The previous post in this blog [LINK] went over my rough intuition of climate, primarily in middle latitudes like those of the US. Most of the broad categories that I described were largely aligned with the Köppen climate classification system (henceforth called the Köppen categorization). However, there is a more recent categorization known as the Trewartha climate classification system (henceforth called the Trewartha categorization) that is supposed to be more representative of middle latitudes like those of the US. Essentially, tropical, desert, and semi-arid climates, as well as polar and ice cap climates, are defined in the same ways between the two categorizations. The differences lie in the definitions of subtropical, continental, and subpolar oceanic/subarctic climates. One benefit of the Trewartha categorization is that it clearly separates
boreal/subpolar climates from other oceanic and continental climates,
whereas the Köppen categorization uses subcategories that could be a little more confusing. However, the definitions of subtropical, oceanic, and continental climates in the Trewartha categorization seem less justifiable to me. Follow the jump to see more details. **Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.**

# Das U-Blog by Prashanth

My Thoughts on Science, Technology, Freedom, and Stuff

## 2023-05-20

### FOLLOW-UP: My Rough Intuition of Climate, Especially in the US

## 2023-04-01

### My Rough Intuition of Climate, Especially in the US

For a long time, I had wondered why the climates of San Francisco, Sacramento, and Los Angeles are so different from those respectively of Richmond, DC, and Atlanta. I had read a few articles on Wikipedia on occasion, so I got a sense that it has to do in part with different ocean currents; this made sense to me, as I had become very comfortable (growing up in the DC area) with the warm waters at beaches along the East Coast in the summer, and I was always surprised by the comparatively much colder waters at beaches along the West Coast whenever I'd visit California even in the summer. I knew though that this wasn't the whole story, and I was surprised to see, for example, that even in South America, South Africa, Western Europe versus East Asia, and Australia, there were very similar contrasts in climates between cities along west versus east coasts in the middle latitudes. This made me more curious about the reasons for these similarities, so I recently went down a rabbit hole of Wikipedia articles to learn more and form an intuition about why different places have different kinds of climates. Follow the jump to see my explanation. **I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.** I'm just putting this out there in case this intuition is helpful to anyone else **as a starting point** to learn more (and I recognize that an incorrect initial intuition could hurt rather than help when trying to learn more).

## 2023-03-01

### Book Review: "Sapiens" by Yuval Noah Harari

I've recently read the book *Sapiens* by Yuval Noah Harari; this was highly recommended to me 7 years ago by a friend in graduate school with whom I had enthusiastic discussions about the material in the book, but I hadn't gotten a chance to read the book until now. This book is supposed to be a history of humans, going from an evolutionary perspective for the first 2 million years since the genus Homo became distinct and then getting into the developments of language, collective myths, agriculture, urbanization, and industry. Even my summary of the flow of the book contains my opinion about its progression (though I should note that the "parts" that I speak of overlap with but aren't identical to the 4 parts that formally divide the book): the beginning part of the book seems to be a serious discussion of evolution, language, and the advent of agriculture, the middle part tries to be serious but has more inconsistencies that I find problematic, and the last part seems more clearly to be more like a "pop-history" type of book with less rigorous speculation (so I read it with a lighter heart even if the author didn't intend it that way and I therefore heavily discounted it in my overall opinion of the book); furthermore, the parts about scientific development since 1500 can be understood more clearly in other books.

There were several things that I learned from the book and several ways in which the book forced me to consider a different perspective. These are as follows, in no particular order. First, I learned that the further development of language to be able to convey detailed information, gossip, abstract ideas, and fictions happened 70,000 years ago and coincided with humans becoming dominant in the food chain and in their spread across the world. Second, I had always uncritically believed in the advent of agriculture and later urbanization as a good thing, to the extent that I've recently sometimes wondered (without being particularly informed about history & sociology) whether clashes between the Mongol invaders & Hindu natives (in contrast to earlier arrivals in India of urban Muslim traders from Mesopotamia) as well as between the expectations of Western government & the reality of the House of Saud represent clashes between urbanized cultures that have developed to a great degree versus nomadic cultures that have endured much harsher conditions and only wish to plunder the cities for wealth without care for more refined aspects of urban cultures; this book forced me to consider that individuals within nomadic tribes had much more varied diets & activities within each day, that the first few millennia of the transition to agriculture may have led to a lot of suffering compared to what came immediately before, and that the domestication of wheat can be reinterpreted as a mutalistic domestication of wheat & humans. Third, I learned that empires might be defined only by the number of culturally distinct tribes under their yokes and the flexibility of their borders in expansion, not by population or area per se. Fourth, while I had some familiarity with how capitalism & European imperialism fueled each other and with the use of scientifically-inspired racism as a justification for European colonization, I didn't have a good sense for how these all tied together until I read this book, especially in the context of scientific voyages only being funded if the scientists could tag along with naval officers ordered to colonize the lands they would reach. Fifth, I appreciated the distinction between ancient & medieval empires which grew in predictable ways by absorbing naval territories versus early modern European empires which grew unpredictably across the world with long-distance seafaring. Sixth, I appreciated the explicit call to attention about how liberal humanistic political philosophies, which profess to be atheistic in themselves & multireligious in the sense of tolerance, cast freedom & political empowerment in terms of a special nature of individuals that is drawn directly from Christian notions of creation & individual souls (though the concepts of creation & individual souls aren't unique to Christianity among traditional religions); this is something that I've pondered before but have typically glossed over, so I appreciated being challenged in this way.

There were a few points that I was happy to see in the sense of agreeing with those worldviews. These include the ideas of collective myths (not only in traditional religions but in the systems of trust that underlie monetary systems & democracies), historical predictions leading to self-fulfilling or self-negating prophecies, the existence of hierarchies of some form in almost all societies larger than about 150 people (as I've wondered, for example, if a solution to the problem of inflation coming from an immediate cash payout to everyone in a universal basic income plan would be to sprinkle it randomly upon different people at different times to ensure that the economic system doesn't stray too far from its previous state and can better respond to bigger numbers of people getting such payouts later even if that creates an effective hierarchy between those who get such payouts at a given time and those who don't), and the ways that even cultures free of external pressures can develop internal contradictions that in turn can lead to continued development of the culture as a unified entity or a split of the culture into multiple descendants. On a lighter note, I also enjoyed seeing the author, in the otherwise problematic speculation about science, point out that science fiction can only rarely, if ever, attempt to describe what would truly be alien experiences to humans, and that most science fiction stories ultimately revolve around myths & social conflicts that in one form or another have been recognizable for millennia, which leads me to the conclusion that there is no reason beyond snobbery to claim that *Star Trek* is science fiction while *Star Wars* supposedly is not (because if science fiction is defined as only portraying truly alien experiences in encounters with new technology or new intelligent species that aren't just thinly-veiled allegories for known interactions among human groups, there may only be a few books, movies, and TV series that may be called "science fiction", perhaps including *Black Mirror* or *2001: A Space Odyssey*, and those works are rare probably exactly because readers or viewers would find them less relatable).

