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# Is Diversification Dead?

January 2021
Read Time: 20 min
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Key Points
• The recent, decade-plus returns of diversified portfolios have disappointed relative to mainstream US stocks and bonds.

• A long-term examination, dating back to the mid-1970s, reveals such a shortfall is not unprecedented and the long-term case for diversification is still powerful.

• Extrapolating the recent superior performance of the 60/40 mix into the future requires the dangerous assumption that US stock market valuations will march to nearly unprecedented levels. Diversification is needed now more than ever.

In 1952, Harry Markowitz published “Portfolio Selection” and helped usher in the era of modern portfolio theory. Harry’s work showed us that a portfolio’s risk is not defined by the riskiness of its individual assets, but by the extent the portfolio’s assets’ price movements correlate, or move together. Harry demonstrated that a technique, now known as mean-variance optimization, derives an efficient frontier of portfolios, each of which maximizes expected return at any given level of risk.1 At a Research Affiliates’ forum, Harry told attendees that “the capital markets offer two free lunches: diversification and mean reversion.”

Diversification is a theoretically powerful concept, but as our friend Jason Hsu is fond of saying: “Diversification is a regret-maximizing strategy.” In a roaring bull market, such as 2009–2020 and 1991–1999, we investors regret every penny we put into diversifiers, while in a secular bear market, such as 2000–2009 and 1968–1982, we regret every penny we did not put into diversifiers.

In a rerun of the 1990s, the 2010s were brutal to diversifying strategies. As it was in 1999, the wisdom of diversification is again under attack. The pressure to ignore diversification grew even stronger last year when the COVID-induced bear market of February–March 2020 was no less savage to diversifying asset classes than to 60/40 investors in the US market, and the stimulus-induced rebound again rewarded the “naïve” 60/40 investor for shunning diversification.

We affirm the wisdom expressed in a quote often misattributed to Yogi Berra: “In theory, there is no difference between theory and practice, while in practice there is.”2 In theory, diversification is a “free lunch” for the patient investor. In practice, an investor was substantially better off if she had invested solely in the classic US 60/40 portfolio—60% in the S&P 500 Index and 40% in the investment-grade bond market—at a near-zero management fee over the last dozen years. Diversified investors were far less rewarded for the same level of risk! This outcome has led many advisors and fiduciaries, on the receiving end of complaints from unhappy clients, to abandon diversification.

## A Longer Look

To gain a good understanding of the merits of diversification, let’s take a patient, long-term survey of the performance characteristics of diversified portfolios compared to the mainstream blend of 60% US stocks and 40% US bonds. We look back to the mid-1970s, when diversifying asset classes first became investible and tracked by market benchmarks.

This long-horizon exercise is important for three reasons:

1. First, most investment portfolios serve future spending needs that extend far longer than the next 10 years. Even a newly minted retiree at age 65 has nearly 20 years of additional life expectancy, and most institutional investment horizons are longer than individuals’ life expectancies: Endowments are ostensibly intended to serve perpetual spending needs, and pension obligations typically extend for a century or more until the last check is cut.3 Human nature, which behavioral finance seeks to model, makes it very difficult, however, for investors to behave as if these long horizons are relevant to their decision making.
2. Second, even an entire decade may not encompass a full market cycle. For example, US investors faced neither a recession nor a bear market (as defined by a 20% drawdown on the S&P 500 using month-end data) during the 2010s.4 Peak to peak, this last cycle extended for over 12 years from October 9, 2007, to February 16, 2020.
3. Third, most importantly, past is not prologue. Whereas 5 or 10 years can seem quite a long time in many aspects of our lives, such a stretch is relatively short in the capital markets, where a recent trend can seem perpetual. Indeed, any long-horizon examination of historical returns shows mean reversion; that is, recent 5- or 10-year winning strategies tend to do poorly in the future and vice versa. We have noted that future 10-, 20-, or 30-year stock market returns are negatively correlated with the preceding 10-, 20-, and 30-year returns, and that the tendency to extrapolate the most recent seemingly long-horizon experience to the future is the “most dangerous shortcut in financial planning” (West and Ko, 2017).

Our survey includes 16 asset classes and their index proxies. We break the asset classes into three categories, or pillars. The “first pillar” is mainstream developed-market equities, whose historical role in a portfolio has been to provide higher returns via participation in real economic growth. The “second pillar,” mainstream US investment-grade bonds, is intended to provide steady, reliable income in a portfolio—pretty anemic today!—and countercyclical dry powder available for investment when the economy and risk markets soften. The purpose of the “third pillar” is to diversify the portfolio and offer the ability to produce better returns than the other two pillars during inflationary periods. Why is inflation a specific function of the third pillar? Because inflationary bouts often lead to rising interest rates and tumbling stock market valuations, a combination that breaks down the complementary, diversifying characteristics of the first and second pillars.

We constructed an equally weighted portfolio that includes all 16 asset classes and a 60/40 portfolio consisting of 60% S&P 500 and 40% Barclays Capital US Aggregate Bond Index. Our survey begins in the first quarter of 1975 with 10 of the 16 asset classes from the first and second pillars and two of the third-pillar asset classes. We added new asset classes to the survey at the inception of their return histories. This gives us four full decades and a fifth stub decade, which comprises the second half of the 1970s. As a simple shorthand, we consider the return difference between the equally weighted portfolio and the 60/40 blend to be the return premium attributable to diversification.

Since 1975, we see comparable results for the more-diversified equally weighted portfolio versus the 60/40 blend. The equally weighted portfolio produced a return of 10.9% a year with annual volatility of 10.0% and a Sharpe ratio of 0.64. The 60/40 blend earned an annualized 10.7% return with annual volatility of 10.4% and a Sharpe ratio of 0.60. If the diversifiers in the equally weighted portfolio can boost the Sharpe ratio of the 60/40 portfolio, certainly they must work even better if an investor just concentrates in these markets. Not so fast! The third-pillar asset classes boost the Sharpe ratio because they offer diversifying and complementary risks in the 60/40 investor’s portfolio.

We can make several high-level observations about the returns of each of the three pillars over the analysis period. The first pillar (essentially developed-market equities) and the second pillar (essentially US investment-grade bonds) produce identical Sharpe ratios of 0.47. The blend of the two pillars in the 60/40 portfolio boosts the Sharpe ratio to 0.60. The remaining third-pillar asset classes did slightly better by producing an annualized return of 10.4%, annual volatility of 10.3%, and a Sharpe ratio of 0.58, essentially matching the result for the 60/40 portfolio. We call attention to the following observations: the 60/40 portfolio has a higher Sharpe ratio than either of the broad stock and bond categories, which perform similarly on a risk-adjusted basis; the third pillar performs as well as the 60/40 portfolio; and the equally weighted portfolio, which combines all three pillars, improves the Sharpe ratio relative to both the 60/40 blend and the third pillar alone.

A very important observation is that the 46-year time span of our survey works in favor of the 60/40 portfolio because our analysis begins at the start of 1975, a bear-market low for both US stocks and bonds, and concludes at the end of 2020, a market high for both of these asset classes. Yet, the diversified equally weighted portfolio still outperforms the 60/40 blend. When we examine the relative performance of the two portfolios based on cumulative wealth creation, the equally weighted portfolio exhibits relentless outperformance over the 46 years. Since 1975, in only one quarter (of the 180 total!) did the 60/40 portfolio show a cumulatively larger value-add than its more-diversified competitor. Ever so briefly in the first quarter of 1999, the 60/40 portfolio caught up the equally weighted mix before falling off substantially. In other words, less than 1% of the history we examine shows a 60/40 strategy as producing superior long-term wealth.

Suppose an asset manager approached you with the recommendation to invest in a strategy that showed such relentless long-term underperformance, simply because the strategy had fared better over the last decade. Given the long history of underperformance, and the theoretical advantage of the diversified strategy, you’d laugh them right out of your life! But that recommendation is precisely what the current crop of diversification naysayers are proclaiming today!

We also looked further to determine how consistent this risk-adjusted edge was for diversification. Let’s call this a Sharpe ratio “hit rate.” We found that in 60% of rolling five-year windows the equally weighted portfolio produced a superior Sharpe ratio relative to 60/40. Collectively, this simple analysis suggests a modestly superior risk-adjusted result for diversification, even during a span in which the 60/40 portfolio benefited from sharply falling yields and rising prices. At a minimum, it is impossible to assert that a 60/40 strategy is superior based on long-term empirical results. So far, so good, for Harry Markowitz!

No investor has a 46-year window between portfolio formation and the portfolio’s first performance review. Indeed, precious few investors have even a five-year evaluation window; the typical performance assessment window is shorter. Therefore, it is unsurprising that the behavioral finance literature notes recency bias as one of the common cognitive errors besetting investors. Recency bias is the tendency to place greater emphasis on the recent past in forming expectations about the future. A related idea is the notion of how an investor’s experience affects her decision making through the tendency to form strong investment beliefs by overweighting personal experience over hard data. And finally, after an investor forms an opinion, confirmation bias encourages her to seek data that support her opinion, while ignoring data that suggest her opinion is wrong.

Keeping such behavioral tendencies in mind, let’s examine decade-by-decade results. For each decade and the stub decade, we calculated the nominal returns for each of the two portfolios and three pillars. We also calculated the returns of the 90-day US Treasury bill index, which reflects the prevailing risk-free rate, and thus can determine the realized excess return of each of the portfolios or pillars. The excess return allows us to compare more effectively the returns for an investor’s bearing risk across decades characterized by vastly different interest rate environments. A 7% return when the 90-day Treasury bill earned 6% is far less impressive than a 7% return when the same Treasury bill netted 1%.

