text: To set alignment in Word, select all of your text, then click the left justification icon in the Home tab or select Format>Paragraph and choose “Left” in the Alignment drop-down box. Whether you are looking for essay, coursework, research, or term paper help, or with any other assignments, it is no problem for us. A comprehensive solution (based on this answer) is to define new column types (say, L, C, and R) that take their width as argument and do the following:. If multiple single-image tables are stacked, they float to align across the page, depending on page width. Pause before and after each command. Now the option is gone. If you place text into a frame with two or more columns, you may see that the lines do not line up from one column to the next, and especially likely when combining a text frame and superimposed image frame. Just wanted to point out that the use of “\vspace{5pt}\\” to get nice spacing in the tables can backfire if the \vspace will cause a new line to be added to that row (if your text is _just_ the wrong length). When the paragaph falls at the top of the page, the text won't line up with the numbers. ; The right-most object is right aligned with it’s parent element. Better is to use “\\[5pt]” which doesn’t have this problem. The text of your command might show in the document for a moment. Stunning gallery layouts with advanced settings for complete customization and control. Some people prefer justified text, others do not. How is this handled now? For example, if you say "select all," the words "select all" show up before your text is selected. right – Content aligns along the right side. This can determine how easy your text is to read. After we discuss tables, we’ll cover some other controls that will help round out your formatting prowess, including adding links, using symbols, creating math equations, and quite a … ; center – Content centers between the left and right edges. Line spacing is usually only single or double-spaced. Make sure there's no space before or after the paragraph, and justified text. Select the font you want to use. This is not always standard, but can improve readability, especially in documents with a lot of text. The @CommonsWare answer is correct. We write high quality term papers, sample essays, research papers, dissertations, thesis papers, assignments, book reviews, speeches, book reports, custom web content and business papers. Achieveressays.com is the one place where you find help for all types of assignments. The inconsistent spacing between words created by justified text can be problematic for people with cognitive concerns such as Dyslexia. Left justified text is also known as ragged right text. In this lesson, we’re going to delve into tables, which are a huge part of laying out well formatted documents. For example, the first line of the last paragraph on page 19 includes no spacing between any of the characters, making the sentence almost unreadable. Justified text. Have a specified foreground and background color. ; The objects will not wrap down as the browser window narrows. This normally happens when after the sentence is over, we do not hit enter, but just keep typing "space" and the sentence starts on the next line. Wiki Configurable text justification settings as part of a justification overhaul that also allows more space to be used inter-char when there are no spaces on the line. Figure 4.6.4b: Where to click to select text justification or left-aligned (“ragged right”) text in the MS Word Home menu tool ribbon To fix the “hockey teeth” gaps resulting from justification such as what you see in parts of this paragraph, turn on hyphenation in MS Word via the Layout tool ribbon: select Automatic in the Hyphenation dropdown menu in the Page Setup section. Thanks! Set the indent for the first line of paragraphs to .2 or so (I started with .3, but that's too much). The most common trim size for paperbacks in the U.S. is 6" x 9" (15.24 x 22.86 cm), but you have several other options.In Word, you set your trim size with the Page Size option. If you want to override it you can use the Text justification option in this section. So a simple solution presents itself: don’t use double spaces if you’re writing justified-alignment text. Our prices depend on the urgency of your assignment, your academic level, the course subject, and the length of the assignment. Notice that it is split into two separate fields, but we want it to wrap to the next line. If you’re using left-alignment, like in most letters, many legal documents, etc., use double spaces, because they look better. If the first word of the text is too long, no text will fit to complete the left-hand side, so beware creating a "ragged left margin" when not enough space remains for text to fit alongside floating tables. Adjust the letter spacing for an entire text layer, or a selection of text. You can also see this when you have two separate text frames side-by-side. from 14 page. Letter spacing lets you define the space between characters in a word, line, or paragraph. FooGallery is a powerful, responsive photo gallery plugin. Buy WoodMart - Responsive WooCommerce WordPress Theme by xtemos on ThemeForest. When you edit a line in a justified (full-width) paragraph, the text is likely to lurch to the left, leaving a blank space between the end of the line and the right margin. No wrap text. ; Each object is equidistant from one another at all times. Have adequate line spacing (at least 1/2 the height of the text) and paragraph spacing (1.5 times line spacing). 4. Highlight some text and click on the "line options" tab. ; Bleed. Acrobat is not a word processor, so there is absolutely no way to adjust leading (line spacing,) so the underlines can't be lined up with the text, so we will remove the lines to make way for a multiline text field. Punctuation spacing is when you include 1 or 2 spaces after punctuation. MDN Understanding WCAG, Guideline 1.4 explanations; Understanding Success Criterion 1.4.8 | Understanding WCAG 2.0 An example of cursively-justified Arabic text, rendered by Tasmeem. Just go to the end of the line and press enter once. ; The objects will stop short of overlapping each other as the browser window narrows. Here is a video that shows how to fix this (version 1.3.3.x): Best, Jolyon By @danfickle. Markdown beautifies text without explicit tags, based on common practices from ASCII e-mail and plain-text documents. Verify that Docs or Slides heard the right command. This happens when the text alignment is "justified" in word. Normally, calibre will preserve the justification in the original document. Content aligns along the left side. #410 Fix for wrong bold setting on list item counters. The Notes field will be an issue. "Markdeep" is farther "down" from "markdown" on the autostyling and beautification path. Most magazines and books are now justified-aligned, so double spacing is probably not appropriate for them. Issue \raggedright, \centering, or \raggedleft to achieve the desired horizontal alignment,. HIGH SCHOOL. If you need professional help with completing any kind of homework, Success Essays is the right place to get it. Also, the justified text is generally not a problem, but the inconsistency in spacing between characters is sometimes problematic. Text ¶ Text can be either justified or not. Justified text has extra spaces between words to give a smooth right margin. Setting trim size and margins A few key terms before you start: Trim size. Your printed book's width and height. UNIVERSITY. Like English, Arabic can be justified by adjusting the spacing between words, but in most styles it can also be justified by calligraphically elongating or compressing the letterforms themselves. Justified text may look nice in newspapers and magazines, but that's because columns are narrower, so spreading the text out evenly across them doesn't result … In earlier versions, to fix this, I checked the Suppress extra line spacing at the top of the page. Double line spacing; Any citation style (APA, MLA, Chicago/Turabian, Harvard) Affordable prices. Cheap essay writing sercice. Thanks @syjer for PR fix (and test!) COLLEGE. ... Variables are used throughout the entire project as a way to centralize and share commonly used values like colors, spacing, or font stacks. and @acieplinski for reporting. from 10 page. The text will align itself along its left side, leaving its right side nonuniform (unjustified). Markdeep combines an easy-to-use and browser-friendly packaging with new unique features such as … from \$ 13 page. ... Internet Explorer 9 and below will render text gray with a nasty text-shadow that we cannot fix. Are NOT fully justified (aligned to both the left and the right margins). The text-align property in CSS is used for aligning the inner content of a block element.. p { text-align: center; } These are the traditional values for text-align: left – The default value. Here are the goals we are trying to achieve: The left-most object is left aligned with it’s parent element. Double-spacing is usually required for academic writing, since it makes it easier to insert comments and corrections. 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Get Full Access to Conceptual Physics - 12 Edition - Chapter 35 - Problem 4e
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# Suppose instead that the person riding on top of the