There were a few specific stories that I liked reading. One was of the Chinese seafarer Zheng He, as it shows that Chinese seafaring technology was as advanced as European seafaring technology around 1500 but China simply didn't have the same ambition to conquer faraway lands through seafaring. The other was of how the accompaniment of Hernán Cortés by Aztec people carrying burning incense sticks near him convinced him that the "primitive" Aztecs were treating him as a deity but was actually because he had terrible body odor due to bad hygiene, as it is a funny story, it shows that the Aztecs, immediately upon encountering Europeans, figured out what other contemporary peoples of Asia & Africa had known for many centuries (leading those peoples to set up quarantine areas for European visitors at ports), namely that Europeans had bad hygiene at that time, and it shows how the self-delusion of European winners of such conflicts (in this case Cortés believing that he was being treated as a deity so the Aztecs must have been "primitive") could persist in "official" historical narratives for many centuries.

Beyond the problematic historiography (especially ignoring the way that so many consequential scientific discoveries were made in Europe individually by people who were independently wealthy while also not clearly explaining which technological discoveries were systematically funded & used by governments, though those parts could be fixed with better writing) and excessively serious-sounding speculation about science in the last few chapters & sprinkled elsewhere in the book, there were three major points of disagreement that I had with the author, in the sense that I believe that these points strike at the fundamental arguments of the book. These are as follows.

First, the author makes a big deal throughout the book about how the global unity in understanding of political, economic, and other norms that has emerged in the last 500 years is unprecedented in all prior years of human existence. My counterargument is that this argument depends too much on the specific way that previous interactions between cultures went or on the fact that certain cultures happened to not interact. It will be based primarily on [Native] American and European cultures before and around the time of their first contact in the middle of the second millennium, as the author makes a big deal about how American tribes were among the groups that were totally isolated from the continuum of groups across Africa, Asia, and Europe (with Australian tribes being among the others). As the author argues, the lack of contact before may well have been because of a combination of technological limitations along with limitations in cultural ambition. However, in a counterfactual situation where, for example, English people looking to start local democratic governments met on a truly equal footing with Iroquois people who embodied the spirit of democracy in their local confederated governments, there is no specific reason to believe that they would have been talking past each other; the author's conception of "global unity" as a phenomenon that developed in the last 500 years with no precedent seems to depend too strongly on peoples having met or being aware of each other's existence and not enough on actual similarities between each other's cultures. Additionally, the example of Hernán Cortés and the incense sticks, along with the example (not in the book) of the origin of the ethnic slur "Indian giver" from a deliberate misunderstanding of Native Americans' attempts to barter with Europeans as gifts that were then demanded to be returned, shows that the author's view of the establishment of "global unity" depended strongly on the actual course of history (in this case Europeans deliberately ignoring what Native Americans were telling them) and not on the broader cultural similarities already present. This dependence on the actual course of history makes this a hindsight-based account that the author supposedly disclaims, making the author rather hypocritical.

Second, in later parts of the book, when the author discusses the reasons for European armies so easily conquering peoples in faraway lands, the author puts a lot of stock in the idea that success was due to the European drive for exploration of the unknown, even before that commitment to exploration started bearing systematic fruit in the forms of scientific discovery or technological advancement; conversely, the author briefly mentions and otherwise glosses over the role of more effective forms of social organization & discipline in those armies. This seems to completely undermine previous chapters in the book that so clearly emphasized the ways that communication of collective myths could lead to new forms of social organization. Perhaps this seeming contradiction can be resolved by interpreting the "drive to explore the unknown" as extending to European armies systematically developing new ways, including new forms of organization of their own armies, of dealing with unknown peoples whom they wish to conquer. However, this seems like a stupid semantic difference and again seems like the author is engaging excessively in analysis from narrow hindsight, contrary to the author's own stated claims. (**UPDATE**: A related point is about how the author implies that the drive for European colonists to learn about unknown cultures & explore how to systematically conquer unknown peoples led them to use what they learned about these cultures to systematically deepen existing divisions or create new divisions. I could agree that Europeans were the first to do this so systematically or so tightly coupled to the seemingly more noble goal of learning things that weren't known to them. However, I cannot agree with the idea that Europeans were the first to exploit & inflame divisions or engage in proxy wars. as ancient Egyptian kingdoms were known to have done this to the Assyrian Empire. Perhaps this can be forgiven if it turns out that this book was published before we knew about how the ancient Egyptians fomented rebellions, civil wars, and proxy wars in the Assyrian Empire, but in any case, the author's seeming unwillingness to directly assert or refute the idea that European colonists' "drive to explore" specifically included exploration of how to organize themselves better & exploit other people's weaknesses more effectively to conquer those other peoples is a much bigger problem with this book.) Moreover, the author does not attempt to explain why peoples were so consistently conquered at all by Europeans from the perspective of those conquered peoples other than simply stating the claim that those peoples couldn't imagine that their knowledge could be incomplete. I think I could do a better job than the author by imagining a counterfactual situation, using the example of Hernán Cortés encountering the Aztecs: even if the Aztecs were similarly driven as the Spanish by exploration of the unknown and had expanded their empire that way before the Spanish landed in America as historically happened, the only way from the perspective of social dynamics that I can see the Aztecs successfully repelling the invasion is by using their knowledge of dealing with unknown peoples to see through Cortés's lies into his true intentions and organize accordingly, yet there is no guarantee that knowing what to do when attempting to conquer unknown peoples would lead an empire to develop knowledge of what to do at the receiving end of a conquest attempt. Finally, in the specific cases of Europeans interacting with Native Americans, the author in a few places briefly acknowledges the role of infectious disease (used by Europeans sometimes accidentally and other times, as in the case of pox blankets in the 1763 Native American siege of the British-occupied Fort Pitt, intentionally) but otherwise glosses over this in favor of explanations based on exploration of the unknown. Yet, as the examples of Cortés as well as the slur "Indian giver" point out, it is quite plausible that Europeans, seeing how easily Native Americans were wiped out by disease, used this as propaganda to better organize themselves and portray Native Americans as weak (independent of specific technology or ideals about exploring the unknown), and I think it is irresponsible for the author to ignore this obvious possibility.

