessibility of modern mathematics - Devlin’s Angle
• Keith Devlin at STANFORD UNIVERSITY

The contents of this metal cylinder could either revolutionize technology or be completely useless— it all depends on whether we can harness the strange physics of matter at very, very small scales. To have even a chance of doing so, we have to control the environment precisely: the thick tabletop and legs guard against vibrations from footsteps, nearby elevators, and opening or closing doors. The cylinder is a vacuum chamber, devoid of all the gases in air. Inside the vacuum chamber is a smaller, extremely cold compartment, reachable by tiny laser beams. Inside are ultra-sensitive particles that make up a quantum computer.

So what makes these particles worth the effort? In theory, quantum computers could outstrip the computational limits of classical computers. Classical computers process data in the form of bits. Each bit can switch between two states labeled zero and one. A quantum computer uses something called a qubit, which can switch between zero, one, and what’s called a superposition. While the qubit is in its superposition, it has a lot more information than one or zero. You can think of these positions as points on a sphere: the north and south poles of the sphere represent one and zero. A bit can only switch between these two poles, but when a qubit is in its superposition, it can be at any point on the sphere. We can’t locate it exactly — the moment we read it, the qubit resolves into a zero or a one. But even though we can’t observe the qubit in its superposition, we can manipulate it to perform particular operations while in this state.

So as a problem grows more complicated, a classical computer needs correspondingly more bits to solve it, while a quantum computer will theoretically be able to handle more and more complicated problems without requiring as many more qubits as a classical computer would need bits.

The unique properties of quantum computers result from the behavior of atomic and subatomic particles. These particles have quantum states, which correspond to the state of the qubit. Quantum states are incredibly fragile, easily destroyed by temperature and pressure fluctuations, stray electromagnetic fields, and collisions with nearby particles. That’s why quantum computers need such an elaborate set up. It’s also why, for now, the power of quantum computers remains largely theoretical. So far, we can only control a few qubits in the same place at the same time.

There are two key components involved in managing these fickle quantum states effectively: the types of particles a quantum computer uses, and how it manipulates those particles. For now, there are two leading approaches: trapped ions and superconducting qubits.

A trapped ion quantum computer uses ions as its particles and manipulates them with lasers. The ions are housed in a trap made of electrical fields. Inputs from the lasers tell the ions what operation to make by causing the qubit state to rotate on the sphere. To use a simplified example, the lasers could input the question: what are the prime factors of 15? In response, the ions may release photons — the state of the qubit determines whether the ion emits photons and how many photons it emits. An imaging system collects these photons and processes them to reveal the answer: 3 and 5.

Superconducting qubit quantum computers do the same thing in a different way: using a chip with electrical circuits instead of an ion trap. The states of each electrical circuit translate to the state of the qubit. They can be manipulated with electrical inputs in the form of microwaves. So: the qubits come from either ions or electrical circuits, acted on by either lasers or microwaves. Each approach has advantages and disadvantages. Ions can be manipulated very precisely, and they last a long time, but as more ions are added to a trap, it becomes increasingly difficult to control each with precision. We can’t currently contain enough ions in a trap to make advanced computations, but one possible solution might be to connect many smaller traps that communicate with each other via photons rather than trying to create one big trap. Superconducting circuits, meanwhile, make operations much faster than trapped ions, and it’s easier to scale up the number of circuits in a computer than the number of ions. But the circuits are also more fragile, and have a shorter overall lifespan.

And as quantum computers advance, they will still be subject to the environmental constraints needed to preserve quantum states. But in spite of all these obstacles, we’ve already succeeded at making computations in a realm we can’t enter or even observe.

by Robert Frost

The way a crow
Shook down on me
The dust of snow
From a hemlock tree

Has given my heart
A change of mood
And saved some part
Of a day I had rued.

• Dust of Snow
• Dust of Snow
• Analysis of Poem ‘Dust of Snow’ by Robert Frost

spruce

spruce

noun

• a type of tree that has long, thin needles instead of leaves and that stays green throughout the year
• an evergreen tree that has short needles for leaves, drooping cones, and light soft wood

Blue spruce

The blue spruce (Picea pungens), also commonly known as green spruce, white spruce, Colorado spruce, or Colorado blue spruce, is a species of spruce tree.

verb

Do you want to spruce up the house with me?