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## Solution for problem 4E Chapter 35

Conceptual Physics | 12th Edition

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Problem 4E

Suppose instead that the person riding on top of the freight car shines a searchlight beam in the direction in which the train is traveling. Compare the speed of the light beam relative to the ground when the train is at rest and when it is in motion. How does the behavior of the light beam differ from the behavior of the ball in Exercise 1? Exercise 1 A person riding on the roof of a freight train throws a ball forward. (a) Neglecting air drag and relative to the ground, is the ball moving faster or slower when the train is moving than when it is standing still? (b) Relative to the freight car, is the ball moving faster or slower when the train is moving than when the train is standing still?

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In the case of a light beam shining from atop a moving freight car, the light beam has the same speed relative to the ground as it has relative to the train while in the case of the behavior of the ball in problem 50, the ball's speed varies in relation to the ground and the moving train.

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November 19, 2021

### Perspectives on Global Monetary Policy Coordination, Cooperation, and Correlation

Vice Chair Richard H. Clarida

At the "Macroeconomic Policy and Global Economic Recovery" 2021 Asia Economic Policy Conference, sponsored by the Federal Reserve Bank of San Francisco Center for Pacific Basin Studies, San Francisco, California (via webcast)

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In my remarks today, I would like to offer some perspectives on global monetary policy correlation and what it can—and cannot—reveal about the prevalence and value of global monetary policy coordination or, in the limit, binding global monetary cooperation.1 In both the Global Financial Crisis (GFC) and the Global Pandemic Collapse (GPC), major central banks around the world responded by cutting policy rates to, and then keeping them at, their effective lower bounds (ELBs); by increasing their balance sheets through ambitious and expansive large-scale asset purchase and lending programs; and by offering forward guidance—both Delphic and Odyssean—on the stance of their future monetary policies.2

As these examples make clear, we certainly do observe that national monetary policies are often correlated, and such examples are not confined to recent experience. Indeed, the international monetary economics literature abounds with historical and empirical studies of correlated global monetary policy cycles, not to mention the evident secular downtrend in global monetary policy rates observed in recent decades. Moreover, global monetary policies often also do appear at least sometimes to be coordinated. There are certainly enough meetings of Group of Seven (G-7), Group of Twenty, and Bank for International Settlements (BIS) central bank governors—virtual and in person—that provide such opportunities. By contrast with evidence of central bank correlation and coordination, the historical record suggests that, since the collapse of the Bretton Woods system 50 years ago, rarely, if ever, do major sovereign central banks actually enter into, at least publicly—let alone respect—binding commitments to pursue formal cooperative policies.3 Here and throughout the lecture, I shall reserve the meaning of the adjective cooperative, as in "cooperative" central bank policies, to define policies that can, at least conceptually, be thought of as policies pursued by sovereign central banks that fully incorporate any spillovers from one central bank's policies to the other central bank's mandates. Cooperative policies defined this way will, in general, differ from noncooperative policies—in the sense of the Nash equilibrium concept—designed to achieve domestic mandates assigned by sovereign institutions while taking other central bank policies as given. In these remarks, I will draw on perspectives informed not only by economic analysis—including my own academic research on the theory and empirics of central banks' policymaking—but also by my observations and experiences serving as Vice Chair of the Federal Reserve Board during what has certainly been an eventful three years.

To begin with the theory, it has long been appreciated that in the sorts of economic models used by central banks around the world, the calibrated gains to international monetary policy cooperation are found to be rather modest relative to a status quo ante in which each country runs a sensible policy, taking as given the sensible policies of the other countries. Today I will make a somewhat different, and less often discussed, case questioning formal global monetary policy cooperation—that, in practice, adopting it could plausibly erode central bank credibility and public support for central bank instrument independence.

International monetary policy coordination, which I think of as including the sharing of information and analysis among central banks regarding the evolution of their individual economies as well as the considerations that govern the setting of their policy instruments—in other words, information about their policy reaction functions can, and in my observation certainly does, enhance the design and effectiveness of monetary policy execution for each country. I shall give some examples later.

But while international monetary policy coordination may enhance the efficiency of monetary policy execution, I am skeptical that in practice there are additional material, reliable, and robust gains that would flow from a formal regime of binding monetary policy cooperation, at least among major G-7 economies with flexible exchange rates, open capital accounts, and central bank mandates that include price stability. Or, more precisely, it seems that whatever gains might exist in theory, they likely do not exceed the full cost of committing to such an arrangement in practice. In such a regime, national monetary policies in each country would be constrained to be set in such a way so as to jointly maximize some metric for global price stability and perhaps also other objectives. The reason is that there can be global externalities to monetary policy that create such theoretical gains to cooperation. However, as Clarida, Galí, and Gertler (2002), among others, have shown in the context of central banks with domestic price-stability mandates, to achieve the theoretical gains to international monetary policy cooperation, policy in each country must be set with reference to an index of inflation in all countries party to the cooperative agreement.

