How To Determine Which Request For Proposal Form: Fill & Download for Free

Please make sure your seat back and folding trays are in their full upright positions…Despite already showing great promise as a mathematician, young Grisha was not offered a position in his prestigious alma mater, Moscow State University. He was Jewish, it was 1970, and that was how it was. Instead, he found a job at the rather obscure Institute for Problems in Information Transmission. It was there that he had happened to hear about the relatively new idea of expander graphs, which are highly connected graphs of fixed degree – a useful property for designing communication networks. Like many other extremal graph-theoretic properties, expander graphs were frustratingly elusive: it was easy to prove that they exist using probabilistic methods, but explicitly constructing them is very difficult. Nobody knew how to do that.The fundamental property of expander graphs demands that if you take at most half the vertices, you will find many other vertices connected to them. This is easy to do in a dense graph, but not easy at all to achieve in a sparse one (such as a large graph of fixed degree). But this expansion property triggered a deep and subtle analogy in Grisha’s mind: it reminded him of Kazhdan’s Property (T) which was introduced by David Kazhdan just a few years before. It’s a completely unrelated property in a seemingly completely unrelated field: infinite-dimensional representations of locally compact groups.Grisha was able to turn that tenuous analogy into a breathtakingly clever explicit construction of expander graphs, by considering quotients of discrete lattices in groups with Property (T). In 1973, he published a first inkling of this idea in an obscure Russian journal, Problemy Peredachi Informatsii. Few people paid attention. Grisha moved on to deepen his study of lattices in Lie groups. He made profound discovery after profound discovery, and in 1978 was awarded the Fields medal, at 32 years old. He was not allowed by the Russian authorities to travel to Helsinki to receive the award. Neither that nor the prestigious medal stopped his growth as a mathematician. A few days ago, on March 18, 2020, Grigory “Grisha” Margulis was awarded the Abel Prize in mathematics, along with Hillel Furstenberg.But back in 1973, his explicit construction of expander graphs lay dormant.A few years later, In 1976, Whitfield Diffie and Martin E. Hellman published a transformative paper titled “New Directions in Cryptography[1] ”. In it, they promoted a new and radical idea: two parties can establish secure communication over an insecure channel, by establishing a shared secret without having couriers quietly meet in obscure cafes, exchanging suitcases. The Diffie-Hellman key exchange was not the very first instance of this idea – British cryptographers were later found to have thought of it several years before – but over time it became the basis for some of the most broadly used key-exchange and secret sharing schemes.The core idea of Diffie-Hellman is simple. Suppose we have some algebraic structure with a fixed element [math]A[/math], publicly known. I pick a random, secret number [math]X[/math] and calculate [math]AX[/math]. You pick a random, secret number [math]Y[/math] and calculate [math]AY[/math]. We both publish [math]AX[/math] and [math]AY[/math]. You take my [math]AX[/math] and multiply it by your secret [math]Y[/math], obtaining [math]AXY[/math]. I do the same thing, taking your [math]AY[/math] and multiplying by my secret [math]X[/math], obtaining [math]AYX[/math]. If the algebraic structure is commutative, the order of elements in a multiplication doesn’t matter, and we now both have a shared secret [math]AXY[/math].Everyone knows [math]A[/math], and everyone sees [math]AX[/math] and [math]AY[/math], but if “division” in the algebraic structure is difficult, there’s no easy way to figure out [math]AXY[/math] from these things. The question now becomes, which algebraic structures have this peculiar property: multiplication is easy, but division or inversion is hard.Diffie and Hellman proposed the multiplicative group of a finite field, or even more simply: the residues modulo a fixed large prime [math]p[/math]. This works quite well, though the Number Field Sieve can break DH in time [math]O(\sqrt[3]{\log(n)})[/math], necessitating somewhat unwieldy key sizes: to achieve “128-bit security”, you should pick the prime [math]p[/math] to be larger than [math]2^{3072}[/math]. This is doable, but cumbersome.Years later, a better algebraic structure was proposed: the group of points on an elliptic curve. This beautiful, devilish trickery replaces the numbers mod [math]p[/math] with the points on an elliptic curve, known for centuries to form a commutative group via a method which corresponds, geometrically, to drawing chords and tangents. The corresponding key-exchange scheme is called Elliptic-curve Diffie–Hellman, but in a surprise twist, I won’t say much more about it because those are not the cryptographic elliptic curves we need to talk about. Elliptic-curve Diffie-Hellman has nothing to do with isogenies. Patience.Meanwhile, both the theory and practice of expander graphs were gaining tremendous momentum. They were found to be fundamental to many areas of theoretical computer science and combinatorics: extremal graph theory (the girth problem), derandomization (turning efficient random algorithms into efficient deterministic ones), network architectures and much more. In particular, a class of graphs was singled out by Lubotzky, Phillips and Sarnak: Ramanujan graphs were defined as [math]d[/math]-regular graphs whose second largest eigenvalue (in absolute value) is less than [math]2\sqrt{d-1}[/math]. This bound is asymptotically best possible (Alon-Boppana), and the association with Ramanujan is related to deep connections between these graphs and the Ramanujan–Petersson conjecture in the theory of modular forms. Around 1986, Lubotzky et al were able to construct such optimal expanders using methods which were a direct continuation of the methods put forward by Margulis in the 1970’s, relying on some of the deepest ideas in representation theory developed in the intervening years. The wonderful, multi-faceted story of this discovery and its consequences is told in an award-winning book by Alex Lubotzky.A few years later, around 1990, Arnold Pizer was able to extend[2] the construction of Lubotzky, Phillips and Sarnak. While still focusing on theta series and Hecke operators, Pizer constructed graphs which can also be viewed as the isogeny graphs of families of supersingular elliptic curves. Elliptic curves are (for our modest purposes here) the solutions of equations that look like [math]y^2=x^3+Ax+B[/math], with some mild condition on the integers [math]A,B[/math] to eliminate singularities (the polynomial on the RHS must not have a repeated root). Those solutions can be taken to be complex numbers, real numbers, rational numbers, or even numbers in finite fields, since the equation can be interpreted in all of these domains. That’s one of the many things that gives elliptic curves their immense depth.Elliptic curves are algebraic curves, and when we have two such curves we like to talk about rational maps (or morphisms) between them. A rational map is simply a function that maps the points of one curve to those of the other curve and is defined (locally) by rational functions (ratios of polynomials). Since elliptic curves are also groups (their points can be added), such morphisms between them often respect the group law; for example, as soon as a morphism maps the neutral [math]0[/math] element of one curve to the [math]0[/math] element of the other, it necessarily preserves addition. Such morphisms are called isogenies, and as a result of their being both rational maps and group homomorphisms, they are rather rare. In the complex context, for example, most elliptic curves only have very few self-isogenies, given by “multiplication-by-[math]N[/math]”: mappings a point [math]P[/math] to [math]P+P+\ldots+P[/math] ([math]N[/math] times) is always a group homomorphism and it is also always a rational map, since the group law of an elliptic curve can be written using rational functions.When you look at elliptic curves over finite fields, on the other hand, the set of isogenies is often richer. It is particularly rich in the case of supersingular curves, and in this case you can take classes of curves (classified by something called their [math]j[/math]-invariant) and connect them when there is an isogeny between them. You obtain in this way a very nice, symmetric graph, and deep results of Eichler and Shimura allowed Pizer to show that those graphs are Ramanujan graphs.(Defining all the terms I’ve used here will take up a small book. I’m sorry).At this point in time, Pizer’s construction was a very cool but mostly theoretical advancement of the deep connection between representation theory and expander graphs. But then quantum happened.In 1994, Peter Shor showed the world[3] that the largely speculative idea of quantum computation may have some very real-life applications: specifically, he demonstrated how quantum computational models can efficiently factor large integers. Among other consequences, this development – if anyone was able to make it practical – would affect the security of the RSA protocols as well as those of protocols based on the hardness of the discrete-log problem, including all known Diffie-Hellman schemes.It took some time for this to be digested, and some more time for anyone to start taking quantum computation as a potentially real thing (though the jury is still divided on this). In response to those developments, in 2016, the US National Institute of Standards and Technology (NIST) issued a Request for Proposals of “Post-quantum cryptographic protocols”. Almost 70 proposals were submitted, many of them both sophisticated and practical, but perhaps the most successful ones are based on Ramanujan graphs, and specifically the isogeny graphs constructed by Pizer in 1990.One of the many features of expander graphs in general, and Ramanujan graphs in particular, are their strong mixing properties. If you start with some fixed vertex on such a graph and perform a random walk along its edges, you will very quickly find yourself in a very random place on the graph. The second eigenvalue, which Ramanujan graphs push to its extreme, controls the mixing time. Graphs with poor mixing require a very long random walk before you are truly at a random place on the graph. Strongly mixing graphs achieve that rapidly (Incidentally, the theory of eigenvalues and mixing times on graphs was used by Larry Page and Sergey Brin in the original design of the PageRank algorithm).The idea of using