Let me spruce up before we go.

adjective

• neat or smart in appearance

He looked very spruce in his new suit.

hemlock

hemlock

noun

• any of a genus (Tsuga) of evergreen coniferous trees of the pine family

Tsuga

cedar

cedar

noun

Cedrus

beech

beech

noun

Beech

aspen

aspen

noun

Aspen

butternut

butternut

noun

Juglans cinerea

Juglans cinerea, commonly known as butternut or white walnut, is a species of walnut native to the eastern United States and southeast Canada.

awe

awe

noun

regard nature’s wonders with awe

Awe: The ‘little earthquake’ that could free your mind

Intentionally seeking the feeling of awe can improve memory, boost creativity and relieve anxious ruminatio

verb

We were awed by the beauty of the mountains.

varicose

varicose

adjective

varicose veins

varicose legs

What are the complications of varicose veins?

• Inflammation or swelling of veins (phlebitis)
• Blood clots

Can varicose veins be prevented?

• Keeping a healthy weight
• Exercising regularly
• Putting your feet up while sitting
• Not crossing your legs while sitting
• Not wearing tight clothing

https://www.hopkinsmedicine.org/health/conditions-and-diseases/varicose-veins

Varicose veins

arboretum

arboretum

noun

Arboretum

fiber

fiber

noun

It’s important to get enough fiber in your diet.

What foods do you recommend as good sources of fiber?

Nylon is a very strong man-made fiber.

Fiber

calorie

calorie

noun

Calorie

mandate

mandate

noun

As anti-vaccine mandate protest enters 5th day in Ottawa, some worry about how it might end

leeway

leeway

noun

They give their students leeway to try new things.

you will be given some leeway in choosing how to carry out the project

pangolin

pangolin

noun

Pangolin

anteater

anteater

noun

Anteater

armadillo

armadillo

noun

Armadillo

chill

chill

noun

He had caught a chill that night, and was now down with a fever.

He closed the windows to keep out the chill.

Her symptoms include chills and a fever.

He caught a chill that turned into a bad cold.

adjective

The chill weather kept us indoors.

Celsius

Celsius

adjective

The temperature reached 23 degrees Celsius.

Celsius

Fahrenheit

Fahrenheit

adjective

It was 70 degrees Fahrenheit outside.

Fahrenheit

nausea

nausea

noun

Some people experience nausea when flying.

Nausea

vomit

vomit

verb

The dog vomited on the floor.

The patient was vomiting blood.

noun

Vomiting

warthog

warthog

noun

Warthog

yacht

yacht

noun

Yacht

TL;DR

TL;DR

abbreviation

• too long; didn’t read — used to say that something would require too much time to read

a TL;DR video

groovy

groovy

adjective

a great movie with groovy special effects

Sisyphean

Sisyphean

adjective

a Sisyphean task

Sisyphus

squall

squall

noun

• a sudden strong gust of wind often with rain or snow

snow squall

Snowsquall

boredom

boredom

noun

• the state of being weary and restless through lack of interest
• the state of being bored

the boredom of a long car trip

Boredom

outage

outage

noun

• a period of interruption especially of electric current

Strong wind gusts may result in broken tree branches as well as isolated power outages.

Power outage

spiky

spiky

adjective

a plant with spiky leaves

spike

spike

noun

railroad spikes

If price spikes continue, people will not be able to afford the new houses they want.

Liberty spikes

Spike protein

asteroid

asteroid

noun

Most asteroids are found between Mars and Jupiter.

Asteroid

sump

sump

noun

Sump pump

Problem

8. String to Integer (atoi)

Java

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5
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10
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26
27
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41

class Solution {
    public int myAtoi(String s) {
        int re = 0;
        long num = 0;
        s = s.trim();
        char[] a = s.toCharArray();
        int len = a.length;
        int[] digits = new int[len];
        int k = 0;
        boolean isNegative = false;
        char c = ' ';
        if (len > 0) {
            boolean hasSignFlag = false;
            if (a[0] == '+' || a[0] == '-')
                hasSignFlag = true;
            if (a[0] == '-')
                isNegative = true;
            if (a[0] == '+')
                isNegative = false;
            for (int i = hasSignFlag ? 1 : 0; i < len; i++) {
                c = a[i];
                if (Character.isDigit(c))
                    digits[k++] = Integer.parseInt(c + "");
                else if (!Character.isDigit(c))
                    break;
                else
                    ;
            }
        }
        for (int m = 0; m < k; m++)
            num += digits[m] * Math.pow(10, k - m - 1);
        if (isNegative)
            num *= -1;
        if (num < Integer.MIN_VALUE)
            num = Integer.MIN_VALUE;
        if (num > Integer.MAX_VALUE)
            num = Integer.MAX_VALUE;
        re = new Long(num).intValue();
        return re;     
    }
}

Submission Detail

• 1082 / 1082 test cases passed.
• Runtime: 5 ms, faster than 26.19% of Java online submissions for String to Integer (atoi).
• Memory Usage: 38.9 MB, less than 80.12% of Java online submissions for String to Integer (atoi).