This is a policy that no central bank would choose were it not bound to the agreement to achieve domestic price stability while taking other central banks' policies as given, which, of course, raises the question of how such an agreement might be enforced. In theory, if central banks were bound to set policy based solely on a policy rule that is a function of observed macroeconomic data, enforcement would be simple. But in practice, no sovereign central bank outsources its policy to an Excel spreadsheet, and, moreover, best-practice policy rules incorporate inputs—such as the neutral rate of interest or expected inflation—that are unobserved and time varying and thus difficult to monitor. Finally, central bank mandates vary across sovereign jurisdictions. For example, the Federal Reserve is mandated to pursue policies that achieve "maximum employment and price stability." The European Central Bank (ECB) is mandated by treaty to pursue price stability but also has a secondary mandate to contribute to achieving the objectives of the European Union, which include balanced economic growth and full employment. Similarly, the Bank of England's primary objective is to achieve the U.K. government's target of 2 percent inflation, but its secondary objective is to support the government's policy aims, including those for employment, growth, and—more recently—environmental sustainability. So even though all major central banks are mandated to achieve "price stability" in some form or fashion, their mandates typically include other obligations that vary across jurisdictions. Because these obligations can, and sometimes do, require central banks to make a tradeoff—for example, in the United States, between inflation and employment in the presence of supply shocks—defining the objective that a cooperative agreement would choose to maximize would be a formidable task.

I believe that, in practice, beyond the issue of enforcement and agreement on terms, there could well be another challenge with policy cooperation that is absent from most theoretical discussions. Simply stated and in the context of a price-stability mandate for each central bank, the problem—as I see it—is the threat to the credibility of the central bank, the challenges to central bank communication, and the resulting potential loss of support for its policy actions from the public when the policy choices that would be required by binding monetary policy cooperation react not only to home inflation deviations from target, but also to deviations of foreign inflation from target. In theoretical models, the commitment to the inflation target is just assumed to be perfect and credible, but in practice, credibility appears to be a function of central bank communication and of the policies actually implemented pushing inflation toward—and, in the absence of shocks, keeping inflation at—target. I suspect that, in practice, central banks would have a hard time maintaining credibility and independence as well as communicating a policy that raised home interest rates aggressively not because home inflation is too high but because foreign inflation is!

While, perhaps for these reasons, we do not have many confirmed sightings of genuine binding monetary policy cooperation, we do observe many examples of monetary policy correlation (see figure 1). Monetary policies will obviously be correlated if countries are subject to common shocks—such as in the GFC and the GPC—but can also be correlated as a consequence of integrated global capital markets. The Clarida, Galí, and Gertler (2002) model provides a simple illustration of how this correlation can come about. In the "home" country, the optimal inflation-targeting monetary policy can be written as a Taylor-type rule:

$R_{t} = r_{t} + (1 + \frac{\lambda}{\alpha})\pi_{t},$

where $r_t$ is the "neutral" interest rate in the home country consistent with price stability and trend growth and $\pi_t$ is inflation in the home country. The ratio $\lambda/\alpha$ captures the extent to which the home central bank trades off its inflation and gross domestic product (GDP) objectives when they are in conflict. The parameter $\alpha$ indexes the priority the central bank places on stabilizing GDP growth relative to its trend path. When $\alpha$ is large, the central bank leans against high inflation less aggressively than it would were $\alpha$ small. With an integrated global capital market, the neutral policy rate at home is a function of trend GDP growth at home as well as expected foreign GDP growth:

$r_{t} = 2E_{t}\{\Delta \bar{y}_{t+1}\} + E_{t}\{\Delta y^{*}_{t+1}\}.$

Now, to the extent the foreign central bank has a comparative advantage in tracking or forecasting foreign output growth—which, of course, it should, since such growth will depend on the foreign central bank's monetary policy—sharing this information with the home central bank can improve that institution's estimate of the home equilibrium real interest rate and thus the effectiveness of its policy rule in meeting its domestic objectives.

The foreign central bank sets its policy rate in an analogous fashion:

$R^{*}_{t} = r^{*}_{t} + (1 + \frac{\lambda}{\alpha^{*}})\pi^{*}_{t}.$

The ratio $\lambda/\alpha^{*}$ captures the extent to which the foreign central bank trades off its inflation and GDP objectives when they are in conflict, and note that this tradeoff may be different than that of the home central bank. With an integrated global capital market, the neutral policy rate abroad is given by

$r^{*}_{t} = 2E_{t}\{\Delta \bar{y}^{*}_{t+1}\} + E_{t}\{\Delta y_{t+1}\}.$

This simple example illustrates one channel through which globally integrated capital markets can introduce policy correlation even in the absence of policy coordination or cooperation. For example, even if home and foreign inflation rates and trend growth rates are uncorrelated across countries, sensible monetary policies can be correlated across countries if neutral policy rates are correlated, as they appear to be in the data (see figure 2, which is based on Holston, Laubach, and Williams, 2017).

A recent working paper, Ferreira and Shousha (2021), models the determinants of the neutral policy rate in the United States and includes in the empirical specification an index of global productivity growth and demographic trends. Their empirical estimates attribute 85 basis points of the decline in U.S. neutral real interest rates since 2000 to global spillovers from the slowdown in global trend growth and demographic trends. So policy correlation can be an outcome of noncooperative monetary policy operating in an integrated global capital market and not evidence, in and of itself, of policy coordination, let alone cooperation. Before continuing, I consider briefly how monetary policy would differ under binding cooperation in the simple example sketched out earlier. Under international monetary policy cooperation, the policy rate is set according to

$R_{t} = r_{t} + (1 + \frac{\lambda}{\alpha})\pi_{t} + \frac{1}{3}(1+\frac{\lambda}{\alpha})\pi^{*}_{t}.$

So, intuitively, while policy in the home country does place more weight on home inflation when setting the interest rate, it also reacts to foreign inflation as well, as I discussed earlier.

Consider the case with home inflation high and foreign inflation low. As can be seen from the preceding equation, the optimal policy under cooperation calls for the home interest rate to be lower—more accommodative—than it would be in the absence of cooperation. In practice, the home central bank's credibility as an inflation targeter to satisfy its domestic price-stability mandate might well suffer if it failed to respond aggressively to high domestic inflation because of, say, serious deflation abroad.

Monetary Policy Spillovers Flow in Both Directions
There is a vast literature that documents the existence of international spillovers from U.S. monetary policy, especially to emerging markets (EMs). Recent research—Hoek, Kamin, and Yoldas (2020)—suggests that the extent and consequences of such spillovers depend importantly on the source of a shift in U.S. monetary policy. Their key finding, illustrated in figure 3, is that Federal Reserve policy rate surprises attributed to stronger U.S. growth generally have only moderate spillovers to EM financial conditions, whereas U.S. policy rate changes attributed to U.S. inflationary pressures trigger more substantial spillovers to EM financial conditions. The authors also find compelling evidence that the magnitude of the cross-border spillovers attributed to U.S. monetary policy shifts depends on a country's macroeconomic vulnerability—as measured by inflation, current account deficit, international reserves, government debt, external debt, and private-sector credit growth, with more vulnerable EM countries experiencing larger spillovers from U.S. monetary policy.