Let’s start with the last decade, the 2010s. The equally weighted portfolio produced a nominal annualized return of 6.4%, the worst return of any of the decades, but its 6% excess return over cash was pretty typical of past experience. Indeed, the equally weighted portfolio has had remarkably steady excess returns over cash. Conversely, the 60/40 mix produced an annualized return of 9.2%, the implication being that the “cost” of diversification was nearly 280 basis points a year. Calendar-year 2020 brought even more pain to the diversified investor. After the dust settled on the nasty COVID-spurred sell-off and ensuing (and heroic) comeback rally, the 60/40 mix produced a return of 15.4% versus only 9.5% for the equally weighted mix. If we add 2020 to the last decade, 11 years of compounding substantially increases the annualized delta. An investor with $100,000 at the beginning of 2010 would have a balance of$294,000 at year-end 2020 if invested in the 60/40 portfolio versus an ending balance of $204,000 if invested in the equally weighted portfolio over the same period. Because the 60/40 portfolio anchors on US large-cap stocks (specifically, the S&P 500), which soundly beat US small-cap and international stocks over the 11-year period, the 60/40 portfolio even edges out the all-equity first pillar, because it equally weights US large- and small-cap and EAFE stocks. What created the relative performance drag in the equally weighted portfolio over the last decade? The answer is the third pillar—those assets whose role is to provide diversification away from equities’ performance. In the 2010s, these asset classes combined to produce a 4.4% annualized return. No wonder some consider the 2010s to be the “Lost Decade of Diversification.” It was! And it continued in 2020 with a measly 5.8% one-year return. Now, let’s look back further to the decade of the 2000s. We observe the opposite performance comparison: the equally weighted portfolio with a 7.0% nominal return outpaced the 60/40 blend’s paltry 2.4% annualized return. Mainstream US stocks entered the period with historically high (nosebleed?) valuations and experienced two horrific bear markets during the decade. Meanwhile, third-pillar diversifiers fared far better than mainstream stocks. Over this decade, diversification paid off. (Later in the article, we will address how valuations fared over our survey period.) Particularly interesting, however, is a comparison of excess returns, because they allow us to assess more effectively the returns for bearing risk across decades with vastly different interest rate environments. We see that the excess return of the equally weighted portfolio in the 2010s (5.9%) meaningfully exceeded its excess return in the 2000s (4.3%). In other words, the disappointment with diversification over the last decade was relative, not absolute! “The disappointment with diversification over the last decade was relative, not absolute!” We’ve seen this movie before. In the decade of the 1990s, the equally weighted excess return was almost 6.0%, practically identical to the experience in the 2010s. This excess return—nothing to scoff at—was indeed the aspirational excess return of many an absolute-return strategy in the 1990s and since, but the 1990s also produced a 9.3% excess return for the 60/40 portfolio. Compared to an excess return of 5.9%, the equally weighted portfolio’s achievement paled by comparison. Accordingly, both of us can state from our personal experience sitting in boardroom “hotseats” that this achievement was viewed with contempt by the end of the 1990s, given the 60/40 portfolio’s stupendous results. We might choose to dismiss this relative-performance fixation and the resulting disappointment in otherwise excellent performance—indeed a diversified portfolio will always be underperforming “something.” It only becomes a problem when that something is an investor’s core holdings, typically resembling a 60/40 blend. We freely acknowledge that the gravitational pull of the peer group and its bias toward the “normal” 60/40 portfolio is a powerful reality for advisors and institutional investors alike. Indeed, one of us (Rob) noted this trade-off in 2003, writing in the Financial Analysts Journal: Suppose in 1988 we found an absolute-return strategy that outpaced U.S. T-bills by 6 percent—each and every year…. Despite higher returns and less risk than a 60/40 portfolio, it underperformed the 60/40 asset mix in six consecutive three-year spans, so it also would probably have been abandoned long before the end of 2000.… Isn’t there something wrong with this picture? (p.4) Little has changed since then. Indeed, human happiness appears to center not on absolute success relative to our own expectations, but to our neighbor’s success. Take the study by economists Sara Solnick and David Hemenway in which they surveyed 257 students, staff, and faculty at the Harvard School of Public Health on their attitudes about absolute position and their position relative to society (Solnik and Hemenway, 1998). Half of the respondents preferred to receive an annual salary of$50,000 when others are making $25,000 than to earn$100,000 a year when others are making $200,000. In other words, the respondents would prefer to have half as much real income as long as they are making twice as much as others. We have noted a similar response from investors during our careers. Clients appear to be pleased when our strategies delivered low returns or even losses as long as the excess returns were high relative to peers’ returns, identical to Solnick and Hemenway’s conclusion. ## The Path to Diversification Disappointment Over the last four decades, the equally weighted portfolio’s excess returns over cash have been fairly consistent, with a low of 4.3% during the 2000s and a high of 7.3% in the 1980s. While no strategy can produce consistent 6.0% excess returns above cash, the equally weighted portfolio has done a fairly good imitation of a constant 6% excess return! In contrast, the 60/40 portfolio has had wider swings, underperforming the risk-free rate in the 2000s and producing stupendous excess returns, over 9%, in the two decades of the 2010s and 1990s. What is the root cause of these swings, and how can understanding the cause help us form expectations for the future? Let’s start with a trivial, but important observation. The equally weighted portfolio of 16 asset classes means no single exposure is likely to dominate our results. Simple arithmetic translates to an exposure of 6.25% per asset class (which we rebalance quarterly in our analysis). Even though we combine related and highly correlated asset classes, such as US large-cap, US large-value, and US small-cap stocks, we still arrive at less than 20% of the portfolio in US stocks and less than 50% in equities worldwide. Conversely, the 60/40 blend holds 60% in a single asset class, US large-cap equities represented by the S&P 500. Intuitively, we can infer that large deltas between the portfolios are likely driven by the return of US equities. To explore further, we can start with an equity return decomposition consistent with the forecasting methodology that underpins our Asset Allocation Interactive (AAI) web tool.5 We break down the 12.0% US large-cap equity market return, proxied by the S&P 500, from the start of our analysis in 1975 into the categories of yield, growth, and change in valuation. Most of the historical return from investing in equities came from dividends and earnings growth. Dividends contributed 2.8% a year. Growth contributed 5.4% to the nominal S&P 500 return, with real fundamental growth at an annualized 1.8% and inflation at an annualized 3.6%. Jack Bogle referred to dividends and growth as the sources of the “return to the investor,” that is, what an investor receives as a buy-and-hold owner of a business. Conversely, valuation changes reflect the price investors are willing to pay for a business, and in the case of the S&P 500, for 500 businesses. Over extraordinarily long horizons, these valuation changes tend to be a very small contributor to overall stock market returns. Since 1870, the 150-year gradual rise in valuation multiples added 0.7% a year to US stock market returns. Over the 46-year time horizon of our survey, however, which encompasses most current investors’ entire personal experience in the capital markets, the number is substantially higher at 3.3% a year, nearly five times valuation’s contribution over the last 150 years. At the beginning of 1975, valuations were very low. At the end of 2020, valuations were in the top 2% of historical observations. The amount of the change in valuation can fluctuate wildly over shorter 10-year periods, which led Jack Bogle to label this component of return as the “return to the speculator.” So, S&P 500 investors had an excellent run over the past 46 years. Should they have expected it based on 1975 observable conditions? The answer may surprise you. In 1975, the dividend yield stood at 5.3%. We estimated the next two components, just as the 1975 investor would have been able to do. Using the trailing 40-year results (i.e., 1935–1974) of the S&P 500, we estimated the components of future growth to be real fundamental growth at 2.2% annualized and inflation at 3.5% annualized. We estimated the speculative component, the change in valuation, as follows. In 1975, the Shiller cyclically adjusted price-to-earnings (CAPE) ratio stood at 8.3, well below the long-term average to that point, which was 15.2. Our AAI forecasting methodology assumes the valuation ratio will mean revert halfway to the historical average, a tacit acknowledgement that “it’s different this time” and that historic averages may not be “fair value.” Over the next 46 years, a halfway-to-historic average reversion takes us to an estimate of 11.8, a gain of 42% as the valuation ratio increases from 8.3 to 11.8. Over the next 46 years, the estimated gain would translate into a 0.8% a year return from the change in valuation. Adding these components gives us a remarkably similar result—an expected 11.8% versus a realized 12.0%! The difference between realized and estimated returns for the period were the contributors to the return. Namely, dividends contributed far less and valuations far more in the live experience. These sources of return are two sides of the same coin. Since 1975, interest rates declined substantially, and dividend yields followed suit. Valuations, the reciprocal of yields, rose far more than forecast. The valuation (CAPE ratio) of the S&P 500 did not end 2020 at 11.8, and its path since 1975 was far from linear. Speculation—guessing what someone will pay for an asset in the future—played a role in that meandering path. Let’s look at how valuations have fluctuated since 1975, especially over the decade-by-decade spans that sear into our minds, occasionally leading us to draw incorrect conclusions. We can deduce the return to the speculator by simply taking the cumulative difference in percentage terms of the beginning and ending CAPE ratios for each decade. The two worst decades for diversification were the 1990s and the 2010s. Both coincided with immense bull markets in US stocks with valuations at peak or near-peak levels. Diversifiers, even faring well, could not keep pace. In December 1999, the S&P 500 CAPE ratio reached an all-time high of 44.2. Twenty years later, at