expander graphs (and specifically, Pizer’s isogeny graphs) for cryptographic purposes goes back at least to Charles, Lauter and Goren[4] in 2006. They describe the use of such graphs to produce a strong hash function: given a sequence of bits as input, perform a walk on a Ramanujan graph, using the input bits to determine which edge to pick at each step. The end vertex is the hash value.A variation of this idea formed the basis of “Supersingular Isogeny Diffie-Hellman”, a key exchange protocol which is among the most promising candidates for a post-quantum world, which we may choose to move into even if quantum computers do not become a viable risk. It’s always good to be safe, and SIDH costs very little: it has short key lengths, is relatively easy to implement, and is very compatible with the earlier Elliptic-curve Diffie-Hellman protocols.This, very briefly and superficially, is the story of isogenies and their connection to cryptography through the wonderful concept of Ramanujan graphs. It’s beautiful to watch those threads of pure and applied mathematics tie together; looked at from a distance, it would have been impossible to believe that cusp forms, Hecke operators and the Ramanujan conjectures would one day let us buy books about the mathematics of Grigory Margulis by tap-tap-tapping on our little screens.Footnotes[1] https://ee.stanford.edu/~hellman/publications/24.pdf[2] American Mathematical Society[3] Shor's algorithm - Wikipedia[4] Cryptographic Hash Functions from Expander Graphs

Hahahaha.You’ve been fed a line of bullshit.First, there is no “inconvenience”. I can go to literally any clinic, doctor, or hospital in the country and all I need to do is to show them my health card:Second, about those taxes: healthcare expenditures are mostly covered by various forms of income taxes, which means that you pay based on what you earn. If you’re a struggling new grad, you pay very little. If you are a rich old fart, you pay more. In both cases, overwhelmingly, people think that it is “worth it”. (And by the way? Just as an aside? In Canada, our taxes are actually lower than your taxes + health insurance + copays.)Here’s a great short video (by an American doctor) which explains the Canadian healthcare system really well:This article is well worth reading: https://www.washingtonpost.com/outlook/2020/08/06/health-insurance-canada-lie/?arc404=trueHere’s the text:In my prior life as an insurance executive, it was my job to deceive Americans about their health care. I misled people to protect profits. In fact, one of my major objectives, as a corporate propagandist, was to do my part to “enhance shareholder value.” That work contributed directly to a climate in which fewer people are insured, which has shaped our nation’s struggle against the coronavirus, a condition that we can fight only if everyone is willing and able to get medical treatment. Had spokesmen like me not been paid to obscure important truths about the differences between the U.S. and Canadian health-care systems, tens of thousands of Americans who have died during the pandemic might still be alive.In 2007, I was working as vice president of corporate communications for Cigna. That summer, Michael Moore was preparing to release his latest documentary, “Sicko,” contrasting American health care with that in other rich countries. (Naturally, we looked terrible.) I spent months meeting secretly with my counterparts at other big insurers to plot our assault on the film, which contained many anecdotes about patients who had been denied coverage for important treatments. One example was 3-year-old Annette Noe. When her parents asked Cigna to pay for two cochlear implants that would allow her to hear, we agreed to cover only one.Clearly my colleagues and I would need a robust defense. On a task force for the industry’s biggest trade association, America’s Health Insurance Plans (AHIP), we talked about how we might make health-care systems in Canada, France, Britain and even Cuba look just as bad as ours. We enlisted APCO Worldwide, a giant PR firm. Agents there worked with AHIP to put together a binder of laminated talking points for company flacks like me to use in news releases and statements to reporters.Here’s an example from one AHIP brief in the binder: “A May 2004 poll found that 87% of Canada’s business leaders would support seeking health care outside the government system if they had a pressing medical concern.” The source was a 2004 book by Sally Pipes, president of the industry-supported Pacific Research Institute, titled “Miracle Cure: How to Solve America’s Health Care Crisis and Why Canada Isn’t the Answer.” Another bullet point, from the same book, quoted the CEO of the Canadian Association of Radiologists as saying that “the radiology equipment in Canada is so bad that ‘without immediate action radiologists will no longer be able to guarantee the reliability and quality of examinations.’ ”Much of this runs against the experience of many Americans, especially the millions who take advantage of low pharmaceutical prices in Canada to meet their prescription needs. But there were more specific reasons to be skeptical of those claims. We didn’t know, for example, who conducted that 2004 survey or anything about the sample size or methodology — or even what criteria were used to determine who qualified as a “business leader.” We didn’t know if the assertion about imaging equipment was based on reliable data or was an opinion. You could easily turn up comparable complaints about outdated equipment at U.S. hospitals.