Third, there is a whole chapter about the history of traditional religions, including animistic, polytheistic, monotheistic, and atheistic religions. My problem is that the author makes too broad claims about religious trends even though there are so few surviving or [relatively] recently extinguished major traditional religions; the sample sizes are so small as to make the claims unconvincing. The author acknowledges similar problems in other contexts elsewhere but not in that chapter.

Beyond these issues, I noted several more minor issues at various points in the book. Although some of these issues personally offended me, I still categorize them as minor because I think that deleting the offending passages from the book would not significantly reduce support for or otherwise qualitatively change the main arguments of the book. These are as follows, in no particular order. Even if some of these points raise questions that have no clear answer, I think the author was irresponsible in not addressing the existence of these questions and clearly stating the lack of a clear answer.

First, the author claims that when big social orders are sustained through collective myths, those collective myths require genuine belief from members of the elite too. Recent news about how Fox News executives & star hosts privately disbelieved claims that the 2020 US presidential election was rigged but knowingly pushed such claims in public just to boost TV ratings & stock prices. On the one hand, perhaps it isn't fair to pin this on the author as this news is much more recent than the publication of the book. On the other hand, I would be curious to see how the author would react to this news now; if the author reacts by claiming to be correct because the degree of true belief among the elite was "always destined to wane at some point" or for some similar reason purely in hindsight, then that tells me that the author's approach is worthless because it would be unfalsifiable.

Second, the author does such a consistently bad job with the history of India that I have to wonder if the author's research about that specific topic consisted exclusively of books written by British colonizers to portray India to their own benefit. Problems include claims that Indo-Aryans "invaded" (as that word is usually used to imply a systematic movement of an army to bring forth a violent clash, yet there is no historical evidence for such singular violent clashes between ancient Central Asian migrants and South Asian natives), the treatment of caste in the Vedas (as even people who aren't apologists for Brahmins or deniers of the history of caste will recognize a lot of subtlety in the way the Vedas used terms associated now with caste, especially as those castes didn't exist in Indian society until after Vedic times), the treatment of caste in general (conflating jati & varna to claim that the "original" 4 varnas over time split into thousands of jatis, when the reality is much more complicated & less clear), the claim that Brahmins could have "learned" from the KKK how to enforce caste divisions (as Brahmins, especially in South India, were already brutally effective in enforcing caste divisions long before the KKK existed), and the claim that India had no national consciousness before the British Empire (which undermines the author's own prior acknowledgment of the Gupta & Maurya Empires as empires by the author's own definition). Another problematic statement by the author that I am willing to forgive given when the book was published (before the political rise of right-wing Hindu nationalism in India in the last 10 years) is the rhetorical question about whether right-wing Hindu nationalists would do away with all symbols of the Mughal Empire, include those of beauty like the Taj Mahal; the author clearly implied that they wouldn't dare to do so, but that view now seems laughably quaint. Finally, a statement by the author about religion in a broader context yields problems in the context of India. In particular, the author claims that polytheistic kings didn't try to convert conquered peoples or make them destroy temples to their own deities, but historical conflicts between Saiva & Vaisnava kingdoms in South India over religion suggest otherwise; perhaps this point can also be forgiven as a rare exception to the rule.

Third, the author's definition of an empire in terms of flexibly expanding borders still leads contemporary readers to imagine borders that strictly control the flow of people across them, which I think is historically misleading given the ease with which people could pass across. Even now, people from Mexico & countries in Central America freely pass through the international border between California & Baja California as seasonal migrant workers although the US border is otherwise very strictly controlled.

Fourth, the author makes claims about the positive cultural developments by empires, but those claims seem incomplete. Additionally, the author tries to distinguish Cyrus of Persia's claim that the empire would benefit all people from the more limited ambitions of Assyrian emperors, but this distinction is not clear at all.

Fifth, the author claims that interest (in the sense of a guaranteed geometric return on an investment) requires the existence of a currency that is not useful for any other reason. I disagree in principle because livestock and crops, which have historically been used as currencies or items of barter, have the potential to multiply over time. That said, this may be a moot point if there is no evidence for societies having charged interest directly on livestock or crops as forms of currency.

Sixth, the explanations of why some peoples did not develop agriculture until much later contact by faraway urbanized peoples seems incomplete. I agree with the idea that some peoples settled in areas where plants & animals simply could not be domesticated; the capability to be domesticated by humans is rare among species. However, this does not explain why many native peoples of America and Australia never developed agriculture even though they lived on grasslands that later turned out to support agriculture very easily; in the case of Australia, the author's omission is especially troubling given that the author explains how humans who moved to Australia 45,000 years ago had no problem with destroying the forests that were already there & replacing them with grasslands.

Seventh, near the end of the book, the author distinguishes ecological destruction from resource scarcity as the more likely cause of future human suffering or extinction. This seems to undermine an earlier chapter in which the author acknowledges that problems with allocating resources to maintain a certain standard of living under certain norms, even if the resources themselves are technically abundant, are more likely to lead to conflict. I wish the author had dealt with this more carefully.

Eighth, the author writes about the seemingly inexorable trend toward globalization and the way that world war has come to seem implausible since World War II. The failure to predict both the retreat from globalization especially since 2014 as well as the Russian military invasion of Ukraine in 2022 are forgivable given when the book was published. However, I'm more troubled that the author's claims about the implausibility of world war are phrased in ways that are unfalsifiable, as the author can always claim either a different definition of "world war" or a finite time period of validity (since the last world war) that the author would not have previously clearly stated.