year-end 2019, the S&P 500 CAPE ratio was over 30, ranking in the most expensive 4% of historical observations. What happened in 2020? During the COVID sell-off, the CAPE ratio ever so briefly dipped to 24.8 before rocketing back to the stratosphere and ending December 2020 at 33.5, ranking in the top 2% of historical observations. So, even though the decade of the 2020s started with a high CAPE, it vaulted higher by roughly another 10%. “The two worst decades for diversification were the 1990s and the 2010s. Both coincided with immense bull markets in US stocks with valuations at peak or near-peak levels. Diversifiers, even faring well, could not keep pace.” For the decade of the 2020s to be able to repeat the experience of the preceding decade and post a 50% rise in US equity valuations, the CAPE ratio would need to increase another 40% over the next nine years to hit the unprecedented level of 45, exceeding the dot-com valuation peak of 44.2. Can this happen? Of course, but relying on it to happen would not be a prudent strategy. If we extrapolate the past decade to form expectations of absolute and relative superiority of a 60/40 strategy, a rise in valuations of this magnitude becomes our baseline. In other words, speculation leading to rising prices becomes an expectation. Importantly, expectations are a midpoint, meaning that half the observations are above the expectation and half below. Investors who extrapolate such a rise in the CAPE ratio must have half of their observations at higher-than-peak levels of 2000! Prognosticators using the recent decade-plus to “prove” a 60/40 strategy is superior to more-diversified approaches need to be mindful that rising S&P 500 valuations played a meaningful role in this win. ## Looking Forward What might we reasonably expect on the investment front over the next decade? Our AAI web tool provides readily available long-term forecasts for over 130 assets and model portfolios, estimated using a conceptual framework that is transparent and robust. The dividend yield on US large-cap stocks is observable today and is far from both the 5.3% yield witnessed at the start of 1975 and the cumulative dividend return of 2.9% since 1975. As of December, 31, 2020, the dividend yield is 1.6%. Our estimate of real fundamental growth is 1.2%, much closer to the long-term average since 1871 as identified by Arnott and Bernstein (2002). Our inflation estimate is 2.3%. This leaves us with the big, speculative wildcard—the change in valuation over the next decade. We can estimate future valuations from two distinct angles. Option A assumes that current valuations are “fair,” so that by the end of the next decade we would end up precisely where we are starting today. The second approach, Option B, assumes some mean reversion in valuations, but not all the way to the historic average. Suppose, following the same method we used to estimate the expected change in valuation from 1975 to 2020, we assume mean reversion halfway to the historic average. If we assume that valuations remain elevated at today’s level (Option A), we arrive at a nominal rate of return for the S&P 500 of 5.1% (as of December 31, 2020). On AAI we label this the Yield & Growth expected return model. If, however, we assume some mean reversion (Option B), things get hairy. If we assume the S&P 500 CAPE ratio falls from 33.5 to 24.6—halfway back to the historic average/median of 16.3—our 10-year nominal forecast falls to 2.0%. A CAPE ratio of 24.6 would fall in the 86th percentile of historic observations and just a shade below its 20-year average of 25.5. Note that we are not talking about a draconian assumption of 1970s’ single-digit multiples! In a later paper, we will explore how remarkably well the simple arithmetic we use here can predict long-term performance for many global asset classes. ## Even Asset Allocation “Clairvoyance” Can Lag When equity valuations rise to peak levels, our experience shows that asset allocation management comes under considerable pressure. Rob has been an asset allocation practitioner since the early 1980s, using models to gauge the attractiveness of assets in order to over- or underweight them in a portfolio. As a thought experiment, we developed, with the obvious benefit of hindsight, a “clairvoyant diversification” portfolio. Every three years, the clairvoyant diversification portfolio looks into the future and invests equally in the top 25% of asset classes that will in fact be the best performers over the next three years. In the 2000s and 2010s, this generally meant the four best asset classes, because for the majority of those years the survey universe included 16 asset classes. In the 1980s and 1990s, a clairvoyant investor would have chosen the three best asset classes, because for the majority of those years the survey universe had only 12 asset classes. In the 1970s, the clairvoyant diversification portfolio was composed of only two or three asset classes. To state the obvious, clairvoyance is obviously not achievable. Nevertheless, please bear with our thought experiment. Over the entire 45-year period ending in 2019,6 clairvoyant diversification produced some amazing results, exactly as we would expect. The portfolio compounded annually at nearly 20%—for 45 years! Volatility was fairly high at 14.8%, just under the volatility of the S&P 500 (15.7%), for two reasons. First, the best performing asset classes over any three-year span are typically the riskier ones with the same often true of the worst performing asset classes, because the riskier markets tend to occupy the extremes in performance. Very rarely do we see low-risk assets, such as short-term Treasuries or core bonds, finish in the top-performing 25% of assets over a three-year period. Second, we are by definition forgoing diversification by concentrating in the top three or four subsequent outperformers, which leads to higher risk. Nonetheless, the clairvoyant diversification portfolio is more than compensated for this risk with a Sharpe ratio of over 1.0, easily surpassing the risk-adjusted figures of the equally weighted and 60/40 portfolios. Realized long-term Sharpe ratios above 1.0 are an exceedingly rare occurrence in the investment management industry, although they remain routine in the world of backtests (and Madoff-style fraud)!7 When we break out returns for clairvoyant diversification by decade, we witness breathtaking outperformance in the late 1970s, the decade of the 1980s, and the decade of the 2000s. In the 2010s, the S&P 500 produced an annualized 13.6% return, outperforming clairvoyant diversification by 1.8% a year—even clairvoyance couldn’t beat the S&P juggernaut! The same occurred in the 1990s when the S&P 500 beat clairvoyant diversification by 1.4% a year. In both cases, over a “long-term” 10-year span, a single asset class beat a portfolio of the three or four best performing asset classes, re-selected every three years to ensure the portfolio was always investing in the winners. “In the 2010s, the S&P 500 produced an annualized 13.6% return, outperforming clairvoyant diversification by 1.8% a year—even clairvoyance couldn’t beat the S&P juggernaut! The same occurred in the 1990s when the S&P 500 beat clairvoyant diversification by 1.4% a year.” Finance theory and consultants recommend diversification because no one can predict the future. But even if we could, we would have still underperformed the S&P 500 in the 1990s and 2010s. What a pity the run was interrupted by the orst decade for 60/40 in the past half-century! ## Conclusion Many of us in the investment management business are fiduciaries, those entrusted with watching over and prudently administering our clients’ money and assets. Even those professionals who do not have a legal fiduciary obligation have a deep-seated desire to do right by their clients. It’s hard work. Capital markets are inherently uncertain and require substantial holding periods for even “proven” strategies to earn their anticipated returns. When, as investors, we encounter disappointment, we first need to ascertain if the performance results are unusual or implausible. We can then make a more informed judgment as to whether to extrapolate these results into the future or to expect benefits from mean reversion. By definition, investors who use the last decade-plus or even the last 30 years—in which two of the three decades ended at peak equity valuations and a perfectly clairvoyant GTAA strategy was unable to keep pace with US stocks—to project the continued superiority of a 60/40 strategy over more-diversified approaches are tacitly relying on prices to rise ever higher. Can we really count on another round of speculative good fortune—indeed, to trust that continued escalation in valuations will successfully serve trillions in future pension obligations? If not, now is not the time to abandon diversification and diversifying asset classes. Certainly, if mean reversion does occur, heeding the lessons of the 2000s we must acknowledge that diversification is needed today more than ever. FEATURED TAGS Learn More About the Author ## Endnotes 1. In the video “Forging the Frontier” on the Research Affiliates website, Harry Markowitz provides more background on his journey to becoming the “Father of Modern Portfolio Theory.” 2. According to the website QuoteInvestigator, this quote first appeared in the Yale Literary Magazine in 1882. 3. Most readers will be shocked to learn that the final benefit payment from the Civil War was paid to the disabled daughter of a veteran of that war in May 2020, 155 years after the end of the war (Miller and McCarthy, 2020), and that the last spousal pension check from the Revolutionary War was paid in 1906, 131 years after the pension had been earned (Harrell-Sesniak, 2012). 4. The economic expansion ran from June 2009 until February 2020, and the bull market was even longer, running from March 2009 until February 2020. The last complete market cycle, peak to peak, lasted from October 2007 until February 2020. 5. Equity Methodology Overview,” available on the Research Affiliates website, offers an opportunity for a deeper dive into the underlying theory of our Asset Allocation Interactive web tool. 6. We are just entering the next three-year clairvoyance window, and thus, we are unable to extend the analysis into 2020. 7. Sadly, Sharpe ratios above 1.0 are an all-too-common occurrence in the confined world of backtesting. Arnott, Harvey, and Markowitz (2019) provides more detail on the dangers of backtests. Even realized Sharpe ratios above 1.0 require serious due diligence as Madoff investors learned the hard way. ## References Arnott, Robert. 2003. “Managing Investments for the Long Term.” Financial Analysts Journal, Editor’s Corner, vol. 59, no 4 (July):4–8. Arnott, Robert, and Peter Bernstein. 2002. “What Risk Premium Is ‘Normal’?” Financial Analysts Journal, vol. 58, no. 2 (March/April):64–85. Arnott, Robert, Campbell R. Harvey, and Harry Markowitz. 2019. “A Backtesting Protocol in the Era of Machine Learning.” Journal of Financial Data Science, vol. 1, no. 1 (Winter):64–74. Also available on SSRN. Harrell-Sesniak, Mary. 2012. “Last Revolutionary War Widow Receives Final Pension—in 1906!” GenealogyBank.com (July 19). Miller, Ryan, and John McCarthy. 2020. “Last Person to Receive Pension from Civil War,$73.13 Monthly, Dies at Age 90.” USA Today.com (June 10).