(Contacted by The Washington Post, an AHIP spokesman said this perspective was “from the pre-ACA past. We are future focused by building on what works and fixing what doesn’t.” He added that the organization “believes everyone deserves affordable, high-quality coverage and care — regardless of health status, income, or pre-existing conditions.” An APCO Worldwide spokesperson told The Post that the company “has been involved in supporting our clients with the evolution of the health care system. We are proud of our work.” Cigna did not respond to requests for comment.)Nevertheless, I spent much of that year as an industry spokesman, my last after 20 years in the business, spreading AHIP’s “information” to journalists and lawmakers to create the impression that our health-care system was far superior to Canada’s, which we wanted people to believe was on the verge of collapse. The campaign worked. Stories began to appear in the press that cast the Canadian system in a negative light. And when Democrats began writing what would become the Affordable Care Act in early 2009, they gave no serious consideration to a publicly financed system like Canada’s. We succeeded so wildly at defining that idea as radical that Sen. Max Baucus (D-Mont.), then chair of the Senate Finance Committee, had single-payer supporters ejected from a hearing.Today, the respective responses of Canada and the United States to the coronavirus pandemic prove just how false the ideas I helped spread were. There are more than three times as many coronavirus infections per capita in the United States, and the mortality rate is twice the rate in Canada. And although we now test more people per capita, our northern neighbor had much earlier successes with testing, which helped make a difference throughout the pandemic.The most effective myth we perpetuated — the industry trots it out whenever major reform is proposed — is that Canadians and people in other single-payer countries have to endure long waits for needed care. Just last year, in a statement submitted to a congressional committee for a hearing on the Medicare for All Act of 2019, AHIP maintained that “patients would pay more to wait longer for worse care” under a single-payer system.While it’s true that Canadians sometimes have to wait weeks or months for elective procedures (knee replacements are often cited), the truth is that they do not have to wait at all for the vast majority of medical services. And, contrary to another myth I used to peddle — that Canadian doctors are flocking to the United States — there are more doctors per 1,000 people in Canada than here. Canadians see their doctors an average of 6.8 times a year, compared with just four times a year in this country.Most important, no one in Canada is turned away from doctors because of a lack of funds, and Canadians can get tested and treated for the coronavirus without fear of receiving a budget-busting medical bill. That undoubtedly is one of the reasons Canada’s covid-19 death rate is so much lower than ours. In America, exorbitant bills are a defining feature of our health-care system. Despite the assurances from President Trump and members of Congress that covid-19 patients will not be charged for testing or treatment, they are on the hook for big bills, according to numerous reports.That is not the case in Canada, where there are no co-pays, deductibles or coinsurance for covered benefits. Care is free at the point of service. And those laid off in Canada don’t face the worry of losing their health insurance. In the United States, by contrast, more than 40 million have lost their jobs during this pandemic, and millions of them — along with their families — also lost their coverage.Then there’s quality of care. By numerous measures, it is better in Canada. Some examples: Canada has far lower rates than the United States of hospitalizations from preventable causes like diabetes (almost twice as common here) and hypertension (more than eight times as common). And even though Canada spends less than half what we do per capita on health care, life expectancy there is 82 years, compared with 78.6 years in the United States.When the pandemic reached North America, Canadian hospitals, which operate under annual global budgets — fixed payments typically allocated at the provincial and regional levels to cover operating expenses — were better prepared for the influx of patients than many U.S. hospitals. And Canada ramped up production of personal protective equipment much more quickly than we did.Of the many regrets I have about what I once did for a living, one of the biggest is slandering Canada’s health-care system. If the United States had undertaken a different kind of reform in 2009 (or anytime since), one that didn’t rely on private insurance companies that have every incentive to limit what they pay for, we’d be a healthier country today. Living without insurance dramatically increases your chances of dying unnecessarily. Over the past 13 years, tens of thousands of Americans have probably died prematurely because, unlike our neighbors to the north, they either had no coverage or were so inadequately insured that they couldn’t afford the care they needed. I live with that horror, and my role in it, every day.