The more we get together,
Together, together,
The more we get together,
The happier we’ll be.

For your friends are my friends,
And my friends are your friends,
The more we get together,
The happier we’ll be!

The More We Get Together

bib

bib

noun

Bib (garment)

apron

apron

noun

Apron

Cantonese

Cantonese

noun

• the dialect of Chinese spoken in Guangzhou and Hong Kong

Cantopop

Cantopop (a contraction of “Cantonese pop music”) or HK-pop (short for “Hong Kong pop music”) is a genre of pop music written in standard Chinese and sung in Cantonese.

caboose

caboose

noun

Caboose

kimchi

kimchi

noun

• a spicy, pungent vegetable dish that consists of one or more pickled and fermented vegetables and especially napa cabbage and radishes with various seasonings (such as garlic, red chili pepper, ginger, scallions, and anchovy paste) and that is the national dish of South Korea

Kimchi

rucksack

rucksack

noun

hikers carrying their food and water in rucksacks

Backpack

backpack

backpack

noun

stuffed her backpack with so many books that she could barely walk

ahoy

ahoy

interjection

• used by a sailor who is calling out to a passing ship or boat

ship ahoy

Ahoy (greeting)

cub

cub

noun

• a young carnivorous mammal (such as a bear, fox, or li

A cub is the young of certain large predatory animals such as big cats or bears; analogous to a domestic puppy or kitten.

panda cub

pup

pup

noun

• a young dog, puppy
• a young animal

seal pups

puppy

puppy

noun

Puppy

bunny

bunny

noun

• informal : rabbit, especially : a young rabbit

Rabbit

foal

foal

noun

• a young animal of the horse family. especially : one under one year

zebra foal

kitten

kitten

noun

• a young cat

Kitten

calf

calf

noun

• the young of the domestic cow

Calf

lamb

lamb

noun

• a young sheep

BY JOEL KOTKIN

For centuries we have used maps to delineate borders that have been defined by politics. But it may be time to chuck many of our notions about how humanity organizes itself. Across the world a resurgence of tribal ties is creating more complex global alliances. Where once diplomacy defined borders, now history, race, ethnicity, religion, and culture are dividing humanity into dynamic new groupings.

Broad concepts — green, socialist, or market-capitalist ideology — may animate cosmopolitan elites, but they generally do not motivate most people. Instead, the “tribe” is valued far more than any universal ideology. As the great Arab historian Ibn Khaldun observed: “Only Tribes held together by a group feeling can survive in a desert.”

Although tribal connections are as old as history, political upheaval and globalization are magnifying their impact. The world’s new contours began to emerge with the end of the Cold War. Maps designating separate blocs aligned to the United States or the Soviet Union were suddenly irrelevant. More recently, the notion of a united Third World has been supplanted by the rise of China and India. And newer concepts like the BRIC nations (Brazil, Russia, India, and China) are undermined by the fact that these countries have vastly different histories and cultures.

The borders of this new world will remain protean, subject to change over time. Some places do not fit easily into wide categories — take that peculiar place called France — so we’ve defined them as Stand-Alones. And there are the successors to the great city-states of the Renaissance — places like London and Singapore. What unites them all are ties defined by affinity, not geography.

1. New Hansa

Denmark, Finland, Germany, Netherlands, Norway, Sweden

In the 13th century, an alliance of Northern European towns called the Hanseatic League created what historian Fernand Braudel called a “common civilization created by trading.” Today’s expanded list of Hansa states share Germanic cultural roots, and they have found their niche by selling high-value goods to developed nations, as well as to burgeoning markets in Russia, China, and India. Widely admired for their generous welfare systems, most of these countries have liberalized their economies in recent years. They account for six of the top eight countries on the Legatum Prosperity Index and boast some of the world’s highest savings rates (25 percent or more), as well as impressive levels of employment, education, and technological innovation.