Research Affiliates. 2014. “Equity Methodology Overview.” Research Affiliates.com.

Solnick, Sara, and David Hemenway. 1998. “Is More Always Better? A Survey on Positional Concerns.” Journal of Economic Behavior & Organization, vol. 37, no. 3:373–383.

West, John, and Amie Ko. 2017. “The Most Dangerous (and Ubiquitous) Shortcut in Financial Planning.” Research Affiliates Publications, Advisor Series (September).


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# Grothendieck group of vector bundles

Let $X$ be a smooth hypersurface in $\mathbb{P}^n$ and let $\mathfrak{P}$ be the completion of $\mathbb{P}^n$ along $X$.

Let $\mathcal{R}$ denote the category of reflexive sheaves on $\mathbb{P}^n$ which are bundles outside a finite set of points in $X$ (the finite set is not fixed, it could/will be different for different sheaves). The motivation for this is Grothendieck's effective Lefschetz condition.

Let $\mathcal{V}$ denote the category of (formal) locally free sheaves on $\mathfrak{P}$ and consider the functor

$$F: \mathcal{R} \to \mathcal{V}.$$ Then by Grothendieck's Lefschetz conditions, this functor is an equivalence of categories. This therefore implies that

$$K_0(\mathbb{P}^n) \cong K_0(\mathfrak{P}).$$

On the other hand, the subgroup $CH^{n}(\mathbb{P}^n) \subset K_0(\mathbb{P}^n)$ maps to zero in $K_0(\mathfrak{P})$. This can be seen in the following way:

take any point $x \in \mathbb{P}^n \setminus X$ and consider the resolution of the skyscraper sheaf $k_x$ $$0 \to F_{\bullet} \to k_x.$$ On restricting this resolution to $\mathfrak{P}$ (restrict to any thickening $X_m$ otherwise), we get an exact sequence $$0 \to F_{\bullet}\otimes O_{\mathfrak{P}} \to 0.$$

Therefore we see that $\sum(-1)^iF_i \mapsto 0$ under the map $K_0(\mathbb{P}^n) \to K_0(\mathfrak{P})$.

Clearly I am making some (stupid) mistake. I would be glad if someone points out what it is. In particular which of the arguments is wrong.

Thanks!

I will first assume that, by $K_0$, you mean the quotient of the free abelian group generated by isomorphism classes of objects in your category, modulo the relation $$\sum_i(-1)^i [F_i] = 0 \quad (A)$$ for every exact sequence $0 \to F_0 \to F_1 \to \dots \to F_{n-1} \to F_n \to 0$.

Well, since the functor does not preserve exactness of such long exact sequences, there is no reason that it should induce an isomorphism of $K_0$ using this definition.

OTOH, if you use as definition of the K-group the quotient generated by relations for short exact sequences, then I do not see why the relation (A) should hold. Of course, in an abelian category, you can decompose a long exact sequence into short exact sequences, and deduce it from the relation . Then the above relation for long exact sequences follows from the relation coming from the short exact sequences. But in an exact category, I do not see how to deduce (A).

• @ABayer: Thanks. I realise the mistake I am making. I was indeed considering relations by short exact sequences. – MOfan Apr 11 '11 at 15:47

Dear all,

First, I apologize for writing this as an "answer" -- it is just a comment on ABayer's answer (but I don't have permission to comment on answers). ABayer's point is right on: the kernels and cokernels in the standard resolution of a skyscraper sheaf of a point are not reflexive. So you cannot break things up into short exact sequences. One case where you do get a short exact sequence is for P^2. But Grothendieck rules out the equivalence in this case (because he requires depth at least 3 or 4, can't remember which at the moment).

Best regards,

Jason

• The text inside <> is supposed to be "not". I am not sure what is wrong with the parser. – Jason Starr Apr 11 '11 at 11:15
• Hi Jason. It seems the computer thinks your method of emphasis looks a lot like an HTML tag. I changed it to a boldface. – S. Carnahan Apr 11 '11 at 16:12

format: plain


text:
Elementary Algebra

$x^{3}$ - 9$x^{2}$ + 27x - 27
$(x - 3)^{3}$ = (x - 3)(x - 3)(x - 3) = Remember that $(a-b)^{2}$ = $a^{2}$ - 2ab + $b^{2}$. So, (x - 3)(x - 3)(x - 3) = ($x^{2}$ - 2$\times$x$\times$3 + $3^{2}$)(x - 3) = ($x^{2}$ - 6x + 9)(x - 3) = Use the distributive property. $x^{2}$(x) +$x^{2}$(-3) -6x(x) -6x(-3) +9(x) +9(-3) = $x^{3}$ - 3$x^{2}$ - 6$x^{2}$ + 18x + 9x - 27 = Simplify. $x^{3}$ - 9$x^{2}$ + 27x - 27


text:
### So It's 28

Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.

### I Like ...

Mr Gilderdale is playing a game with his class. What rule might he have chosen? How would you test your idea?

# Sort Them Out (1)

## Sort Them Out (1)

It's best to do this with $3$ other friends. You can get copies of these cards here: .docpdf

Share all the cards out equally among the four of you.
Work out what the answer is to each question on your cards.
Do you notice anything?
What do you notice?
Try sorting or arranging the cards in some way.

### Why do this problem?

The intention of this problem is that children will work together as a team. It should help in developing mathematical language as well as providing interesting practice in addition and subtraction and possibly offer opportunities to share calculation strategies. It gives ample opportunities to practise calculations using numbers to $20$ and should support children to remember these facts so that they are able to recall and use them whenever they need to do so.

### Possible approach

This sheet will need to be printed out and cut into the fifteen separate cards. You must give out all the cards in the set!

Give out a set of cards to each group of four to six children. (The questions asked in the set do vary in difficulty so you may like to choose who to give specific cards.) To begin with, give children time to work individually on the questions on their own cards. After a suitable period of time, encourage them to ask each other about the questions and answers on other cards. At this stage, members of the group can help each other with the questions, but discourage them from simply giving someone else an answer.

Challenge each group to organise or arrange the cards in some way. At this point, stand back and try not to intervene as they work together. You could invite them to display their cards in their chosen arrangement and then give some time for everyone to move around the room to look at the way each group has sorted the cards.

A short plenary should provide an opportunity for the groups to explain the organisation they chose and for such remarks as "I knew my answer was wrong because it didn't fit in with anyone else's".

### Key questions

Tell me about the number you have found.


text:
# Robot Engine / Isaac SDK Bridge¶

The Robot Engine / Isaac SDK Bridge Extension is used to connect Omniverse Isaac Sim with NVIDIA Isaac SDK™. Isaac SDK is a developer toolbox for accelerating the development and deployment of AI-powered robots. This extension enables message communication between the two platforms over TCP. These messages include simulated sensor data, drive commands, ground truth state of simulated assets and scenario management.

This extension is enabled by default. If it is ever disabled, it can be re-enabled from the Extension Manager by searching for omni.isaac.robot_engine_bridge.

## API Documentation¶

See the API Documentation for complete usage information.

## User Interface¶

The Robot Engine Bridge contains an IsaacSim-side Isaac SDK application that sends and receives messages, these messages are converted to/from native SDK Cap’n Proto messages for use in the simulator.

To start the IsaacSim-side IsaacSDK application:

• Open the Robot Engine Bridge UI window from the Isaac Utils -> Robot Engine Bridge menu item.

• Click Create Application to start the default Isaac Sim application.

Once Pressed the specified application in the Application Json Path field will be launched. Custom application can be specified by modifying this path. The application can be stopped by pressing the Destroy Application button.

The default isaacsim application contains a interface node, an input isaac::alice::TcpSubscriber component on port 55001 and a output isaac::alice::TcpPublisher component on port 55000

Expand to see the full app config File
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41  { "name": "isaacsim", "modules": [], "graph": { "nodes": [ { "name": "interface", "components": [ { "name": "ledger", "type": "isaac::alice::MessageLedger" }, { "name": "input", "type": "isaac::alice::TcpSubscriber" }, { "name": "output", "type": "isaac::alice::TcpPublisher" }, { "name": "time_sync", "type": "isaac::alice::TimeSynchronizer" } ] } ], "edges": [] }, "config": { "interface": { "input": { "port": 55001, "host": "localhost" }, "output": { "port": 55000 } } } }

The json path can also be provided at runtime by using the --carb/isaac/robot_engine_bridge/json="path/to/app.json" commandline argument

Note

• The set of Isaac SDK components currently supported is limited to: isaac::alice::MessageLedger, isaac::alice::TcpSubscriber, isaac::alice::TcpPublisher and isaac::alice::TimeSynchronizer and at least one of each should be present to work correctly. Multiple subscribers and publishers can be created and used by changing the default inputComponent and outputComponent fields in the components below.

## Robot Engine Bridge Components¶

All of the properties below can be edited in the “Raw USD Properties” section of the properties tab. Relationships can also be directly modified there.

The following properties are shared between all components

Expand to See Properties

Name

Type

Default

Description

nodeName

string

interface

Name of the node in the application to publish/subscribe to

enabled

bool

True

Enable or disable this component from sending/receiving data

timeOffset

double

0

Time offset applied to acquisition time when sending to SDK

The Scene loader component is a special component that allows the user to load USD files once the application has been created and play is pressed to start simulation. It is controlled via the collapsible UI in the main Robot Engine Extension UI.