A recent paper—Curcuru, De Pooter, and Eckerd (2018)—examines and compares the consequences of ECB policy actions for U.S. financial conditions and the consequences of Federal Reserve policy actions for euro-area financial conditions. Their main findings are summarized in the two panels in figure 4. The left panel presents evidence of statistically significant spillovers from Federal Open Market Committee (FOMC) policy announcements to euro-area bond markets. But, as shown in the right panel, the authors find that the spillovers from ECB policy announcements to U.S. yields are roughly as large as those from FOMC announcements to bund yields.

While I will certainly acknowledge that both fundamental and financial shocks originating in the United States, including shifts in the direction of U.S. monetary policy, can and do propagate globally (see, for example, Miranda-Agrippino and Rey, forthcoming), the evidence suggests that causality can and does run in both directions. It is not difficult to recall external events that have triggered spillovers from foreign sovereign markets to the U.S. economy that have had observable implications for U.S. monetary policy.

The Mexican peso crisis of 1994–95 and the Asian financial crisis of 1997–98 resulted in substantial declines in economic activity in emerging markets but had only a modest effect on the U.S. economy. One reason for this outcome was "safe haven" flows into the dollar assets that pushed down U.S. bond yields. But the Russian default of August 1998, followed by the collapse of Long-Term Capital Management, had more substantial effects on global markets and posed greater risk to the U.S. economy, and these events triggered a policy response by the Federal Reserve that cut the federal funds rate 75 basis points between September and November of that year. When in the next year the global financial system stabilized, the Federal Reserve reversed those actions and returned the federal funds rate to the level prevailing before those events.

In more recent times, global shocks have also been consequential for U.S. economic prospects and monetary policy. Examples include the 2011–13 euro-zone crisis and the China devaluation and capital flight episode of 2015–16, when worries about a hard landing and renminbi depreciation, respectively, roiled world markets. Both of these shocks originated in economies with large footprints in the global economy and financial system, and, as a result, they induced substantial disruptions in global financial markets. During both episodes, U.S. stock markets fell and the dollar appreciated, especially during 2015 and 2016. As ever, these adverse global shocks triggered safe-haven flows that pushed U.S. Treasury yields down. But despite the drop in Treasury yields, overall financial conditions in the United States tightened, weighing on aggregate demand.

U.S. monetary policy responded to these global "headwinds," helping stave off actual contractions of U.S. activity. During the 2011–13 euro-zone crisis, the Federal Reserve was already pursuing very accommodative policies in the wake of the GFC, but the introduction of the maturity extension program in September 2011 and the announcement in September 2012 of the third installment of quantitative easing were in part aimed at offsetting the effect of these global headwinds on U.S. aggregate demand. Regarding the aforementioned China devaluation episode of 2015–16, Federal Reserve statements and transcripts from that time indicate that concerns about these developments and their effect on the U.S. economy were a factor that contributed to the delay in implementing previously signaled policy rate increases.4 Again, as in the 1997–98 episode discussed earlier, once the "storm had passed," the previously signaled policy normalization process commenced in 2017.

The Federal Reserve and ECB Framework Reviews
The monetary policy framework reviews conducted by the Federal Reserve and the ECB provide another example of monetary policy correlation. In February 2019, the Federal Reserve System launched a review of its monetary policy strategy, tools, and communication practices. A key motivation for the Fed's review as well as the ECB's review was the substantial decline in estimates of the neutral real interest rate, r*, that, over the longer run, is consistent with price stability. This decline has critical implications for monetary policy because it leaves central banks with less conventional policy space to offset adverse shocks to aggregate demand.5

As discussed in Clarida (2020), the Fed's review, from the outset, built on three pillars: a series of Fed Listens events, a flagship research conference, and a series of rigorous briefings for the Committee commencing in July 2019 and running through January 2020.6 While our plans to conclude the review earlier in 2020 were, like so many things, delayed by the arrival of the pandemic, in August 2020, the Federal Reserve did announce, with unanimous support, an evolution of its monetary policy strategy to flexible average inflation targeting.7

Similar to the Federal Reserve, the ECB launched a review of its monetary policy strategy in January 2020.8 Since the time of its previous strategy review in 2003, like the Fed, the ECB observed profound structural changes in the global and euro-area economies that have driven down neutral interest rates and increased the incidence and duration of episodes in which nominal policy interest rates are close to the ELB. So the ECB's review sought to adapt its monetary policy to the current economic environment and to ensure that its policy could remain effective at the ELB. In its review, the ECB heard from a wide variety of European organizations and citizens, including through a series of ECB Listens events.9 The ECB's Governing Council engaged in a series of deliberations, also informed by extensive staff background analysis on a range of topics. In July 2021, with unanimous support from its Governing Council, the ECB announced its new strategy.10 Key elements of that strategy are a symmetric 2 percent inflation target, a view that the ELB requires especially forceful or persistent monetary policy measures to keep inflation expectations from drifting lower, and the affirmation that the full range of measures it has used in recent years will remain in its toolkit.

The similarities in the two framework evolutions are due to the fact that powerful common global forces are driving down neutral policy rates and limiting the effectiveness of monetary policy in downturns to offset declines in aggregate demand. This asymmetry of policy effectiveness caused by the ELB imparts a secular downward bias to inflation that, if not offset, could de-anchor inflation expectations below the central bank's price-stability objectives. In addition, the similarities in the processes according to which these two framework reviews were conducted reflected also a convergence in thinking about best practices for such reviews that was facilitated by a working group on central bank frameworks set up by the BIS.

Conclusion
In this speech, I have offered some perspectives on global monetary policy correlation and what it can—and cannot—reveal about the prevalence and value of global monetary policy coordination or, in the limit, binding global monetary cooperation. I have argued that, while there are several recent and historical examples where we certainly do observe that national monetary policies are often correlated, adopting formal global monetary policy cooperation could plausibly erode central bank credibility and public support for central bank independence. But I also observe that international monetary policy coordination, defined as including the sharing of information and analysis among central banks regarding the evolution of their individual economies and information about their policy reaction functions, can enhance the design and effectiveness of monetary policy execution for each country.

Thank you very much for your time and attention. I look forward to my conversation with Sylvain.