Depends what level. That's the reason why I didn't answer the other questions: I don't know what exactly is meant by the goal of "judging" a C++ programmer.But I bet the OP really wants to hear some hard C++ questions, so maybe the ones below will interest them. But there are a lot of topics I won't cover because there are large parts of the language that I don't understand very well. For example, I know almost nothing about all the concurrency related features (threads/atomics/futures and so on) or the new array_view/string_view stuff that's being proposed for C++17.The questions below are ranked in roughly ascending order of how confusing I think they are.1. If class A can be converted to class B either through a converting constructor in B or a conversion operator in A, as in the following scenario, which one will be called? What happens if A::operator B() is declared explicit? What happens if A::operator B() is declared private?struct A; struct B {  B(const A&); }; struct A {  operator B(); }; int main() {  // request implicit conversions from A to B } Answer: The overload resolution process is used to determine which of the two functions will be called. If the expression to be converted is of type non-const A, then the conversion operator wins because it binds the argument to the implicit object parameter of type A&, which is less cv-qualified than the const A& parameter of B's converting constructor. If the expression to be converted is of type const A, the conversion operator can't be used because it's not const, and the converting constructor is called.If the conversion operator is explicit, it's removed from overload resolution, and the converting constructor is called in both cases. If the conversion operator is private, the access check is applied after overload resolution, so for a non-const argument, you get an error; the compiler doesn't choose B::B(const A&) instead. (This behaviour differs from Java, in which a private function is invisible outside its class.)2. Write the function template std::make_tuple.Answer: A possible answer is given here: std::make_tuple - cppreference.comThis question tests both familiarity with this key standard library component, including exactly what its effect is, and facility with writing variadic templates, which often confuse even experienced C++ programmers. The key is that std::make_tuple deduces its template parameters from the argument types and instantiates std::tuple with the deduced types, but first decays them. Were this not the case, you would run into all sorts of issues, such as forming references to the arguments and storing those references in the tuple, which is not supposed to happen.I'd be especially impressed if you got the std::reference_wrapper case as well---that's something I have a tendency to forget about, since I never use it.3. Why doesn't a using-declaration work to solve the diamond proble​m?Answer: Given in the SO thread. The key here is that a using-declaration only affects name lookup, not implicit conversions.4. Why does only the last line of main produce an error? That is, why is it that f(&s) works but f(nullptr) doesn't? And how could you make f(nullptr) compile?struct S {  friend void f(S*) {} }; int main() {  S s;  f(&s);  f(nullptr); } Answer: A function declared by a friend declaration can't be found during ordinary unqualified or qualified name lookup until after a matching declaration is made in the nearest enclosing namespace scope. That's why f(nullptr) doesn't compile, and you can fix it by adding a global declaration after the definition of S:void f(S*);The reason why f(&s) works without this global declaration is that the argument is of type S*, not std::nullptr_t, and since it's a pointer to class type, this triggers argument-dependent lookup on f. Argument-dependent lookup, unlike other forms of name lookup, is allowed to find friend functions that have not been declared at namespace scope yet.5. Why is there an undefined reference to S::m2 but not to S::m1 in the below code:#include <iostream> struct S {  static constexpr const char* m1 = "hello, world!\n";  static constexpr const char m2[] = "hello, world!\n"; }; int main() {  std::cout << S::m1;  std::cout << S::m2; } Answer: This question tests your understanding of the concept of odr-use. You can find discussion here, with both the intuitive reasoning and the standardese: Undefined reference to static constexpr string (except if it's a pointer)I would like to point out that this is not as obscure as it looks. Linker issues from not defining static members happen all the time, and often confuse less experienced C++ programmers.Despite how arcane these questions are, there are many people who frequent the C++ tag on Stack Overflow who could answer all of them without breaking a sweat. (Look for people with gold C++ badges.)