Expand to See Properties

Name

Type

Default

Description

requestChannel

string

scenario_control

Channel name that receives json message with information on what usd stage to load

inputComponent

string

input

The name of the component on which commands are received

outputComponent

string

output

Then name of the component on which reply is sent

string

cameraControl

string

camera_switch

Channel name that receives a camera name to switch the viewport to

### Differential Base¶

This component is used to control a differential base

Expand to See Properties

Name

Type

Default

Description

accelerationSmoothing

float

1.0

A smoothing factor for the estimated acceleration

chassisPrim

relationship

USD Prim Path for Chassis Prim

inputChannel

string

base_command

The name of the channel on which commands are received

inputComponent

string

input

The name of the component on which commands are received

leftWheelJointName

string

Name of left wheel joint

maxSpeed

float2

(1.5, 1.0)

The maximal allowed linear and angular speed in m/s and rad/s

maxTimeWithoutCommand

float

0.2

The maximum allowed time duration which the robot will continue with the last sent speed command in the absence of speed commands

outputChannel

string

output

The name of the channel on which state is published

outputComponent

string

base_state

The name of the component on which state is published

rightWheelJointName

string

Name of right wheel joint

robotFront

float3

(1,0,0)

Vector representing the robots forward direction

wheelBase

float

0.5

Distance between wheel and center of chassis in meters

float

0.1

Radius of driven wheels in meters

### Holonomic Base¶

This component is used to control a holonomic base with three wheels such as Kaya

Expand to See Properties

Name

Type

Default

Description

accelerationSmoothing

float

1.0

A smoothing factor for the estimated acceleration

articulationPrim

relationship

USD Prim Path for Articulation

inputChannel

string

base_command

The name of the channel on which commands are received

inputComponent

string

input

The name of the component on which commands are received

maxSpeed

float2

(1.5, 1.0)

The maximal allowed linear and angular speed in m/s and rad/s

maxTimeWithoutCommand

float

0.2

The maximum allowed time duration which the robot will continue with the last sent speed command in the absence of speed commands

outputChannel

string

output

The name of the channel on which state is published

outputComponent

string

base_state

The name of the component on which state is published

robotFront

float3

(1,0,0)

Vector representing the robots forward direction

wheel1JointName

string

Name of first wheel joint

wheel2JointName

string

Name of second wheel joint

wheel3JointName

string

Name of third wheel joint

wheelBase

float

0.125

Distance from wheel to chassis center in meters

float

0.04

Radius of the wheel in meters

### Vehicle Control¶

This component is used to control a PxVehicle with acceleration and steering commands

Expand to See Properties

Name

Type

Default

Description

inputChannel

string

vehicle_command

The name of the channel on which commands are received

inputComponent

string

input

The name of the component on which commands are received

outputChannel

string

vehicle_state

The name of the channel on which state is published

outputComponent

string

output

The name of the component on which state is published

vehiclePrim

relationship

USD Prim path to the PxVehicle

usePID

bool

false

Enable PID controller for acceleration

historyLength

int

100

Length of buffer used to smooth acceleration data

controllerPIDValues

float3

(1,1,1)

PID controller values

### Joint Control¶

This component publishes and subscribes the joint positions of an articulation

Expand to See Properties

Name

Type

Default

Description

articulationPrim

relationship

USD Prim path to the articulation being controlled

inputChannel

string

joint_position

The name of the channel on which commands are received

inputComponent

string

input

The name of the component on which commands are received

outputChannel

string

joint_state

The name of the channel on which state is published

outputComponent

string

output

The name of the component on which state is published

### Scissor Lift¶

This component controls a vertical translational joint on an articulation

Expand to See Properties

Name

Type

Default

Description

articulationPrim

relationship

USD Prim path to the articulation being controlled

inputChannel

string

lift_command

The name of the channel on which commands are received

inputComponent

string

input

The name of the component on which commands are received

liftJointName

string

lift_joint

Name of the translational joint on the articulation being controlled

liftSpeed

float

0.02

Speed at which the joint raises and lowers in m/s

outputChannel

string

lift_state

The name of the channel on which state is published

outputComponent

string

output

The name of the component on which state is published

### Surface Gripper¶

This component simulates a suction cup gripper

Expand to See Properties

Name

Type

Default

Description

inputChannel

string

io_command

The name of the channel on which commands are received

inputComponent

string

input

The name of the component on which commands are received

gripperEntity

string

gripper

name returned by the component

outputChannel

string

io_state

The name of the channel on which state is published

outputComponent

string

output

The name of the component on which state is published

parentPrim

relationship

Rigid body prim representing the suction cup gripper

gripThreshold

float

1

forceLimit

float

1e10

Max force before joint breaks

torqueLimit

float

1e10

Max torque before joint breaks

bendAngle

float

0

Max angle the joint can bend before hitting a limit

stiffness

float

1e10

Stiffness of the joint

damping

float

1e3

damping parameter for the joint

offsetPosition

float3

(0,0,0)

offset from the parent prim where the joint is created

offsetRotation

float4

(1,0,0,0)

quaternion representing rotation offset of the joint from the parent prim

### Two Finger Gripper¶

This component simulates a gripper with two finger joints

Expand to See Properties

Name

Type

Default

Description

articulationPrim

relationship

The articulation that contains the two finger joints

inputChannel

string

io_command

The name of the channel on which commands are received

inputComponent

string

input

The name of the component on which commands are received

gripperEntity

string

gripper

name returned by the component

outputChannel

string

io_state

The name of the channel on which state is published

outputComponent

string

output

The name of the component on which state is published

closedDistance

float

0

Distance between fingers when closed

openDistance

float

0.04

Distance between fingers when open

leftFingerJoint

string

left_finger

The name of the left finger joint

rightFingerJoint

string

right_finger

The name of the right finger joint

speed

float

0

Zero speed will instantly close/open based on joint parameters

### Rigid Body Sink¶

This component publishes the rigid body poses for a list of primitives

Expand to See Properties

Name

Type

Default

Description

rigidBodyPrims

relationship

list of relationship targets to rigid bodies that need to be published

outputChannel

string

bodies

The name of the channel on which state is published

outputComponent

string

output

The name of the component on which state is published

### Teleport¶

This component controls the pose for a list of primitives

Expand to See Properties

Name

Type

Default

Description

teleportPrims

relationship

list of relationship targets to rigid bodies that need to be teleported

inputChannel

string

teleport

The name of the channel on which commands are received

inputComponent

string

input

The name of the component on which commands are received

### Scenario From message¶

This component can create new prims, teleport them and publish their rigid body pose

Expand to See Properties

Name

Type

Default

Description

inputChannel

string

scenario_actors

The name of the channel on which commands are received

inputComponent

string

input

The name of the component on which commands are received

teleportInputComponent

string

input

name returned by the component

teleportInputChannel

string

teleport

Name of the node in the application to publish/subscribe to

rigidBodySinkOutputChannel

string

bodies

The name of the channel on which state is published

rigidBodySinkOutputComponent

string

output

The name of the component on which state is published

### Camera¶

This component publishes RGB, Depth and Segmentation data

Expand to See Properties

Name

Type

Default

Description

rgbOutputChannel

string

color

The name of the channel on which color state is published

rgbOutputComponent

string

output

The name of the component on which color state is published

depthOutputChannel

string

depth

The name of the channel on which depth state is published

depthOutputComponent

string

output

The name of the component on which depth state is published

segmentationOutputChannel

string

segmentation

The name of the channel on which segmentation state is published

segmentationOutputComponent

string

output

The name of the component on which segmentation state is published

boundingBox2DOutputChannel

string

bbox

The name of the channel on which 2D bbox state is published

boundingBox2DOutputComponent

string

output

The name of the component on which 2D bbox state is published

boundingBox3DOutputChannel

string

bbox3d

The name of the channel on which 3D bbox state is published

boundingBox3DOutputComponent

string

output

The name of the component on which 3D bbox state is published

boundingBox3DClassList

string

The list of class names to filter

rgbEnabled

bool

true

true if rgb data is published

depthEnabled

bool

false

true if depth data is published

segmentationEnabled

bool

false

true if segmentation data is published

boundingBox2dEnabled

bool

false

true if 2D bbox data is published

boundingBox3dEnabled

bool

false

true if 3D bbox data is published

cameraPrim

relationship

USD Prim path to the active camera in the viewport

resolution

int2

(1280, 720)

Resolution of the viewport

### Lidar¶

This component publishes lidar data

Expand to See Properties

Name

Type

Default

Description

lidarPrim

relationship

USD Prim path to lidar sensor

outputChannel

string

rangescan

The name of the channel on which state is published

outputComponent

string

output

The name of the component on which state is published

### Contact Monitor¶

This component publishes a message if a specified object collides with another.

Expand to See Properties

Name

Type

Default

Description

targetPrim

relationship

USD Prim path to rigid body to report contacts for

ignoredPrims

relationship

USD Prims paths that will be ignored when reporting contacts

outputChannel

string

collision

The name of the channel on which state is published

outputComponent

string

output

The name of the component on which state is published

forceThreshold

float

100000

Contact force threshold above which contacts are reported

### Occupancy Grid Map¶

This component publishes occupancy grid map

Expand to See Properties

Name

Type

Default

Description

parentPrim

relationship

If specified, used to compute mapping location

outputChannel

string

occupancy_map

The name of the channel on which state is published

outputComponent

string

output

The name of the component on which state is published

offset

float3

(0,0,0)

Mapping location offset, if parentPrim is not specified this is the offset from global origin

cellSize

float

0.1

Size of each pixel in meters

degreesPerRay

float

5

Number of degrees per ray when doing a 360 sweep

surfaceOffset

float

0.02

Offset from surface in meters to compute next sweep

occupancyThreshold

float

1

Probability used to determine occupancy

maxRays

float

1000000

Max number of rays to compute before terminating

mapSize

int2

(256,256)

Map size in pixels, the mapping area is computed using cell size

occupiedValue

float

1.0

Value returned for a region that is occupied

unoccupiedValue

float

0.0

Value returned for a region that is unoccupied

unknownValue

float

0.5

Value returned for a region that could not be mapped from the start location

debugDraw

bool

false

Enable Debug Drawing

### Ultrasonic¶

This component publishes ultrasonic data

Expand to See Properties

Name

Type

Default

Description

ultrasonicPrim

relationship

USD Prim path to ultrasonic sensor array

outputChannel

string

uss_envelopes

The name of the channel on which state is published

outputComponent

string

output

The name of the component on which state is published

### Polyline Visualizer¶

This component visualizes polyline

Expand to See Properties

Name

Type

Default

Description

inputChannel

string

sight_plan

The name of the channel on which commands are received

inputComponent

string

input

The name of the component on which commands are received

width

float

1.0

Width of polyline

color

color4f

(1, 1, 1, 1)

Color of polyline

offset

float3

(0,0,0)

Translation offset for polyline

parentPrim

relationship

If specified will be used as origin transform for polyline

### Sample Applications¶

Below are some sample Isaac SDK applications leveraging Omniverse Isaac Sim. For more details, please refer to Isaac SDK Documentation.