2. The Border Areas

Belgium, Czech Republic, Estonia, Hungary, Iceland, Ireland, Latvia, Lithuania, Poland, Romania, Slovakia, U.K.

These countries are seeking to find their place in the new tribal world. Many of them, including Romania and Belgium, are a cultural mishmash. They can be volatile; Ireland has gone from being a “Celtic tiger” to a financial basket case. In the past, these states were often overrun by the armies of powerful neighbors; in the future, they may be fighting for their autonomy against competing zones of influence.

3. Olive Republics

Bulgaria, Croatia, Greece, Italy, Kosovo, Macedonia, Montenegro, Portugal, Slovenia, Spain

With roots in Greek and Roman antiquity, these lands of olives and wine lag behind their Nordic counterparts in virtually every category: poverty rates are almost twice as high, labor participation is 10 to 20 percent lower. Almost all the Olive Republics—led by Greece, Spain, and Portugal—have huge government debt compared with most Hansa countries. They also have among the lowest birthrates: Italy is vying with Japan to be the country with the world’s oldest population.

4. City-States

London

It’s a center for finance and media, but London may be best understood as a world-class city in a second-rate country.

Paris

Accounts for nearly 25 percent of France’s GDP and is home to many of its global companies. It’s not as important as London, but there will always be a market for this most beautiful of cities.

Singapore

In a world increasingly shaped by Asia, its location between the Pacific and Indian oceans may be the best on the planet. With one of the world’s great ports, and high levels of income and education, it is a great urban success story.

Tel Aviv

While much of nationalist-religious Israel is a heavily guarded borderland, Tel Aviv is a secular city with a burgeoning economy. It accounts for the majority of Israel’s high-tech exports; its per capita income is estimated to be 50 percent above the national average, and four of Israel’s nine billionaires live in the city or its suburbs.

5. North American Alliance

Canada, United States

These two countries are joined at the hip in terms of their economies, demographics, and culture, with each easily being the other’s largest trade partner. Many pundits see this vast region in the grip of inexorable decline. They’re wrong, at least for now. North America boasts many world-class cities, led by New York; the world’s largest high-tech economy; the most agricultural production; and four times as much fresh water per capita as either Europe or Asia.

6. Liberalistas

Chile, Colombia, Costa Rica, Mexico, Peru

These countries are the standard–bearers of democracy and capitalism in Latin America. Still suffering low household income and high poverty rates, they are trying to join the ranks of the fast-growing economies, such as China’s. But the notion of breaking with the U.S.—the traditionally dominant economic force in the region—would seem improbable for some of them, notably Mexico, with its close geographic and ethnic ties. Yet the future of these economies is uncertain; will they become more state-oriented or pursue economic liberalism?

7. Bolivarian Republics

Argentina, Bolivia, Cuba, Ecuador, Nicaragua, Venezuela

Led by Venezuela’s Hugo Chávez, large parts of Latin America are swinging back toward dictatorship and following the pattern of Peronism, with its historical antipathy toward America and capitalism. The Chávez-influenced states are largely poor; the percentage of people living in poverty is more than 60 percent in Bolivia. With their anti-gringo mindset, mineral wealth, and energy reserves, they are tempting targets for rising powers like China and Russia.

8. Stand-Alones

Brazil

South America’s largest economy, Brazil straddles the ground between the Bolivarians and the liberal republics of the region. Its resources, including offshore oil, and industrial prowess make it a second-tier superpower (after North America, Greater India, and the Middle Kingdom). But huge social problems, notably crime and poverty, fester. Brazil recently has edged away from its embrace of North America and sought out new allies, notably China and Iran.

France

France remains an advanced, cultured place that tries to resist Anglo-American culture and the shrinking relevance of the EU. No longer a great power, it is more consequential than an Olive Republic but not as strong as the Hansa.

Greater India

India has one of the world’s fastest-growing economies, but its household income remains roughly a third less than that of China. At least a quarter of its 1.3 billion people live in poverty, and its growing megacities, notably Mumbai and Kolkata, are home to some of the world’s largest slums. But it’s also forging ahead in everything from auto manufacturing to software production.

Japan

With its financial resources and engineering savvy, Japan remains a world power. But it has been replaced by China as the world’s No. 2 economy. In part because of its resistance to immigration, by 2050 upwards of 35 percent of the population could be over 60. At the same time, its technological edge is being eroded by South Korea, China, India, and the U.S.