Overall, I think this is still an interesting book, though I wasn't incredibly impressed by it (unlike, for example, my friend from graduate school). I'd recommend it with the caveats discussed above. In any case, as this was among the first books to go on the reading list (for books unrelated to my work) that I made for myself in graduate school, I'm glad to have finally read it.

## 2023-02-01

### My Time at the 2023 TRB Annual Meeting

Last month, I attended the 2023 TRB Annual Meeting. I meant to write this post immediately afterwards, but I became busy with work and with preparing for international travel which I started soon after that and finished just yesterday. Thus, for the first time in the history of this blog, there was a month with no new post. I'll try my best going forward to continue posting at least once each month, but sometimes, these issues can come up.

The conference, which was held in DC, was a lot of fun. The graduate student researcher working with me was able to present our work as a poster. Additionally, it was my first experience attending an external conference (i.e. one not hosted by UC Davis) since changing fields to get into transportation policy research, so it was an excellent opportunity for professional networking. Because I had previously attended the APS March Meeting for 3 years (2017-2019), I was prepared for a conference of a similar size and scope; in particular, I intentionally avoided overloading my schedule with presentation sessions, made time to take breaks, and made time to meet people informally. Additionally, as transportation is a much more policy-oriented field than physics and is not confined to academia, I made sure to attend TRB committee meetings, different organizations' receptions in the evenings, and other events technically outside of the conference itself. I certainly professionally got what I wanted out of it, and I look forward to attending again in the future.

## 2022-12-02

### Fundamental Theorem of Calculus for Functionals

I happened to think more about the idea of recovering a functional by somehow integrating its functional derivative. In the process, I realized that certain ideas that I would have to consider make this post a natural follow-up to a recent post [LINK] about mapping scalars to functions. This will become clear later in this post.

For a single variable, a function \( f(x) \) has an antiderivative \( F(x) \) such that \( f(x) = \frac{\mathrm{d}F}{\mathrm{d}x} \). One statement of the fundamental theorem of calculus is that this implies that \[ \int_{a}^{b} f(x)~\mathrm{d}x = F(b) - F(a) \] for these functions. In turn, this means \( F(x) \) can be extracted directly from \( f(x) \) through \[ F(x) = \int_{x_{0}}^{x} f(x')~\mathrm{d}x' \] in which \( x_{0} \) is chosen such that \( F(x_{0}) = 0 \).

For multiple variables, a conservative vector field \( \mathbf{f}(\mathbf{x}) \) in which \( \mathbf{f} \) must have the same number of components as \( \mathbf{x} \) can be said to have a scalar antiderivative \( F(\mathbf{x}) \) in the sense that \( \mathbf{f} \) is the gradient of \( F \), meaning \( \mathbf{f}(\mathbf{x}) = \nabla F(\mathbf{x}) \); more precisely, \( f_{i}(x_{1}, x_{2}, \ldots, x_{N}) = \frac{\partial F}{\partial x_{i}} \) for all \( i \in \{1, 2, \ldots, N \} \). (Note that if \( \mathbf{f} \) is not conservative, then it by definition cannot be written as the gradient of a scalar function! This is an important point to which I will return later in this post.) In such a case, a line integral (which, as I will emphasize again later in this post, is distinct from a functional path integral) from vector point \( \mathbf{a} \) to vector point \( \mathbf{b} \) of \( \mathbf{f} \) can be computed as \( \int \mathbf{f}(\mathbf{x}) \cdot \mathrm{d}\mathbf{x} = F(\mathbf{b}) - F(\mathbf{a}) \); more precisely, this equality holds along any contour, so if a contour is defined as \( \mathbf{x}(s) \) for \( s \in [0, 1] \), no matter what \( \mathbf{x}(s) \) actually is, as long as \( \mathbf{x}(0) = \mathbf{a} \) and \( \mathbf{x}(1) = \mathbf{b} \) hold, then \[ \sum_{i = 1}^{N} \int_{0}^{1} f_{i}(x_{1}(s), x_{2}(s), \ldots, x_{N}(s)) \frac{\mathrm{d}x_{i}}{\mathrm{d}s} \mathrm{d}s = F(\mathbf{b}) - F(\mathbf{a}) \] must also hold. This therefore suggests that \( F(\mathbf{x}) \) can be extracted from \( \mathbf{f}(\mathbf{x}) \) by relabeling \( \mathbf{x}(s) \to \mathbf{x}'(s) \), \( \mathbf{a} \) to a point such that \( F(\mathbf{a}) = 0 \), and \( \mathbf{b} \to \mathbf{x} \). Once again, if \( \mathbf{f}(\mathbf{x}) \) is not conservative, then it cannot be written as the gradient of a scalar field \( F \), and the integral \( \sum_{i = 1}^{N} \int_{0}^{1} f_{i}(x_{1}(s), x_{2}(s), \ldots, x_{N}(s)) \frac{\mathrm{d}x_{i}}{\mathrm{d}s} \mathrm{d}s \) will depend on the specific choice of \( \mathbf{x}(s) \), not just the endpoints \( \mathbf{a} \) and \( \mathbf{b} \).