Carter robot navigating in the Warehouse simulation environment. For more details, please refer to Isaac SDK Carter Navigation sample.

### Cart Delivery¶

The robot delivering a cart in the Warehouse simulation environment. For more details, please refer to Isaac SDK Cart Delivery sample.

### Shuffle Box with Simulator¶

The UR10 robot arm with suction gripper picks and drops a pink box in the simulation environment. For more details, please refer to Isaac SDK Shuffle Box sample.

### Simple Joint Control¶

The UR10 robot arm is manipulated in the simulation environment via joint commands from Isaac SDK. For more details, please refer to Isaac SDK Joint Control sample.

## Code Snippets¶

The following python-only examples show how to interface with the Isaac SDK when running in pure python mode. Make sure that the steps in the Setup section have been completed.

This sample loads any usd stage and starts the Omniverse Isaac Sim side sdk application to publish and receive TCP messages.

./python.sh standalone_examples/api/omni.isaac.robot_engine_bridge/load_stage.py --usd_path /Samples/Isaac_SDK/Scenario/carter_warehouse_with_forklifts.usd

### Pose Estimation¶

This sample demonstrates how to do two pass rendering for a pose estimation data generation scenario. In the first pass, the object of interest, a cube, is rendered occluded in the environment. In the second the object is rendered unoccluded with the environment hidden.

The textures in the environment, camera pose and object pose are all randomized. While running the current FPS is output to the screen.

./python.sh standalone_examples/api/omni.isaac.robot_engine_bridge/pose_estimation.py

### Publishing custom messages¶

This sample demonstrates how to build a custom Composite proto message and publish on a given port. This sample can be used as a template for publishing any data with a custom structure using one of the provided Isaac SDK proto messages. All proto messages supported by Isaac SDK are available via the pyalice python interface

./python.sh standalone_examples/api/omni.isaac.robot_engine_bridge/custom_message.py

## Tutorials & Examples¶

The following example showcases how to best use this extension:

Examples

• Isaac SDK Vehicle Control Example: Isaac Examples > Isaac SDK > Isaac SDK Vehicle Control


text:
# To J. D. Hooker   2 November [1858]

Down Bromley Kent

Nov. 2d

My dear Hooker

I have pleasure in enclosing draft for 330 for Mr Ralfs1—if you fail in getting amount I will add £2s2 more.—

I hardly like sending Circulars to friends who are not naturalists, & all Naturalists you will think of, as J. Lubbock, Bunbury, Leighton of Shrewsbury, C. Spence Bate of Plymouth.2 Ld. Overstone professes to care for Science3 I would send him one.— You can send one to Miss Carr care of The Lady Cranworth4 Holwood Bromley, Kent She is interested in such subjects & kind, I fancy pretty rich.— Add my name in M.S. to List5 T. C. Eyton Esq Eyton Wellington Salop. I can think of no one else. Mr Peters (& I do not know his address) a rich Banker cares, or did care, much about Microscopes.—6

If I can think of any one else I will let you know.—

Many thanks for information about Museum; you seem progressing well.—7 I suppose Kensington Gore must be borne; but I must think a very large proportion of those who now visit Brit. Mus. go there on their feet & will not go by Railway, even if not costing more than an Omnibus.—8 I confess I feel rather frightened at change but sincerely hope that it is vain fear; & as Government is moving, I thoroughily agree to wisdom of naturalists moving.

Moor Park did me wonderful good & I walked one day 4$\frac{1}{2}$ miles! with only a few rests!

My dear Hooker | Ever yours | C. Darwin

The following addresses would be worth I think a penny stamp. Revd. W. Darwin Fox Delamere Rectory Northwich E. Vernon Harcourt Esq 9 Hastings E. Darwin Esq 57. Queen Anne St J. M. Herbert Esq 10 County Court Judge Hereford Wollaston 10 Hereford St Park Lane11 J. G. Perry Esq 12 12 Westbourne St Hyde Park Gardens H. C. Watson13 Thames Ditton W. H. Benson Esq 14 28 Grt. George St Portman Sqe

N.B. If many give or you think I ought to give more than 330 tear up my draft & I will at once send 550.15

You may add my name in M.S. for all these to list of Committee, if you like.—

## Footnotes

CD’s contribution to a fund set up to aid the botanist John Ralfs. Hooker’s role in organising the fund through the Philosophical Club of the Royal Society is described in L. Huxley ed. 1918, 1: 414. The minutes of the Philosophical Club meeting on 28 October 1858 (Royal Society) read: Mr Huxley brought before the members of the Club individually the case of Mr. Ralfs well known for his Researches on the Desmidiae, who with failing health had been unexpectedly deprived of the small annuity upon which he depended Mr H. stated that he should be glad to receive subscriptions towards the purchase of an annuity for him— Dr. Hooker supported Mr. Huxley’s appeal. Ralfs’s work in compiling a comprehensive monograph on desmids (microscopic algae), to be published in a new edition of Andrew Pritchard’s A history of infusoria, living and fossil (1841), was delayed by ill-health, encroaching blindness, and poverty (DNB). Hooker’s appeal for £500 to buy Ralfs an annuity was subscribed twice over (L. Huxley ed. 1918, 1: 414). Instead of using the whole sum for an annuity much larger than intended, the Philosophical Club of the Royal Society proposed investing the excess capital to serve as a general scientific charitable fund. CD was a signatory of a preliminary draft drawn up on 16 December 1858 to put this proposal before the Royal Society’s council (see Correspondence vol.7, Appendix VII).
CD refers to John Lubbock, Charles James Fox Bunbury, William Allport Leighton, and Charles Spence Bate.
Both CD and Samuel Jones Loyd, Baron Overstone, were members of the Athenæum Club (see Correspondence vol. 6, letters to W. E. Darwin, [26 February 1856], to J. D. Hooker, 11 May [1856], and to T. H. Huxley, 27 May [1856]).
Robert Monsey Rolfe and his wife Laura (Lord and Lady Cranworth) lived at Holwood Hill, Keston, Kent. Lord Cranworth was lord chancellor, 1852–8. Miss Carr was probably a sister of Lady Cranworth (née Laura Carr).
Hooker evidently intended to circulate a printed list of subscribers to potential donors. The addition of CD’s name, would explain why the circular was to be sent to the people named in the letter.
Probably William Peters of 35 Nicholas Lane, London and Ashford Lodge, Sussex. Peters was a partner in the banking company Masterman, Peters & Co. (Post Office London directory 1858).
Hooker had evidently mentioned to CD the possibility of the new museum being located in Kensington Gore, a road on the northern boundary of the government site in Kensington. See also letter to J. D. Hooker, 29 [October 1858].
Edward William Vernon Harcourt was an expert on the ornithology of Madeira and had formerly corresponded with CD on that topic (see Correspondence vol. 6).
John Maurice Herbert had been a close friend of CD’s since their undergraduate days at Cambridge.
Thomas Vernon Wollaston, a specialist on Madeiran entomology, was a friend and a scientific correspondent of CD’s.
John George Perry was a London surgeon and astronomer. He is listed in CD’s Address book (Down House MS).
Hewett Cottrell Watson, well known to both CD and Hooker, was particularly interested in the geographical distribution of plants. He had corresponded with CD since 1847 about botanical problems arising out of CD’s work.
William Henry Benson was a conchologist and entomologist. He had supplied CD with observations on shells in India in 1855 (Correspondence vol. 5, letter from W. H. Benson, 5 December 1855).
A payment of £3 3s. for ‘Ralfs charity’ was recorded in CD’s Account book (Down House MS) on1 November 1858.

## Bibliography

Correspondence: The correspondence of Charles Darwin. Edited by Frederick Burkhardt et al. 27 vols to date. Cambridge: Cambridge University Press. 1985–.

DNB: Dictionary of national biography. Edited by Leslie Stephen and Sidney Lee. 63 vols. and 2 supplements (6 vols.). London: Smith, Elder & Co. 1885–1912. Dictionary of national biography 1912–90. Edited by H. W. C. Davis et al. 9 vols. London: Oxford University Press. 1927–96.

Post Office London directory: Post-Office annual directory. … A list of the principal merchants, traders of eminence, &c. in the cities of London and Westminster, the borough of Southwark, and parts adjacent … general and special information relating to the Post Office. Post Office London directory. London: His Majesty’s Postmaster-General [and others]. 1802–1967.

## Summary

On moving the natural history collection of the British Museum to Kensington.

Subscription for John Ralfs.