South Korea

South Korea has become a true technological power. Forty years ago its per capita income was roughly comparable to that of Ghana; today it is 15 times larger, and Korean median household income is roughly the same as Japan’s. It has bounced back brilliantly from the global recession but must be careful to avoid being sucked into the engines of an expanding China.

Switzerland

It’s essentially a city-state connected to the world not by sea lanes but by wire transfers and airplanes. It enjoys prosperity, ample water supplies, and an excellent business climate.

9. Russian Empire

Armenia, Belarus, Moldova, Russian Federation, Ukraine

Russia has enormous natural resources, considerable scientific-technological capacity, and a powerful military. As China waxes, Russia is trying to assert itself in Ukraine, Georgia, and Central Asia. Like the old tsarist version, the new Russian empire relies on the strong ties of the Russian Slavic identity, an ethnic group that accounts for roughly four fifths of its 140 million people. It is a middling country in terms of household income — roughly half of Italy’s — and also faces a rapidly aging population.

10. The Wild East

Afghanistan, Azerbaijan, Kazakhstan, Kyrgyzstan, Pakistan, Tajikistan

This part of the world will remain a center of contention between competing regions, including China, India, Turkey, Russia, and North America.

11. Iranistan

Bahrain, Gaza Strip, Iran, Iraq, Lebanon, Syria

With oil reserves, relatively high levels of education, and an economy roughly the size of Turkey’s, Iran should be a rising superpower. But its full influence has been curbed by its extremist ideology, which conflicts not only with Western countries but also with Greater Arabia. A poorly managed economy has turned the region into a net importer of consumer goods, high-tech equipment, food, and even refined petroleum.

12. Greater Arabia

Egypt, Jordan, Kuwait, Palestinian Territories, Saudi Arabia, United Arab Emirates, Yemen

This region’s oil resources make it a key political and financial player. But there’s a huge gap between the Persian Gulf states like Saudi Arabia and the United Arab Emirates and the more impoverished states. Abu Dhabi has a per capita income of roughly $40,000, while Yemen suffers along with as little as 5 percent of that number. A powerful cultural bond—religion and race—ties this area together but makes relations with the rest of the world problematic.

13. The New Ottomans

Turkey, Turkmenistan, Uzbekistan

Turkey epitomizes the current reversion to tribe, focusing less on Europe than on its eastern front. Although ties to the EU remain its economic linchpin, the country has shifted economic and foreign policy toward its old Ottoman holdings in the Mideast and ethnic brethren in Central Asia. Trade with both Russia and China is also on the rise.

14. South African Empire

Botswana, Lesotho, Namibia, South Africa, Swaziland, Zimbabwe

South Africa’s economy is by far the largest and most diversified in Africa. It has good infrastructure, mineral resources, fertile land, and a strong industrial base. Per capita income of $10,000 makes it relatively wealthy by African standards. It has strong cultural ties with its neighbors, Lesotho, Botswana, and Namibia, which are also primarily Christian.

15. Sub-Saharan Africa

Angola, Cameroon, Central African Republic, Congo-Kinshasa, Ethiopia, Ghana, Kenya, Liberia, Malawi, Mali, Mozambique, Nigeria, Senegal, Sierra Leone, Sudan, Tanzania, Togo, Uganda, Zambia

Mostly former British or French colonies, these countries are divided between Muslim and Christian, French and English speakers, and lack cultural cohesion. A combination of natural resources and poverty rates of 70 or 80 percent all but assure that cash-rich players like China, India, and North America will seek to exploit the region.

16. Maghrebian Belt

Algeria, Libya, Mauritania, Morocco, Tunisia

In this region, spanning the African coast of the Mediterranean, there are glimmers of progress in relatively affluent countries like Libya and Tunisia. But they sit amid great concentrations of poverty.

17. Middle Kingdom

China, Hong Kong, Taiwan

China may not, as the IMF recently predicted, pass the U.S. in GDP within a decade or so, but it’s undoubtedly the world’s emerging superpower. Its ethnic solidarity and sense of historical superiority remain remarkable. Han Chinese account for more than 90 percent of the population and constitute the world’s single largest racial-cultural group. This national cultural cohesion, many foreign companies are learning, makes penetrating this huge market even more difficult. China’s growing need for resources can be seen in its economic expansion in Africa, the Bolivarian Republics, and the Wild East. Its problems, however, are legion: a deeply authoritarian regime, a growing gulf between rich and poor, and environmental degradation. Its population is rapidly aging, which looms as a major problem over the next 30 years.