For continuous functions, the generalization of a vector \( \mathbf{x} \), or more precisely \( x_{i} \) for \( i \in \{1, 2, \ldots, N\} \), is a function \( x(t) \) where \( t \) is a continuous dummy index or parameter analogous to the discrete index \( i \). This means the generalization of a scalar field \( F(\mathbf{x}) \) is the scalar functional \( F[x] \). What is the generalization of a vector field \( \mathbf{f}(\mathbf{x}) \)? To be precise, a vector field is a collection of functions \( f_{i}(x_{1}, x_{2}, \ldots, x_{N}) \) for all \( i \in \{1, 2, \ldots, N \} \). This suggests that its generalization should be a function of \( t \) and must somehow depend on \( x(t) \) as well. It is tempting therefore to write this as \( f(t, x(t)) \) for all \( t \). However, although this is a valid subset of the generalization, it is not the whole generalization, because vector fields of the form \( f_{i}(x_{i}) \) are collections of single-variable functions that do not fully capture all vector fields of the form \( f_{i}(x_{1}, x_{2}, \ldots, x_{N}) \) for all \( i \in \{1, 2, \ldots, N \} \). As a specific example, for \( N = 2 \), the vector field with components \( f_{1}(x_{1}, x_{2}) = (x_{1} - x_{2})^{2} \) and \( f_{2}(x_{1}, x_{2}) = (x_{1} + x_{2})^{3} \) cannot be written as just \( f_{1}(x_{1}) \) and \( f_{2}(x_{2}) \), as \( f_{1} \) depends on \( x_{2} \) and \( f_{2} \) depends on \( x_{1} \) as well. Similarly, in the generalization, one could imagine a function of the form \( f = \frac{x(t)}{x(t - t_{0})} \mathrm{exp}(-(t - t_{0})^{2}) \); in this case, it is not correct to write it as \( f(t, x(t)) \) because the dependence of \( f \) on \( x \) at a given dummy index value \( t \) comes through not only \( x(t) \) but also \( x(t - t_{0}) \) for some fixed parameter \( t_{0} \). Additionally, the function may depend not only on \( x \) per se but also on derivatives \( \frac{\mathrm{d}^{n} x}{\mathrm{d}t^{n}} \); the case of the first derivative \( \frac{\mathrm{d}x}{\mathrm{d}t} = \lim_{t_{0} \to 0} \frac{x(t) - x(t - t_{0})}{t_{0}} \) illustrates the connection to the aforementioned example. Therefore, the most generic way to write such a function is effectively as a functional \( f[x; t] \) with a dummy index \( t \). The example \( f = \frac{x(t)}{x(t - t_{0})} \mathrm{exp}(-(t - t_{0})^{2}) \) can be formalized as \( f[t, x] = \int_{-\infty}^{\infty} \frac{x(t')}{x(t' - t_{0})} \mathrm{exp}(-(t' - t_{0})^{2}) \delta(t - t')~\mathrm{d}t' \) where the dummy index \( t' \) is the integration variable while the dummy index \( t \) is free. (For \( N = 3 \), the condition of a vector field being conservative is often written as \( \nabla \times \mathbf{f}(\mathbf{x}) = 0 \). I have not used that condition in this post because the curl operator does not easily generalize to \( N \neq 3 \).)

If a functional \( f[x; t] \) is conservative, then there exists a functional \( F[x] \) (with no free dummy index) such that \( f \) is the functional derivative \( f[x; t] = \frac{\delta F}{\delta x(t)} \). Comparing the notation between scalar fields and functionals, \( \sum_{i} A_{i} \to \int A(t)~\mathrm{d}t \) and \( \mathrm{d}x_{i} \to \delta x(t) \), in which \( \delta x(t) \) is a small variation in a function \( x \) specifically at the index value \( t \) and nowhere else. This suggests a generalization of the fundamental theorem of calculus to functionals as follows. If \( a(t) \) and \( b(t) \) are fixed functions, then \( \int_{-\infty}^{\infty} \int f[x; t]~\delta x(t)~\mathrm{d}t = F[b] - F[a] \). More precisely, a path from the function \( a(t) \) to the function \( b(t) \) at every index value \( t \) can be parameterized by \( s \in [0, 1] \) by the map \( s \to x(t, s) \) which is a function of \( t \) for each \( s \) such that \( x(t, 0) = a(t) \) and \( x(t, 1) = b(t) \); this is why I linked this post to the most recent post on this blog. With this in mind, the fundamental theorem of calculus becomes \[ \int_{-\infty}^{\infty} \int_{0}^{1} f[x(s); t] \frac{\partial x}{\partial s}~\mathrm{d}s~\mathrm{d}t = F[b] - F[a] \] where, in the integrand, the argument \( x \) in \( f \) has the parameter \( s \) explicit but the dummy index \( t \) implicit; the point is that this equality holds regardless of the specific parameterization \( x(t, s) \) as long as \( x \) at the endpoints of \( s \) satisfies \( x(t, 0) = a(t) \) and \( x(t, 1) = b(t) \). This also means that \( F[x] \) can be recovered if \( b(t) = x(t) \) and \( a(t) \) is chosen such that \( F[a] = 0 \), in which case \[ F[x] = \int_{-\infty}^{\infty} \int_{0}^{1} f[x'(s); t]~\frac{\partial x'}{\partial s}~\mathrm{d}s~\mathrm{d}t \] (where \( x(t, s) \) has been renamed to \( x'(t, s) \) to avoid confusion with \( x(t) \)). If \( f[x; t] \) is not conservative, then there is no functional \( F[x] \) whose functional derivative with respect to \( x(t) \) would yield \( f[x; t] \); in that case, with \( x(t, 0) = a(t) \) and \( x(t, 1) = b(t) \), the integral \( \int_{-\infty}^{\infty} \int_{0}^{1} f[x(s); t] \frac{\partial x}{\partial s}~\mathrm{d}s~\mathrm{d}t \) does depend on the specific choice of parameterization \( x(t, s) \) with respect to \( s \) and not just on the functions \( a(t) \) and \( b(t) \) at the endpoints of \( s \).

As an example, consider from a previous post [LINK] the nonrelativistic Newtonian action \[ S[x] = \int_{-\infty}^{\infty} \left(\frac{m}{2} \left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^{2} + F_{0} x(t) \right)~\mathrm{d}t \] for a particle under the influence of a uniform force \( F_{0} \) (which may vanish). The first functional derivative is \[ f[x; t] = \frac{\delta S}{\delta x(t)} = F_{0} - m\frac{\mathrm{d}^{2} x}{\mathrm{d}t^{2}} \] and its vanishing would yield the usual equation of motion. The action itself vanishes for \( x(t) = 0 \), which will be helpful when using the fundamental theorem of calculus to recover the action from the equation of motion. In particular, one can parameterize \( x'(t, s) = sx(t) \) such that \( x'(t, 0) = 0 \) and \( x'(t, 1) = x(t) \). This gives the integral \( \int_{0}^{1} \left(F_{0} - ms\frac{\mathrm{d}^{2} x}{\mathrm{d}t^{2}}\right)x(t)~\mathrm{d}s = F_{0} x(t) - \frac{m}{2} x(t) \frac{\mathrm{d}^{2} x}{\mathrm{d}t^{2}} \). This is then integrated over all \( t \), so the first term is identical to the corresponding term in the definition of \( S[x] \), and the second term becomes the same as the corresponding term in the definition of \( S[x] \) after integrating over \( t \) by parts and setting the boundary conditions that \( x(t) \to 0 \) for \( |t| \to \infty \). (Other boundary conditions may require more care.) In any case, the parameterization \( x'(t, s) = sx(t) \) is not the only choice that could fulfill the boundary conditions; the salient point is that any parameterization fulfilling the boundary conditions would yield the correct action \( S[x] \).