References

Campbell, Jeffrey R., Charles L. Evans, Jonas D.M. Fisher, and Alejandro Justiniano (2012). "Macroeconomic Effects of Federal Reserve Forward Guidance (PDF)," Brookings Papers on Economic Activity, Spring, pp. 1–54.*

Clarida, Richard H. (2020). "The Federal Reserve's New Monetary Policy Framework: A Robust Evolution," speech delivered at the Peterson Institute for International Economics, Washington (via webcast), August 31.

——— (2021). "The Federal Reserve's New Framework and Outcome-Based Forward Guidance," speech delivered at "SOMC: The Federal Reserve's New Policy Framework," a forum sponsored by the Manhattan Institute's Shadow Open Market Committee, New York (via webcast), April 14.

Clarida, Richard, Jordi Galí, and Mark Gertler (2002). "A Simple Framework for International Monetary Policy Analysis," Journal of Monetary Economics, vol. 49 (July), pp. 879–904.

Curcuru, Stephanie E., Michiel De Pooter, and George Eckerd (2018). "Measuring Monetary Policy Spillovers between U.S. and German Bond Yields," International Finance Discussion Papers 1226. Washington: Board of Governors of the Federal Reserve System, April.

European Central Bank (2020). "ECB Launches Review of Its Monetary Policy Strategy," press release, January 23.

——— (2021a). "Strategy Review," webpage, https://www.ecb.europa.eu/home/search/review/html/index.en.html.

——— (2021b). "ECB's Governing Council Approves Its New Monetary Policy Strategy," press release, July 8.

Ferreira, Thiago, and Samer Shousha (2021). "Supply of Sovereign Safe Assets and Global Interest Rates," International Finance Discussion Papers 1315. Washington: Board of Governors of the Federal Reserve System, April.

Hoek, Jasper, Steve Kamin, and Emre Yoldas (2020). "When Is Bad News Good News? U.S. Monetary Policy, Macroeconomic News, and Financial Conditions in Emerging Markets (PDF)," International Finance Discussion Papers 1269. Washington: Board of Governors of the Federal Reserve System, January.

Holston, Kathryn, Thomas Laubach, and John C. Williams (2017). "Measuring the Natural Rate of Interest: International Trends and Determinants," Journal of International Economics, vol. 108 (May, S1), pp. S59–75.

Miranda-Agrippino, Silvia, and Hélène Rey (forthcoming). "The Global Financial Cycle," in Gita Gopinath, Elhanan Helpman, and Kenneth Rogoff, eds., Handbook of International Economics, vol. 5. Oxford, United Kingdom: North‑Holland.

Powell, Jerome H. (2020). "New Economic Challenges and the Fed's Monetary Policy Review," speech delivered at "Navigating the Decade Ahead: Implications for Monetary Policy," a symposium sponsored by the Federal Reserve Bank of Kansas City, held in Jackson Hole, Wyo., August 27.

1. The views expressed are my own and not necessarily those of other Federal Reserve Board members or Federal Open Market Committee participants. I would like to thank Burcu Duygan-Bump, Chiara Scotti, and Paul Wood for assistance in preparing these remarks. Return to text

2. The terms "Delphic" and "Odyssean" are used to describe different types of forward guidance. Delphic forward guidance forecasts macroeconomic performance and likely monetary policy actions under that forecast, whereas Odyssean forward guidance publicly commits the FOMC to a future action. For more details, see Campbell and others (2012). Return to text

3. While adopting the euro—and, more broadly, joining a currency union—is the ultimate binding commitment to cooperative monetary policy, the focus of my remarks today is on countries that are not members of a currency union. Return to text

4. The minutes and transcripts of these FOMC meetings are accessible on the Board's website at https://www.federalreserve.gov/monetarypolicy/fomccalendars.htm. Return to text

5. For a detailed discussion of the elements that motivated the launch of the review, see Clarida (2020, 2021) and Powell (2020). Return to text

6. Summaries of these discussions can be found in the minutes of these FOMC meetings, which are accessible on the Board's website at https://www.federalreserve.gov/monetarypolicy/fomccalendars.htm. Return to text

7. The Statement on Longer-Run Goals and Monetary Policy Strategy is available on the Board's website at https://www.federalreserve.gov/monetarypolicy/review-of-monetary-policy-strategy-tools-and-communications-statement-on-longer-run-goals-monetary-policy-strategy.htm. Return to text

* This reference was updated on November 22, 2021. Return to text

Last Update: November 22, 2021


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# A ball at rest is dropped from a height of 12 m. It looses $25 \%$ of its Kinetic energy on striking the ground and bounces back to a height 'h'. Then value of 'h' is

$(a)\;3\;m \quad (b)\;6\;m \quad (c)\;9\;m \quad (d)\;12\;m$

(c) 9m


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# Proving smoothness of a variety

I want to understand how smoothness is checked for algebraic varieties given as zero sets of functions.

Let me consider a "textbook" example to help fix ideas. Let the algebraic variety be given by these three polynomial equations,

$x_0x_3 -x_1x_2=0$

$x_1^2 -x_0x_2=0$

$x_2^2- x_1x_3=0$

Firstly I observe that if I assume $x_3$ and $x_2$ to be nonzero then the third equation follows from the first two.

Does that allow me to totally forget about the third equation?

Now I can write down the Jacobian matrix for the above set. I guess then I am supposed to check if the rank of that Jacobian evaluated on the common zeroset = or not to the number of functions.

But I can't figure out how this calculation has to be done. How do I determine the rank and further so on the common zero set?

I would be grateful if someone can walk me through the steps in this specific example.

-
From a (basic) differential topology viewpoint a strategy could be to show that $0$ is a regular value of your polynomials (hence their zero-set is a smooth manifold) and then checking that the intersections are transversal. (Of course assuming a real or complex manifold) Unfortunately this does not work for your example... –  Jose27 Sep 14 '11 at 7:05
Dear Anirbit, The third equation lets you solve for $x_2$, and eliminate it from the other equations, so that you are left with two equations in the three variables $x_0,x_1,x_3$. Now try solving these two equations just as you would have when you were in high school, and draw the resulting curve. Is it smooth? Regards, –  Matt E Sep 14 '11 at 12:02
I predict you intended the last relation to be $x_2^2 - x_1 x_3$, because then this would be a standard example. Am I right? –  David Speyer Sep 14 '11 at 21:13
Dear Anirbit, One difficulty with checking smoothness via Jacobians is that presumably you are interested in the variety cut out by your equations, not some possibly non-reduced scheme. For this, you need to compute the radical of the ideal generated by your equations before you move on to computing Jacobians. (E.g. $x = 0$ cuts out a smooth hyperplane, while $x^2 = 0$ cuts out the same hyperplane, but won't satisfy th e Jacobian condition.) Regards, –  Matt E Sep 14 '11 at 21:41
@David Yes. I meant it to be $x_2^2$. Corrected it! –  Anirbit Sep 17 '11 at 23:15

Taking Matt E's hint into account you could rewrite this as $x_0x_3-x_1^2x_3=0$ and $x_1^2-x_0x_1x_3=0$.