## Letter details

Letter no.
DCP-LETT-2351
From
Charles Robert Darwin
To
Joseph Dalton Hooker
Sent from
Down
Source of text
DAR 114: 252
Physical description
6pp

## Please cite as

Darwin Correspondence Project, “Letter no. 2351,” accessed on 22 September 2020, https://www.darwinproject.ac.uk/letter/DCP-LETT-2351.xml

Also published in The Correspondence of Charles Darwin, vol. 7

letter


text:
Comparing the incomparable — expanding the time map concept

We are surrounded by discrete events: posts and comments, purchases from the online store, page visits are all examples of discrete events. Some of these events happen periodically; others happen sporadically. Some happen once in a while; others are generated every microsecond. There are many ways to visualise such streams, the most naive one being a one-dimensional plot in which events are placed on a time axis, as demonstrated in the figure below for three simple cases.

But what happens if there are too many such events? How do we gain a global overview? Recently, I have learned about “Time maps” — a creative way to perform such a visualization, which I will describe below using examples from WordPress.com. I will expand this approach to accommodate for bringing cases with drastically different time scales to the same domain, for easier comparison and pattern discovery.

Time maps

Instead of plotting the absolute time values, or time differences, Max Watson from District Data Labs suggests representing each data point in two dimensions — in terms of time passed before and after the events. Thus, all the data points from case A in the figure above will be represented by the pairs $(1; 1)$ as all of them occur at regular intervals of 1 second. Most of the dots in case B from the same figure will be represented by the same pair, $(1; 1)$. However, the pause between the event at time marks 3 and 6 seconds and not 1. Thus, the third point will be represented by $(1; 3)$ and the fourth one — by $(3; 1)$. Following are the time maps for the three toy cases from the figure above:

Note that all the dots in case A overlap. Now we can identify several interesting regions in these time maps. Events that happen at equal intervals lie on the diagonal line and overlap. Events that occur at approximately equal intervals will be distributed over the diagonal. The upper right portion of the map represent events that happen at a slow pace while the lower-left quadrant of the graph represents fast events. Long interruptions followed by recovery manifest as large deviations from the diagonal — the interruption itself (high $\Delta t after$) is shown as a dot above the diagonal and the event after the interruption (high $\Delta t before$) appears as a dot to the right of the diagonal.

Let us examine some real-life events. I’ll start with publishing posts or pages on some of the WordPress.com blogs.

Following is a time map of a particularly busy site with more than 7,700 posts. On the right, you may see post and pages time map in which each point is colored according to the GMT in which the post was published. Note the logarithmic scale of the plot that shows events at time intervals of several orders of magnitude — from seconds to weeks. As you may see, the vast majority of publishing events occur at intervals of several minutes to one hour. The lines of purple dots represent breaks of approximately eight hours in publishing new posts — something that is consistent with a medium size editorial team. To the right, I present a landscape representation of the same time map. The dark blue bulb in the graph on the left shows that the vast majority of the events occur at regular intervals between several minutes and one hour.

Time map of another busy site, with its 6511 posting events, shows a slightly different publishing pattern.

In this news site, most of the events occurred at one-minute intervals. The typical service break in this site is either one or two days long, as represented by the two distinct strips.
Using the time map approach is relatively easy to identify bots. For example, the owners of the following site have uploaded more than 1,000 posts such that most of them were published at a one-minute interval — a pattern that is unlikely to be performed by a human.

What about smaller blogs? This Norwegian blogger posted less than 150 posts with a typical time interval between a day and week.

I have analyzed time maps of different administrative tasks performed by  several random WordPress.com users.
This user, for example, is a moderately active blogger with a single blog. Their time map is characteristic of sporadic events with intervals of several seconds to one day. We may see that this user is very disciplined and does not leave the blog for more than a day.

On the other hand, another user, who has two blogs presents a completely different time map with thousands of events within the range of micro- and milliseconds. Notice that the two types of events (the super-fast ones and the “normally paced” ones) are well separated and that the super-fast events occur at very specific hours. Does this mean that that user’s computer is affected by a virus? Does that user use an aggressive script to monitor their site? This is a topic for another post.

Bringing the maps to the common scale

You may have noticed the vast differences in time scales that we saw in the maps above. Sometimes, we are interested in map’s shape, so that we can compare similar patterns that occur at different time scales. To achieve this task, I first sort time difference values in each case (user, blog, log record, etc.). Next, I transform these values such that the median time difference (the one that is larger than half of the observed points) gets the value of 0. Time differences larger then the median receive positive values and time differences smaller than the median receive negative values. In technical terms: I first compute the empirical cumulative distribution of the time differences, and then perform logit transform.

The following figure shows the normalised maps for four blogs I have mentioned at the beginning of this document . We may see that the elliptical shape of the dot cloud represents a series of events that occur somewhat regularly at a relatively low frequency. Round-shaped graphs represent cases in which the typical variability in inter-event periods is comparable to that period. Bot-induced time maps are still easily detected in this representation

Let us now examine the normalised time maps of the two users mentioned above:

One interesting thing to note is the fact that the lower-left region in the second user’s graph is now significantly enlarged because it contains a much larger amount of data points. As with the case of the blog time maps, the approximately round shape of the first user’s time map indicates that the variability of the between-event interval is comparable in its size to the interval. Note, though, that when observing first user’s raw map, it was clear that that user’s events mostly occur not faster than once a second, and that there the maximal interval is never larger than one day. Such a notion, which may be valuable in many cases, is absent from the normalised graph.

Conclusions

Time maps, as developed by Max Watson from District Data Labs provide a valuable insight in the analysis of event streams. It is evident that these maps reveal interesting activity patterns. Analysis of such patterns may be used in spam and fraud detection.

The proposed normalisation of time maps provides a tool for comparing the shapes of drastically different maps. The following figure, for example, presents time maps of Premium and Business purchases performed in Japanese Yens between June 1st and Dec 20th, 2015 :.

I selected the Yen as the least frequent currency used in WordPress.com. Had I analysed the same data for USD, we would see a completely different raw map, but comparable normalised one:

2 thoughts on “Comparing the incomparable — expanding the time map concept”