I considered that example because I wondered whether any special formulas need to be considered if \( f[x; t] \) depends explicitly on first or second derivatives of \( x(t) \), as might be the case in nonrelativistic Newtonian mechanics. That example shows that no special formulas are needed because even if the Lagrangian explicitly depends on the velocity \( \frac{\mathrm{d}x}{\mathrm{d}t} \), the action \( S \) only explicitly depends as a functional on \( x(t) \), so proper application of functional differentiation and regular integration by parts will ensure proper accounting of each piece.

This post has been about the fundamental theorem of calculus saying that the 1-dimensional integral of a function in \( N \) dimensions along a contour, if that function is conservative, is equal to the difference between the two endpoints of its scalar antiderivative. This generalizes easily to infinite dimensions and continuous functions instead of finite-dimensional vectors. There is another fundamental theorem of calculus saying that the \( N \)-dimensional integral in a finite volume of the scalar divergence of an \( N \)-dimensional vector function, if that volume has a closed orientable surface, is equal to the \( N - 1 \)-dimensional integral of the inner product of that function with the normal vector (of unit 2-norm) at every point on the surface across the whole surface, meaning \[ \int_{V} \sum_{i = 1}^{N} \frac{\partial f_{i}}{\partial x_{i}}~\mathrm{d}V = \oint_{\partial V} \sum_{i = 1}^{N} f_{i}(x_{1}, x_{2}, \ldots, x_{N}) n_{i}(x_{1}, x_{2}, \ldots, x_{N})~\mathrm{d}S \] where \( \sum_{i = 1}^{N} |n_{i}(x_{1}, x_{2}, \ldots, x_{N})|^{2} = 1 \) for every \( \mathbf{x} \). From a purely formal perspective, this could generalize to something like \( \int_{V} \int_{-\infty}^{\infty} \frac{\delta f[x; t]}{\delta x(t)}~\mathrm{d}t~\mathcal{D}x = \oint_{\partial V} \int_{-\infty}^{\infty} f[x; t]n[x; t]~\mathrm{d}t~\mathcal{D}x \) having generalized \( \frac{\partial}{\partial x_{i}} \to \frac{\delta}{\delta x(t)} \), \( \prod_{i} \mathrm{d}x_{i} \to \mathcal{D}x \), and \( n_{i}(\mathbf{x}) \to n[x; t] \) where \( n[x; t] \) is normalized such that \( \int_{-\infty}^{\infty} |n[x; t]|^{2}~\mathrm{d}t = 1 \) for all \( x(t) \) on the surface. However, this formalism may be hard to further develop because the space has infinite dimensions. Even when working in a countable basis, it might not be possible to characterize an orientable surface enclosing a volume in an infinite-dimensional space; the surface is also infinite-dimensional. While the choice of basis is arbitrary, things become even less intuitive when choosing to work in an uncountable basis.

## 2022-11-01

### Mapping Scalars to Functions

In just over a year, I've written three posts for this blog about functionals, specifically about their application to probability theory [LINK], finding their stationary points [LINK], and the use of their stationary points in classical mechanics [LINK]. As a reminder, a functional is an object that maps a space of functions to a space of numbers. This got me thinking about what the reverse, namely an object that maps a space of numbers to a space of functions, looks like. To be clear, this is not the same as an ordinary function which, as an element in a space of functions, maps a space of numbers to a space of numbers.

As I thought about it more, I realized that this is a bit easier to understand and therefore more commonly encountered than a functional. An extremely glib way to describe such an object is a function of multiple variables. However, it may be more enlightening to describe this in further detail to avoid potentially deceptive images that may arise from that glib description.

In the discrete case, the matrix elements \( A_{ij} \) can be described as a map from integers to vectors, in which an integer \( j \) is associated with a vector whose elements indexed by an integer \( i \) are \( A_{ij} \). This is the essential idea behind seeing the columns of the matrix with elements \( A_{ij} \) as a collection of vectors. Formally, this maps \( i \to (j \to A_{ij}) \) where the map \( j \to A_{ij} \) defines a vector indexed by the free variable \( i \).

Similarly, in the continuous case, the function elements \( f(x, y) \) can be described as a map from numbers to functions, in which a number \( y \) is associated with a function whose
elements indexed by a number \( x \) are \( f(x, y) \). Formally, this maps \( x \to (y \to f(x, y)) \) where the map \( y \to f(x, y) \) defines a function indexed
by the free variable \( x \). These ideas are foundational to the development of more abstract notions of functions, like lambda calculus.

## 2022-10-02

### Technological Restrictions on E-Books and Culture Wars on Books in 2022

Despite the long title, this post will be fairly short. This blog used to publish a lot more often (I had a lot more free time in high school & college) and focus a lot more on issues related to free software, free culture, and things like that, yet even after looking through posts on this blog from its early years (which, aligning with the stereotype of an adult looking through essays written in high school, made me cringe at the quality of writing even if I agreed with some of the basic opinions), I actually couldn't find any posts specifically about the effects of so-called digital rights management (DRM) on E-books.

In any case, I was motivated to write this because I recently listened to an episode [LINK] of a podcast associated with The Daily Show in which the guests discussed recent instances of conservative politicians in the US preventing public schools & libraries from teaching or carrying books that offend those politicians' cultural sensibilities. I distinctly remember reading in high school & college about warnings of the consequences of putting DRM on E-books, including making it easier to ban such books. At that time, I and many others felt it would be ridiculous for politically motivated book bans to take effect in the US especially given respect for the First Amendment to the US Constitution. Leaving aside whether such book bans from public schools & libraries technically violate that amendment if such bans don't go beyond those domains, it is disheartening to see a direct example of censorship so closely connected to technological restrictions that are politically motivated (not due to fundamental technical limitations). It will be interesting to see whether authors of banned books encourage or tolerate people scanning & sharing unauthorized PDF files of the books for free; this wouldn't be unprecedented, given that the huge markup of textbooks in the US compared to other countries has led many textbook authors to encourage students to buy cheaper editions from other countries.