The Jacobian is $\left(\begin{matrix} x_3 & -2x_1x_3 & x_0-x_1^2 \\ -x_1x_3 & 2x_1-x_0x_3 & -x_0x_1\end{matrix}\right)$.

Since your equations aren't homogeneous, I'm assuming this isn't meant to be projective, so in particular $(0,0,0)$ is on your variety and also makes the Jacobian have $0$ in every entry so it isn't smooth.

Maybe you were hoping to see something more illuminating like how to check it has full rank when it actually does have full rank on the zero set?

-
So is your conclusion that the variety I wrote down is NOT smooth? And yes, I was looking for a method to check that the rank version of thinking. I would think that one has to show that there exists a subset of the equations such that the Jacobian on the zero-set is of the same rank as the number of equations in that subset I guess thats what you are referring to as the Jacobian having a "full rank" on the zero set? –  Anirbit Sep 14 '11 at 17:12
It is not smooth. If it were, then the Jacobian will have constant rank $n-r$ where $n$ is the number of variables and $r$ is the number of equations (so the rank is the dimension of the variety if it is a complete intersection). –  Matt Sep 14 '11 at 17:57
@Anirbit: Dear Anirbit, You can easily figure out explicitly what variety you wrote down; it is reducible, a union of three one-dimensional components, two of which pass through the origin. Hence it is not smooth at the origin. Regards, –  Matt E Sep 14 '11 at 21:42
@Matt E: As I understand it Anirbit actually doesn't care if this particular variety is smooth or not he (or she) is much more interested in how a general argument would go if one were to use the Jacobian criterion. –  Matt Sep 14 '11 at 23:55
@Matt I had a typo in my equations. Sorry for the trouble. I can see Griffiths in his book saying that this variety is smooth. I can more or less understand his claim going through the Jacobian argument but I am not fully sure. May be you can elaborate. –  Anirbit Sep 17 '11 at 23:17


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/* ---- Google Analytics Code Below */

## Saturday, October 26, 2019

### Cross Entropy

What is this?  New to me as a term.  I record here for my own refernce and pass it out to others.   In particular the suggestion here is that classification may be improved or even optimized this way.   See Jason's other publications, some mentioned here.

A Gentle Introduction to Cross-Entropy for Machine Learning   by Jason Brownlee

Cross-entropy is commonly used in machine learning as a loss function.

Cross-entropy is a measure from the field of information theory, building upon entropy and generally calculating the difference between two probability distributions. It is closely related to but is different from KL divergence that calculates the relative entropy between two probability distributions, whereas cross-entropy can be thought to calculate the total entropy between the distributions.

Cross-entropy is also related to and often confused with logistic loss, called log loss. Although the two measures are derived from a different source, when used as loss functions for classification models, both measures calculate the same quantity and can be used interchangeably.

In this tutorial, you will discover cross-entropy for machine learning.

After completing this tutorial, you will know:

How to calculate cross-entropy from scratch and using standard machine learning libraries.
Cross-entropy can be used as a loss function when optimizing classification models like logistic regression and artificial neural networks.
Cross-entropy is different from KL divergence but can be calculated using KL divergence, and is different from log loss but calculates the same quantity when used as a loss function.
Discover bayes opimization, naive bayes, maximum likelihood, distributions, cross entropy, and much more in my new book, with 28 step-by-step tutorials and full Python source code.

Let’s get started: ... "


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USF Home > College of Arts and Sciences > Department of Mathematics & Statistics

Mathematics & Statistics

# Colloquia — Spring 2012

## Friday, April 27, 2012

Title
Speaker

Time
Place

The Schwarzian norm and distortion of conformal mappings
Peter Duren
University of Michigan
3:00pm-4:00pm
PHY 130
Dmitry Khavinson

Abstract

The Schwarzian derivative of a locally univalent analytic function is $$Sf=\left(f''/f'\right)'-\frac12\left(f''/f'\right)^2.$$ The Schwarzian norm of a function defined in the unit disk $$\mathbb{D}$$ is given by $$\|Sf\|=\sup\limits_{z\in\mathbb{D}}\left(1-|z|^2\right)^2|Sf(z)|.$$ We begin by discussing some basic properties of the Schwarzian derivative and its classical applications to problems of differential equations and conformal mapping. A long-celebrated theorem of Nehari says that if $$\|Sf\|\le 2$$ then $$f$$ is (globally) univalent in $$\mathbb{D}$$. A weaker bound $$\|Sf\|\le 2\left(1+\delta^2\right)$$ for some $$\delta > 0$$ is known to imply the sharp lower bound $$d(\alpha,\beta)\ge\pi/\delta$$ on the hyperbolic distance between any pair of distinct points $$\alpha$$ and $$\beta$$ in $$\mathbb{D}$$ where $$f(\alpha)=f(\beta)$$. We will outline a proof that any upper bound on the Schwarzian norm is equivalent to both an upper bound and a lower bound on two-point distortion. It seems remarkable that such precise geometric information is encoded in the Schwarzian norm. Finally, we propose to describe a generalization of Nehari's theorem to harmonic mappings, or rather their lifts to a minimal surface.

## Friday, April 20, 2012

Title
Speaker

Time
Place

Exact solution of the six-vertex model: the Riemann-Hilbert approach
Pavel Bleher
Indiana University-Purdue University Indianapolis
3:00pm-4:00pm
PHY 130
Razvan Teodorescu

Abstract

We will review the Riemann-Hilbert approach to the exact solution of the sixvertex model with domain wall boundary conditions in different phase regions. This is a joint project with Karl Liechty.