18. The Rubber Belt

Cambodia, Indonesia, Laos, Malaysia, Philippines, Thailand, Vietnam

These countries are rich in minerals, fresh water, rubber, and a variety of foodstuffs but suffer varying degrees of political instability. All are trying to industrialize and diversify their economies. Apart from Malaysia, household incomes remain relatively low, but these states could emerge as the next high-growth region.

19. Lucky Countries

Australia, New Zealand

Household incomes are similar to those in North America, although these economies are far less diversified. Immigration and a common Anglo-Saxon heritage tie them culturally to North America and the United Kingdom. But location and commodity-based economies mean China and perhaps India are likely to be dominant trading partners in the future.

Kotkin is a Distinguished Presidential Fellow at Chapman University of Orange County, Calif., and an adjunct fellow with the Legatum Institute in London. Legatum provided research for this article.

The New World Order: A Map BY JOEL KOTKIN ON 9/26/10 AT 7:00 AM EDT

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Merriam-Webster

Word of the Year: Vaccine

Insurrection

The violent events of January 6th led to a striking spike in lookups for insurrection, driven by headlines like this: “Military Joint Chiefs condemn ‘sedition and insurrection’ at US Capitol” (CNN, 1/13/21). Another, smaller spike took place in July, when the U.S. House select committee hearings on the event began.

Perseverance

The word perseverance shot to the top of our lookups in February after NASA’s Mars rover “Perseverance” landed safely on the red planet on February 18th. The rover got its name thanks to student Alexander Mather, the seventh grader who submitted the winning entry to NASA’s “Name the Rover” essay contest. The name, chosen out of 28,000 entries from K-12 students from every U.S. state and territory, is apt: Perseverance traveled almost 300 million miles over seven months—a remarkable journey.

The word perseverance is defined as “continued effort to do or achieve something despite difficulties, failure, or opposition; the action or condition or an instance of persevering; steadfastness.” It’s ultimately from Latin persevērāre, meaning “to persist in a course of action or an attitude in spite of opposition, keep on.”

Woke

Lookups for woke saw dramatic increases several times in 2021—first in April (they were up 1,373% on April 6th), then in May (up 1,279% on the 4th), and again in June (up 354% on the 24th). But it was during November’s election that the most dramatic increases occurred: lookups for woke were up 4,303% on November 8th in response to use of the word in election coverage.

The use of woke drawing interest is a new one, compared to the word’s historic use as the past tense form of wake: originating in in African American English (AAE), the new use of woke is defined in our dictionary as “aware of and actively attentive to important facts and issues (especially issues of racial and social justice).” The semantics of the newer use are, however, evolving, and it’s increasingly used as a general pejorative for anyone who is or appears to be politically left-leaning; the spikes in lookups mentioned above all correlate to instances in which reactions to racial and social injustice were dismissed as being “woke.”

The neologism wokelash combines this pejorative use with the word backlash to pin Republican political gains on a perceived adverse reaction to “woke” politics.

Nomad

When the movie Nomadland snagged three of this year’s Oscar Awards on April 25th, including Best Picture, Best Actress in a Leading Role, and Best Directing, lots of people turned to Merriam-Webster.com to look up the word nomad—enough people to make lookups jump 4,625%.

Nomad can refer specifically to a member of a people who have no fixed residence but who move from place to place, usually seasonally and within a well-defined territory, or more broadly to someone who roams about. Nomad comes from the Latin word meaning “member of a wandering pastoral people,” and ultimately from the Greek verb that means “to pasture, graze (flocks).”

Infrastructure

Lookups for infrastructure spiked 677% on April 7th, following President Biden’s announcement of his $2.3 trillion dollar plan. Subsequent spikes came in June with the announcement of a bipartisan agreement in congress on the legislation, and in November following former President Trump’s comments disparaging Republican legislators who voted for the bill and again after President Biden’s signing ceremony.

Sustained interest in this term was evident in public debates over what constitutes infrastructure and whether the word can be used to refer only to physical building and improvements to such things as roads and bridges, as opposed to some other elements of the proposal, such as broadband internet development and elder care funding. The debate was over the definition of the word itself.

The Latin roots of infrastructure mean simply “underneath or below the structure.” This word came to English from French, and is a relatively young word in English, dating to the mid-20th century. It was initially widely used in the context of building military bases, railroads, and airfields for use by NATO forces following World War II.