## 2022-09-17

### Book Review: "From the War on Poverty to the War on Crime" by Elizabeth Hinton

I've recently read the book *From the War on Poverty to the War on Crime* by Elizabeth Hinton. This book is a history of the progression through the titular subjects in the US, starting with the Kennedy presidency and ending with the Reagan presidency. It shows how while some problems associated with the war on poverty did come from good but conflicting intentions when implementing social welfare programs, many more problems came from halfhearted implementation of social welfare programs with the intent & through the lens of fighting crime leading ultimately to replacement of those programs with more explicit expansions of policing to fight crime especially in response to high-profile riots in large cities in the 1960s & 1970s.

The introduction makes clear that the war on crime started with the presidency of Lyndon Johnson and arguably even earlier with the Kennedy administration's efforts to combat juvenile delinquency, so the association of the war on crime primarily with the presidencies Nixon, Ford, and Reagan (as Carter is often left out of popular discussions about this despite being equally responsible in these ways) is because those presidents (and Carter) cut funding for social welfare programs that Johnson initially saw as integral to the success of programs combating crime but began to back away from by the end of his presidency, increased funding for policing, and shifted focus to arrests & imprisonments as ways to prevent future crimes. The author discusses how law enforcement agencies started to develop biased metrics for crime & collect data in biased ways to justify racist theories about the supposedly inherent pathologies of black Americans in cities, even as some politicians at that time wondered if law enforcement agencies should be collecting crime statistics given the conflict of interest. The author emphasizes the bipartisan white American political consensus about crime after 1960 to show that it wasn't just Reagan or other Republicans in the 1980s who focused on punishment through veiled racism. The author also discusses how black American community leaders in & after the 1960s wanted to partner with law enforcement agencies to develop effective strategies together to deal with local problems at the root of local crimes, but conservative politicians (from both parties) deliberately moved away from such partnerships toward the federal block grant funding model that would incentivize states to conduct law enforcement in the most heavy-handed & punitive way possible especially in urban black neighborhoods, which led many black Americans to stop trusting law enforcement. My only criticism specifically about the introduction (that doesn't have to do with the rest of the book) is that the author's language about conditional probabilities is quite sloppy, which is problematic in the context of discussions about biases in data collection & statistical analysis by law enforcement agencies about crime.

The rest of the book simply goes through the history in detail. In the introduction, I wasn't sure who the target audience of the book was supposed to be given frequent references to gaps in academic literature in the main text, but the narrative became more clear through the rest of the book.

There are a few points that I credit the book for. These are as follows.

First, the author repeats points effectively to reinforce the narrative. This makes the narrative easy to follow, and the narrative is clearly well-sourced.

Second, I didn't know that the war on crime dated back to the Kennedy administration. I can claim to have learned that from this book.

Third, I didn't know that close federal cooperation directly with local governments also dated back to the Kennedy administration at the latest. I can claim to have learned that from this book. This is an issue that has been on my mind for a while in the context of empowering cities whose political views oppose those of their state governments which want to disempower them (because while there is a clear federal relationship among the federal, state, and tribal governments in the US Constitution, the US Constitution doesn't govern states' internal affairs, and many states treat their constituent cities as fully subordinate to state governments in all matters).

Fourth, in my view, the author correctly recognizes that policies to combat crime should be evaluated for effectiveness several years afterwards, as there are no quick fixes. The author therefore evaluates whether different parts of the war on crime had effects on crime 10-20 years later instead of just a year later, as the latter would have been a political cheap shot.

Fifth, I appreciate the author's recognition that the US made the same mistakes in both military & social welfare strategies domestically as in invasions of other countries. The author makes clear that many politicians at that time recognized this in the context of wars in Southeast Asia. This adds another dimension to arguments from the book *How Everything Became War and the Military Became Everything* by Rosa Brooks (which I have reviewed on this blog [LINK]) which, from what I remember, focused a little more on more recent invasions of Afghanistan & Iraq.

However, there are many more points in the book which I find problematic. These are as follows.

First, at several points, the author claims that news media & politicians often overstated the prevalence of violent crime, but the author's claims that federal law enforcement statistics about crime were less biased before the 1960s are undercut by the author's acknowledgment that such statistics were collected much more sparsely and in a more ad hoc way before the 1960s. Even leaving aside this point, the author doesn't (in the main text, leaving out the endnotes) name or cite enough specific sources to effectively argue against the dominant narrative of that time, which the book simply restates while feebly arguing against, or intuitively explain why certain crime statistics may initially look alarming but may actually imply rarity of such incidents in practice for an individual. For example, it could be argued that a certain homicide rate given in homicides per million people per year could look big at first but could be argued to imply that an individual has a very low chance of being killed by another person on a given day. (I don't know enough about crime statistics to know what specific number could plausibly fit this description.)

Second, at several points, the author basically wags a finger against what the federal government did, but in some cases where solutions are longer-term and therefore more obvious like investments in unleaded plumbing, better street lighting, or better sidewalks, the author barely identifies these, and in other cases where issues like imminent or continuing riots are spiraling out of control, the author doesn't convincingly argue in favor of specific alternatives except in perhaps 1 or 2 isolated cases. Part of the problem, as becomes clear in the epilogue, may be that the federal government prematurely dismissed such alternatives before seriously trying them, but then the author should have spent more space arguing for such programs on their own potential specific merits instead of giving most of the space to the arguments of contemporary politicians that, by dominating the narrative, may unintentionally seem more convincing than the author may have wanted.

Third, the author argues at some points that poor black Americans should have been empowered from the bottom-up but at other points that they should have gotten similar top-down federal assistance as poor white Americans (which did happen in the war on poverty, albeit at much less monetary amounts per person). This seems incoherent, and the author makes no attempt to explain why these views are compatible with each other.

Fourth, the author ignores the issues of continued slum clearance, urban highway construction, and the dynamics of white flight from urban cores in much of the narrative. I've read in many other places how critical these concerns were in the context of urban crime, so it is surprising to see no mention of these concerns in the book.

Fifth, the author seems to unduly dismiss the challenges that the Carter administration faced in rebuilding damaged urban neighborhoods in the face of high interest rates & high inflation in the 1970s. Perhaps the argument would have been stronger if the author could have found examples of the Carter administration spending scarce resources on less dire issues.