## Thursday, April 19, 2012

Title
Speaker

Time
Place

How the Genome Folds
Erez Lieberman Aiden
Harvard Society of Fellows/
11:00am-12:00pm
PHY 120
Nagle Lecture Committee

Abstract

I describe Hi-C, a novel technology for probing the three-dimensional architecture of whole genomes. Developed together with collaborators at the Broad Institute and UMass Medical School, Hi-C couples proximity-dependent DNA ligation and massively parallel sequencing.

Our lab employs Hi-C to construct spatial proximity maps of the human genome. Using Hi-C, it is possible to confirm the presence of chromosome territories and the spatial proximity of small, gene-rich chromosomes. Hi-C maps also reveal an additional level of genome organization that is characterized by the spatial segregation of open and closed chromatin to form two genome-wide compartments. At the megabase scale, the conformation of chromatin is consistent with a fractal globule, a knot-free conformation that enables maximally dense packing while preserving the ability to easily fold and unfold any genomic locus. The fractal globule is distinct from the more commonly used globular equilibrium model. Our results demonstrate the power of Hi-C to map the dynamic conformations of whole genomes.

## Friday, April 13, 2012

Title
Speaker

Time
Place

Construction of Riemann surfaces by using parallel translations
Kiyoshi Ohba
Ochanomizu University
Japan
3:00pm-4:00pm
PHY 130
Masahiko Saito

Abstract

A Riemann surface is a connected orientable surface with a complex structure. The problem of how to parametrize the complex structures on a fixed topological surface originated with B. Riemann. In this talk, I will introduce some methods of constructing Riemann surfaces by cutting the complex plane along line segments and pasting by parallel translations, and observe visually the deformation of complex structures of Riemann surfaces.

## Friday, March 30, 2012

Title
Speaker

Time
Place

Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall’s work 25 years later
Andrei Martínez-Finkelshtein
Almería, SPAIN
3:00pm-4:00pm
PHY 120
E. Rakhmanov

Abstract

In 1986 J. Nuttall published in Constructive Approximation a paper where he studied the behavior of the denominators (“generalized Jacobi polynomials”) and the remainders of the Padé approximants to a special class of algebraic functions with 3 branch points. I will try to look at this problem 25 years later from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the other, they satisfy complex orthogonality relations, and thus are suitable for the Riemann-Hilbert asymptotic analysis. Along with the names mentioned in the title, this talk features also a special appearance by Riemann surfaces, quadratic differentials, compact sets of minimal capacity, special functions and other characters.

## Friday, March 23, 2012

Title
Speaker

Time
Place

$$n$$-ary algebras: from Physics to Mathematics
Abdenacer Makhlouf
University of Haute Alsace
France
3:00pm-4:00pm
PHY 130

Abstract

Lie algebras and Poisson algebras have played an extremely important role in mathematics and physics for a long time. Their generalizations, known as $$n$$-Lie algebras and Nambu algebras also arise naturally in physics in many different contexts. In this talk, I will review some basics on $$n$$-ary algebras, present some key constructions and discuss the representation theory and cohomology theory.

## Friday, March 9, 2012

Title
Speaker

Time
Place

Multi-Dimensional Obliquely Reflected BSDEs Arising from Real Options
Shanjian Tang
Fudan University
China
3:00pm-4:00pm
PHY 130
Yuncheng You

Abstract

The definition of real options is recalled. The price of real options is associated to an optimal switching problem for backward stochastic differential equations (BSDEs). Interestingly, the associated dynamical programming equation for such kind of optimal switching problem is a multi-dimensional BSDE with oblique reflection.

## Friday, February 24, 2012

Title
Speaker

Time
Place

Fourier bases on fractal Hilbert Spaces
Keri Kornelson
University of Oklahoma
Norman, OK
3:00pm-4:00pm
PHY 130
Catherine Bénéteau

Abstract

The study of Bernoulli convolution measures, which are supported on Cantor subsets of the real line, dates back to the 1930's, and experienced a resurgence with the connection between the measures and iterated function systems. We will use this IFS approach to consider the question of Fourier bases on the $$L^2$$ spaces with respect to Bernoulli convolution measures.

There are some interesting phenomena that arise in this setting, e.g., one can sometimes scale or shift the Fourier frequencies of an orthonormal basis by an integer and obtain another ONB.

We also describe properties of the unitary operator mapping between two such Fourier bases. This operator exhibits a fractal-like self-similarity, leading us to call it an “operator-fractal”.

## Monday, February 13, 2012

Title
Speaker

Time
Place

An introduction to handlebody-knot theory
Atsushi Ishii
Tsukuba University
Japan
3:05pm-3:55pm
PHY 108
Masahiko Saito

Abstract

A handlebody-knot is a handlebody embedded in the $$3$$-sphere. A handlebody-link is a disjoint union of handlebodies embedded in the $$3$$-sphere. A handlebody-knot is a $$1$$-component handlebody-link. Two handlebody-links are equivalent if one can be transformed into the other by an isotopy of the $$3$$-sphere. I will explain how two handlebody-links can be distinguished. We may decompose handlebody-links to distinguish them, or we may use invariants. This talk is an introduction to handlebody-knot theory.

## Friday, February 10, 2012

Title
Speaker

Time
Place

Some Combinatoric and Comedic Consequences of the Proof of the Pizza Conjecture
Rick Mabry*
Louisiana State University in Shreveport
Shreveport, LA
4:00pm-5:00pm
PHY 130
Athanassios Kartsatos

Abstract

The recent proof of the “Pizza Conjecture” by Deiermann and Mabry was accomplished by wishful thinking. Unable to solve the original problem, the would-be solvers generalized the problem in a fairly extreme way and hoped for the best. The resulting problem was ultimately solved by reducing a nice, continuous (calculus) problem to a gruesome, discrete (combinatorial) one. The solution made some news, and, in a comedy of errors, cheesy comments flooded the internet in the aftermath. This talk will highlight some mathematical, gastronomical, and comical slices of what ensued, and look at a more recent pizza theorem and its combinatorial solution.

Title
Speaker

Time
Place

Sparse Ramsey Hosts
Kevin Milans
University of South Carolina
Columbia, SC
3:00pm-4:00pm
PHY 130
Brendan Nagle

Abstract

In Ramsey Theory, we study conditions under which every partition of a large structure yields a part with additional structure. For example, Van der Waerden's theorem states that every $$s$$-coloring of the integers contains arbitrarily long monochromatic arithmetic progressions, and the Hales-Jewett Theorem guarantees that every game of tic-tac-toe in high dimensions has a winner. Ramsey's Theorem implies that for any target graph $$G$$, every $$s$$-coloring of the edges of some sufficiently large host graph contains a monochromatic copy of $$G$$. In Ramsey's Theorem, the host graph is dense (in fact complete). We explore conditions under which the host graph can be sparse and still force a monochromatic copy of $$G$$.