Cicada

Lookups for cicada were high for much of the spring and summer, as adult members of Brood X (as the particular group of periodic cicadas emerging in 2021 was called) made their way to the Earth’s surface, their larvae having been living underground for the past 17 years. Lookups were especially pronounced in May and June, when we saw increases of 1,442% and 264%, respectively.

The word cicada is defined as “any of a family (Cicadidae) of homopterous insects which have a stout body, wide blunt head, and large transparent wings,” and came to English from Latin, by way of New Latin—that is, Latin as used since the end of the medieval period especially in scientific description and classification.

Murraya

Murraya spiked in lookups on July 8th, after Zaila Avant-garde spelled it correctly to win the Scripps National Spelling Bee.

We define Murraya as “a genus of tropical Asiatic and Australian trees (family Rutaceae) having pinnate leaves and flowers with imbricated petals.”

All champion spellers study etymology in order to know the patterns of spellings in English. Zaila showed that she recognized that Murraya was an eponym, a word based upon a name, by asking the judges at the Bee: “Does this word contain the English name Murray, which could be the name of a comedian?”

The word comes from New Latin, and was named not for Bill Murray, but for Johan A. Murray, an 18th century Swedish botanist. (Zaila got a surprise congratulations from Bill Murray on “Jimmy Kimmel Live” after her win.)

The Bee always brings fresh attention to rarely looked-up words in the dictionary, and year-to-year comparisons aren’t useful when discussing words that typically get very few clicks per year. It’s safe to say that Murraya has never had so many lookups as in the days following the Bee, and may never again.

Cisgender

Cisgender saw dramatic spikes in lookups twice in 2021: the first time was in May, when lookups increased by 5,836% following the release of a CIA recruitment video in which a CIA member used the word to describe herself; the second in October, when lookups were up 20,811% after a student’s op-ed in a college newspaper used the word. Both events drew condemnation from those on the political right, as well as curiosity about the term itself.

Cisgender is defined as “of, relating to, or being a person whose gender identity corresponds with the sex the person had or was identified as having at birth.” The term is contrasted with transgender, and dates, per currently available evidence, to the mid-1990s. See more on the history and use of cisgender.

Guardian

When Cleveland’s American League baseball team chose Guardians as its new name in July, lookups of the word guardian jumped 3,142%.

The team considered more than a thousand names before settling on Guardians as a replacement for its much-criticized Indians moniker. The new name alludes to eight statuary guardians that flank both sides of Cleveland’s Hope Memorial Bridge. The Guardians of Traffic statues are 43 feet tall and, like the Greek God Hermes (the god of travel, among other things; his Roman counterpart is Mercury) they bear winged helmets on their heads.

The pertinent meaning of the word guardian here is “one that guards; custodian.” The word comes from the Middle English word gardein or wardein, source too of warden. Both words ultimately come from Anglo-French warder, “to guard.”

Meta

When one of the most prominent companies in the world chooses an abstract word as its new name, you can expect millions of people to seek more information about that word. This is exactly what occurred in October, when lookups of meta jumped 10,860% in response to Facebook’s announcement that the company would now be called “Meta.”

Meta is defined as “showing or suggesting an explicit awareness of itself or oneself as a member of its category; cleverly self-referential,” and as “concerning or providing information about members of its own category.” It’s an adjective constructed from a prefix: meta- has a number of meanings including “occurring later than or in succession to”; “situated behind or beyond”; and “later or more highly organized or specialized form of.”

The name change is part of a shift in the company’s focus to something called the metaverse. Here’s a guide to that still-emerging term.

Collins

‘NFT’, the abbreviation of ‘non-fungible token’, the unique digital identifier that records ownership of a digital asset which has entered the mainstream and seen millions spent on the most sought-after images and videos, has been named Collins Word of the Year 2021.

Oxford Dictionaries

Vax is our 2021 Word of the Year. When our lexicographers began digging into our English language corpus data it quickly became apparent that vax was a particularly striking term. A relatively rare word in our corpus until this year, by September it was over 72 times more frequent than at the same time last year. It has generated numerous derivatives that we are now seeing in a wide range of informal contexts, from vax sites and vax cards to getting vaxxed and being fully vaxxed, no word better captures the atmosphere of the past year than vax.

• Oxford Word of the Year 2021
• Merriam-Webster’s Word of the Year 2021
• The Collins Word of the Year 2021 is…
• Get your crypto at the ready: NFTs are big in 2021