Sixth, in the introduction, the author claims that the war on crime specifically wasn't a reincarnation of the Jim Crow era, but later in the book, the author at many points implies & comes close to explicitly saying exactly that. This seems inconsistent, though to be fair, it was obvious to me that the author would ultimately argue that the war on crime was related to the Jim Crow era, so this inconsistency only threw me off in the introduction.

Seventh, I found it interesting that the author used scare quotes around the term "evil empire" and called the Cold War "Reagan's Cold War". It could be argued that the latter was a more specific reference to how the Reagan administration waged the Cold War in the 1980s as a smaller part of the broader conflict over decades, but based on the author's other stated & implied views through the book, I see it more likely as evidence of the author having far left-wing sympathies, because the latter term in the broader context of the author's views through the book sounds like the author believes the Reagan administration was too bellicose toward the USSR and was too taken by American propaganda over decades to admit that some parts of Soviet propaganda especially about race relations could be true (which is debatable in the context of internal affairs in the USSR).

Eighth, the author seems to argue that the implementation of the ban of handguns but not shotguns was racist because it failed to separate bans on weapons (which should mostly be about destroying those weapons and removing sources of weapons production, though perhaps some further consequences could be appropriate for repeat offenders) from the harsh punishment of offenders. I agree with this in the context of having excessive punishments and in the sense that, even now, it is clear in the US that white Americans are much more often than black Americans given the benefit of the doubt with respect to usage of firearms in self-defense or the possession of firearms per the Second Amendment to the US Constitution. Moreover, it is worth remembering that while courts of law in the US didn't start to systematically recognize an individual right to bear arms until the 2000s (many decades after the 1960s), that doesn't mean that everyone was prosecuted equally for bearing firearms before the 2000s; it is much more plausible that there were racial disparities in enforcement of such laws. However, I think the author doesn't do a good job of acknowledging how the much greater difficulty in using a shotgun to commit violent crimes in urban settings compared to using a handgun for the same purpose makes a handgun ban more sensible without such a ban (separate from the consequences to humans who violate such bans) necessarily being racist per se.

Ninth, at several points, the author seems to go beyond merely describing large-scale riots led by black Americans as a predictable consequence of oppression to being an apologist for such riots. I do at least acknowledge the merit in describing such predictable consequences from a sociological perspective. Also, I admit that it took me a while to understand the author's point that the problem with claiming in the 1960s & later that black Americans are culturally pathological is that such claims were coming from white American politicians who wanted to claim that such cultural pathology was the root of poverty & crime (and not the other way around). Finally, while I still believe that any human culture can have pathological elements in which some elements come from a history of being oppressed & therefore traumatized while other elements may be evidence of being a privileged or oppressive group (so there is no such thing as being a purely "good" or "oppressed" group or a purely "bad" or "oppressive" group), I can understand the author's desire to call white American racism as the dominant group in the US something closer to "cultural pathology" and the cultural pathologies that may result directly from centuries of oppression "trauma" as qualitatively distinct things within the context of US history. Having said those things, I don't think the author did a good enough job of acknowledging that even if some ideal form of reparations for these harms could be formulated & implemented overnight, the lasting effects of these traumas could continue to have negative consequences for different localities & the US as a whole for many decades, and especially now with the US having so many residents & citizens who come from other places & don't identify as white or black, I'm not sure how many Americans would have the patience to wait decades for those things to resolve even if they are much more understanding & supportive of the need for reparations than most white Americans would have been in the 1960s (as the author shows how white Americans in the 1960s wanted quick fixes to the eruptions of violence in response to oppression, which led to further oppression through brutal crackdowns by law enforcement).

Overall, I still recommend this book because the basic historical narrative is well-written & well-sourced; I do feel like I learned a little bit and I had a lot to think about. I just think that readers should be aware of the author's biases as I've discussed above.

## 2022-08-07

### Review: KDE neon 5.25

It has been a long time since I've reviewed a Linux distribution on this blog; the last one was of Linux Mint 19 "Tara" from 4 years ago [LINK]. In a more recent post about problems that I had with a scanner that required me to install Linux Mint 20 "Ulyana" MATE because the existing operating system was damaged beyond repair [LINK], I explained that I had come to trust the consistency & stability of Linux Mint enough and liked it enough that, in conjunction with the lack of novelty in Linux distributions compared to 10 years prior, I no longer felt motivated to do such reviews. Thus, it may seem strange that I should do a review like this now. In truth, the motivation wasn't hugely compelling, but I thought it might make for a nice post on this blog as I didn't have much else in mind. I thought of checking out a showcase of KDE, namely KDE neon, because it had been a long time since I tried KDE and I was getting a little concerned that the odd artifacts I was starting to see in Linux Mint 20 "Ulyana" MATE when hovering over right-click menus might be the tip of an iceberg of problems. While the latter concern has thankfully not come to pass even after several months of experiencing these more minor issues, I figured it might be nice to see what KDE is like now.

Default desktop, before changes |

## 2022-07-10

### FOLLOW-UP: Some Recent Troubles with pCloud and Google Chat

Almost exactly 11 months ago, I wrote a post [LINK] about problems I was experiencing with pCloud and Google Chat. I don't have any updates about Google Chat (or Google Meet), but I do have an update regarding pCloud. In the previous post, I noted that for files & folders protected by standard encryption (as opposed to stronger zero-knowledge encryption, for which this problem doesn't exist in pCloud), some files & folders require multiple attempts to transfer; I also speculated that pCloud may have been secretly deleting files & folders. After having spent more time using pCloud, I've been able to verify that my files & folders protected by standard encryption have transferred properly, and I think I've figured out why they initially seemed to require multiple attempts to transfer.

As I understand, pCloud creates a temporary folder, essentially like a cache, on the local hard drive, to transfer files before they are uploaded to pCloud. The process of transferring from the local cache to pCloud is limited by upload speeds, which are quite slow (as I mentioned in the previous post). Additionally, once the pCloud program is closed and the remote drive is unmounted, file transfer stops, so many folders & files that the user might think were transferred might not have been transferred. A good way to verify this is to leave the desktop application for pCloud open, monitor how many files as well as what total amount of data still remain to be transferred, and only close the application when all folders & files have been transferred; I like to think of it as a practice similar to leaving a torrent open for uploading after it has finished downloading.