We write $$H \stackrel{s}{\to} G$$ if every $$s$$-edge-coloring of $$H$$ contains a monochromatic copy of $$G$$. The $$s$$-color Ramsey number of $$G$$ is the minimum $$k$$ such that some $$k$$-vertex graph $$H$$ satisfies $$H \stackrel{s}{\to} G$$. The degree Ramsey number of $$G$$ is the minimum $$k$$ such that some graph $$H$$ with maximum degree $$k$$ satisfies $$H \stackrel{s}{\to} G$$. Chvátal, Rödl, Szemerédi, and Trotter proved that the Ramsey number of bounded-degree graphs grows only linearly, sharply contrasting the exponential growth that generally occurs when the bounded-degree assumption is dropped. We are interested in the analogous degree Ramsey question: is the $$s$$-color degree Ramsey number of $$G$$ bounded by some function of $$s$$ and the maximum degree of $$G$$? We resolve this question in the affirmative when $$G$$ is restricted to a family of graphs that have a global tree structure; this family includes all outerplanar graphs. We also investigate the behavior of the $$s$$-color degree Ramsey number as $$s$$ grows. This talk includes results from three separate projects that are joint with P. Horn, T. Jiang, B. Kinnersley, V. Rödl, and D. West.

## Wednesday, February 8, 2012

Title
Speaker

Time
Place
Speaker

Automata generating free products of groups of order $$2$$
Dmytro Savchuk
SUNY Binghamton
Binghamton, NY
3:00pm-4:00pm
PHY 118
Nataša Jonoska

Abstract

We construct a family of automata with $$n$$ states, $$n>3$$, acting on a rooted binary tree that generate the free products of cyclic groups of order $$2$$. Groups generated by automata is a fascinating class of groups that includes counterexamples to several famous conjectures in group theory. I will start from discussing the definition and main properties of these groups. Then I will give a short exposition of the history of the question, explain the construction and main ideas behind the proof, which involve the notion of a dual automaton.

This is a joint result with Yaroslav Vorobets of Texas A&M University.

## Monday, February 6, 2012

Title
Speaker

Time
Place

Cyclic Sieving and Cluster Multicomplexes
University of Southern California
Los Angeles, CA
3:05pm-3:55pm
PHY 108
Brian Curtin

Abstract

Let $$X$$ be a finite set, $$C=\langle c\rangle$$ be a finite cyclic group acting on $$X$$, and $$X(q)\in N[q]$$ be a polynomial with nonnegative integer coefficients. Following Reiner, Stanton, and White, we say that the triple $$(X,C,X(q))$$ exhibits the *cyclic sieving phenomenon* if for any integer $$d>0$$, the number of fixed points of $$c^d$$ is equal to $$X(\zeta^d)$$, where $$\zeta$$ is a primitive $$|C|^{\mathrm{th}}$$ root of unity. We explain how one can use representation theory to prove instances of the cyclic sieving phenomenon involving the action of tropical Coxeter elements on (complexes closely related to) cluster complexes. The representation theory involves cluster monomial bases of geometric realizations of finite type cluster algebras.

## Friday, February 3, 2012

Title
Speaker

Time
Place

Searching for Structure in Graph Theory: Chromatic Index and Immersion
Jessica McDonald
Simon Fraser University
Burnaby, British Columbia
3:00pm-4:00pm
PHY 130
Brendan Nagle

Abstract

Searching for structure is a fundamental theme in graph theory. The celebrated Goldberg-Seymour Conjecture is an example of this — it asserts that all multigraphs with “high” chromatic index contain a “dense” subgraph. In this talk we discuss edge-colouring in this context, and use the method of Tashkinov trees to gain new insights. Namely, we extend a classical characterization result of Vizing, and prove an approximation bound towards the Goldberg-Seymour Conjecture. We also consider the important containment relation of immersion. In particular, motivated by the Graph Minors Project of Robertson and Seymour and by Hadwiger's Conjecture, we explore conditions under which graphs and digraphs contain clique immersions. The results we obtain are in analogue to the clique subdivision theorem of Bollobas-Thomason and Komlos-Szemeredi.

## Wednesday, February 1, 2012

Title
Speaker

Time
Place

Duality and equivalence of graphs in surfaces
Iain Moffatt
University of South Alabama
Mobile, AL
3:00pm-4:00pm
PHY 108
Masahiko Saito

Abstract

This talk revolves around two fundamental constructions in graph theory: duals and medial graphs. There are a host of well-known relations between duals and medial graphs of graphs drawn in the plane. (This can also be thought of in terms of knot diagrams and their graphs.) By considering these relations we will be led to the working principle that duality and equality of plane graphs are equivalent concepts. It is then natural to ask what happens when we change our notion of equality. In this talk we will see how isomorphism of abstract graphs corresponds to an extension of duality called twisted duality, and how twisted duality extends the fundamental relations between duals and medial graphs from graphs in the plane to graphs in other surfaces. We will then go on to see how this group action leads to a deeper understanding of the properties of, and relationships among, various graph polynomials, including the chromatic polynomial, the Penrose polynomial, and topological Tutte polynomials.

## Friday, January 20, 2012

Title
Speaker

Time
Place

Freak Waves in the Ocean
Victor L'vov
Weizmann Institute of Science
Rehovot, Israel
3:00pm-4:00pm
PHY 130


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# Solution: Predict the sigma bond hybridizations indicated by the arrows in the following molecule.

###### Problem

Predict the sigma bond hybridizations indicated by the arrows in the following molecule.


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# Spahn higher inductive type (Rev #2, changes)

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Categorially a higher inductive type in an extensional type theory is an initial algebra of an endofunctor. In intensional type theory this analogy fails.

In particular $W$-type s in an extensional type theory correspond to initial algebras ofpolynomial functor s. Also this is not true for intensional type theories.

This failure can be (at least for $W$-types and some more general cases) remedied by replacing “initial” by “homotopy initial”. This is the main result (to be found in §2) of

• Steve Awodey, Nicola gambino, Kristina Sojakova, inductive types in homotopy type theory, arXiv:1201.3898

Revision on February 25, 2013 at 20:44:37 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.