1. Thank you. I actually saw this typo, but it somehow managed to escape 🙂

Like


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# outdoor dining set nl

(D.P) - Determine whether A is diagonalizable. Every Diagonalizable Matrix is Invertible Is every diagonalizable matrix invertible? Diagonalizable matrix From Wikipedia, the free encyclopedia (Redirected from Matrix diagonalization) In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix. If so, give an invertible matrix P and a diagonal matrix D such that P-1AP = D and find a basis for R4 consisting of the eigenvectors of A. A= 2 1 1 0 0 1 4 5 0 0 3 1 0 0 0 2 Sounds like you want some sufficient conditions for diagonalizability. Counterexample We give a counterexample. The eigenvalues are immediately found, and finding eigenvectors for these matrices then becomes much easier. Solution. If so, give an invertible matrix P and a diagonal matrix D such that P-AP = D and find a basis for R4 consisting of the eigenvectors of A. A= 1 -3 3 3 -1 4 -3 -3 -2 0 1 1 1 0 0 0 Determine whether A is diagonalizable. A matrix is diagonalizable if and only if for each eigenvalue the dimension of the eigenspace is equal to the multiplicity of the eigenvalue. Given a partial information of a matrix, we determine eigenvalues, eigenvector, diagonalizable. As an example, we solve the following problem. I do not, however, know how to find the exponential matrix of a non-diagonalizable matrix. How can I obtain the eigenvalues and the eigenvectores ? It also depends on how tricky your exam is. A matrix is diagonalizable if and only of for each eigenvalue the dimension of the eigenspace is equal to the multiplicity of the eigenvalue. If is diagonalizable, find and in the equation To approach the diagonalization problem, we first ask: If is diagonalizable, what must be true about and ? For the eigenvalue $3$ this is trivially true as its multiplicity is only one and you can certainly find one nonzero eigenvector associated to it. Determine whether the given matrix A is diagonalizable. A matrix can be tested to see if it is normal using Wolfram Language function: NormalMatrixQ[a_List?MatrixQ] := Module[ {b = Conjugate @ Transpose @ a}, a. b === b. a ]Normal matrices arise, for example, from a normalequation.The normal matrices are the matrices which are unitarily diagonalizable, i.e., is a normal matrix iff there exists a unitary matrix such that is a diagonal matrix… (because they would both have the same eigenvalues meaning they are similar.) If so, find a matrix P that diagonalizes A and a diagonal matrix D such that D=P-AP. True or False. How do I do this in the R programming language? Can someone help with this please? Here you go. ...), where each row is a comma-separated list. A is diagonalizable if it has a full set of eigenvectors; not every matrix does. Definition An matrix is called 8‚8 E orthogonally diagonalizable if there is an orthogonal matrix and a diagonal matrix for which Y H EœYHY ÐœYHY ÑÞ" X Thus, an orthogonally diagonalizable matrix is a special kind of diagonalizable matrix: not only can we factor , but we can find an matrix that woEœTHT" orthogonal YœT rks. Therefore, the matrix A is diagonalizable. Here are two different approaches that are often taught in an introductory linear algebra course. Thanks a lot Since this matrix is triangular, the eigenvalues are 2 and 4. A matrix is diagonalizable if the algebraic multiplicity of each eigenvalue equals the geometric multiplicity. For example, consider the matrix $$\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$$ In order to find the matrix P we need to find an eigenvector associated to -2. Solution If you have a given matrix, m, then one way is the take the eigen vectors times the diagonal of the eigen values times the inverse of the original matrix. In this post, we explain how to diagonalize a matrix if it is diagonalizable. Once a matrix is diagonalized it becomes very easy to raise it to integer powers. Given the matrix: A= | 0 -1 0 | | 1 0 0 | | 0 0 5 | (5-X) (X^2 +1) Eigenvalue= 5 (also, WHY? A matrix that is not diagonalizable is considered “defective.” The point of this operation is to make it easier to scale data, since you can raise a diagonal matrix to any power simply by raising the diagonal entries to the same. I have a matrix and I would like to know if it is diagonalizable. If the matrix is not diagonalizable, enter DNE in any cell.) (a) (-1 0 1] 2 2 1 (b) 0 2 0 07 1 1 . f(x, y, z) = (-x+2y+4z; -2x+4y+2z; -4x+2y+7z) How to solve this problem? All symmetric matrices across the diagonal are diagonalizable by orthogonal matrices. I am currently self-learning about matrix exponential and found that determining the matrix of a diagonalizable matrix is pretty straight forward :). How to solve: Show that if matrix A is both diagonalizable and invertible, then so is A^{-1}. In other words, if every column of the matrix has a pivot, then the matrix is invertible. This MATLAB function returns logical 1 (true) if A is a diagonal matrix; otherwise, it returns logical 0 (false). By solving A I x 0 for each eigenvalue, we would find the following: Basis for 2: v1 1 0 0 Basis for 4: v2 5 1 1 Every eigenvector of A is a multiple of v1 or v2 which means there are not three linearly independent eigenvectors of A and by Theorem 5, A is not diagonalizable. A matrix is said to be diagonalizable over the vector space V if all the eigen values belongs to the vector space and all are distinct. In fact if you want diagonalizability only by orthogonal matrix conjugation, i.e. If is diagonalizable, then which means that . Matrix diagonalization is the process of performing a similarity transformation on a matrix in order to recover a similar matrix that is diagonal (i.e., all its non-diagonal entries are zero). One method would be to determine whether every column of the matrix is pivotal. That should give us back the original matrix. Consider the $2\times 2$ zero matrix. \] We can summarize as follows: Change of basis rearranges the components of a vector by the change of basis matrix $$P$$, to give components in the new basis. The answer is No. In that The zero matrix is a diagonal matrix, and thus it is diagonalizable. If so, find the matrix P that diagonalizes A and the diagonal matrix D such that D- P-AP. So, how do I do it ? Determine whether the given matrix A is diagonalizable. In the case of $\R^n$, an $n\times n$ matrix $A$ is diagonalizable precisely when there exists a basis of $\R^n$ made up of eigenvectors of $A$. Meaning, if you find matrices with distinct eigenvalues (multiplicity = 1) you should quickly identify those as diagonizable. [8 0 0 0 4 0 2 0 9] Find a matrix P which diagonalizes A. Calculating the logarithm of a diagonalizable matrix. Johns Hopkins University linear algebra exam problem/solution. Solved: Consider the following matrix. ), So in |K=|R we can conclude that the matrix is not diagonalizable. A= Yes O No Find an invertible matrix P and a diagonal matrix D such that P-1AP = D. (Enter each matrix in the form ffrow 1), frow 21. There are many ways to determine whether a matrix is invertible. Now writing and we see that where is the vector made of the th column of . (Enter your answer as one augmented matrix. Beware, however, that row-reducing to row-echelon form and obtaining a triangular matrix does not give you the eigenvalues, as row-reduction changes the eigenvalues of the matrix … Find the inverse V −1 of V. Let ′ = −. Does that mean that if I find the eigen values of a matrix and put that into a diagonal matrix, it is diagonalizable? But eouldn't that mean that all matrices are diagonalizable? Determine if the linear transformation f is diagonalizable, in which case find the basis and the diagonal matrix. I know that a matrix A is diagonalizable if it is similar to a diagonal matrix D. So A = (S^-1)DS where S is an invertible matrix. Get more help from Chegg. If A is not diagonalizable, enter NO SOLUTION.) Not all matrices are diagonalizable. In this case, the diagonal matrix’s determinant is simply the product of all the diagonal entries. A matrix $$M$$ is diagonalizable if there exists an invertible matrix $$P$$ and a diagonal matrix $$D$$ such that \[ D=P^{-1}MP. But if: |K= C it is. Then A′ will be a diagonal matrix whose diagonal elements are eigenvalues of A. Given a matrix , determine whether is diagonalizable. 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Transformation f is diagonalizable row is a comma-separated list I am currently self-learning about matrix exponential found... Is triangular, the diagonal matrix, it is simply the product of all the diagonal D!


text:
# On star-symmetric polynomials with classical behaviour

I will discuss sequences of polynomials of a single variable that are orthogonal with respect to a vector of weights defined in the complex plane. Such polynomial sequences satisfy a recurrence relation of finite (and fixed) order higher than 2. They share a number of properties that mimic those of standard orthogonality on $L_2$ spaces. The main focus will be on polynomial sequences possessing a three-star symmetry and whose multiple orthogonality is preserved under the action of the derivative operator.


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# How do you solve 16 = t + 7?

Apr 29, 2016

$t = 9$

#### Explanation:

To find the value of $t$, you must get $t$ on one side of the equals sign and all of the other numbers in the other side.

To cancel out $+ 7$ from the right side of the sum, minus $7$ from both side of the sum. To get the right answer, you must do whatever you do on one side of the equals sign to the other side of the equals sign.

$16 - 7 = t$

$16 - 7$ is equal to $9$ so:

$9 = t$

Or this is more commonly written like:

$t = 9$


text:
Largest remainder method

Wikipedia has an article on:

The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems. It is a contrast to the highest averages method.

Method

The largest remainder method requires the number of votes for each party to be divided a quota representing the number of votes required for a seat, and this gives a notional number of seats to each, usually including an integer and either a fraction or alternatively a remainder. Each party receives seats equal to the integer. This will generally leave some seats unallocated: the parties are then ranked on the basis of the fraction or equivalently on the basis of the remainder, and parties with the larger fractions or remainders are each allocated one additional seat until all the seats have been allocated. This gives the method its name.

Quotas

There are several possibilities for the quota. The most common are the Hare quota and the Droop quota.

The Hare Quota is defined as follows

${\displaystyle \frac{\mbox{total} \; \mbox{votes}}{\mbox{total} \; \mbox{seats}}}$

The Hamilton method of apportionment is actually a largest-remainder method which is specifically defined as using the Hare Quota. It is used for legislative elections in Namibia and in the territory of Hong Kong. It was historically applied for congressional apportionment in the United States during the nineteenth century.

The Droop quota is the integer part of

${\displaystyle 1+\frac{\mbox{total} \; \mbox{votes}}{1+\mbox{total} \; \mbox{seats}}}$

and is applied in elections in South Africa.

The Hare quota tends to be slightly more generous to less popular parties and the Droop quota to more popular parties. Which is more proportional depends on what measure of proportionality is used.

${\displaystyle \frac{\mbox{total} \; \mbox{votes}}{2+\mbox{total} \; \mbox{seats}}}$

is rarely used since it suffers from the problem that it may result in more candidates being elected than there are seats available; this will certainly happen if there are only two parties. In such a case, it is usual to increase the quota until the number of candidates elected is equal to the number of seats available, in effect changing the voting system to a highest averages system with the Jefferson apportionment formula.

The largest remainder method is the only apportionment that satisfies the quota rule; in fact, it is designed to satisfy this criterion. However, it comes at the cost of paradoxical behaviour. The Alabama paradox is defined as when an increase in seats apportioned leads to decrease in the number of seats a certain party holds. Suppose we want to apportion 25 seats between 6 parties in the proportions 1500:1500:900:500:500:200. The two parties with 500 votes get three seats each. Now allocate 26 seats, and it will be found that the these parties get only two seats apiece.

With 25 seats, we get:

Party A B C D E F Total
Votes 1500 1500 900 500 500 200 5100
Seats             25
Hare Quota             204
Quotas Received 7.35 7.35 4.41 2.45 2.45 0.98
Automatic seats 7 7 4 2 2 0 22
Remainder 0.35 0.35 0.41 0.45 0.45 0.98
Surplus seats 0 0 0 1 1 1 3
Total Seats 7 7 4 3 3 1 25

With 26 seats, we have:

Party A B C D E F Total
Votes 1500 1500 900 500 500 200 5100
Seats             26
Hare Quota             196
Quotas Received 7.65 7.65 4.59 2.55 2.55 1.02
Automatic seats 7 7 4 2 2 1 23
Remainder 0.65 0.65 0.59 0.55 0.55 0.02
Surplus seats 1 1 1 0 0 0 3
Total Seats 8 8 5 2 2 1 26

Examples

These examples take an election to allocate 10 seats where there are 100,000 votes.

Hare quota

Party Yellows Whites Reds Greens Blues Pinks Total
Votes 47,000 16,000 15,800 12,000 6,100 3,100 100,000
Seats             10
Hare Quota             10,000
Votes/Quota 4.70 1.60 1.58 1.20 0.61 0.31
Automatic seats 4 1 1 1 0 0 7
Remainder 0.70 0.60 0.58 0.20 0.61 0.31
Highest Remainder Seats 1 1 0 0 1 0 3
Total Seats 5 2 1 1 1 0 10

Droop quota

Party Yellows Whites Reds Greens Blues Pinks Total
Votes 47,000 16,000 15,800 12,000 6,100 3,100 100,000
Seats             10
Droop Quota             9,091
Votes/Quota 5.170 1.760 1.738 1.320 0.671 0.341
Automatic seats 5 1 1 1 0 0 8
Remainder 0.170 0.760 0.738 0.320 0.671 0.341
Highest Remainder Seats 0 1 1 0 0 0 2
Total Seats 5 2 2 1 0 0 10