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In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains a Souslin subtree, if there is an inaccessible cardinal. This is stronger than Komjaths theorem that asserts the same consistency from two inaccessible cardinals. We will show that our large cardinal assumption is optimal, i.e. if every Kurepa tree has an Aronszajn subtree then $omega_2$ is inaccessible in the constructible universe textsc{L}. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree $T$ such that if $U subset T$ is a Kurepa tree with the inherited order from $T$, then $U$ has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: assume $textrm{MA}_{omega_2}$ holds and $omega_2$ is not a Mahlo cardinal in $textsc{L}$. Then there is a Kurepa tree with the property that every Kurepa subset has an Aronszajn subtree. Our work entails proving a new lemma about Todorcevics $rho$ function which might be useful in other contexts.
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We address the problem of which planar sets can be drawn with a pencil and eraser. The pencil draws any union of black open unit disks in the plane $mathbb{R}^2$. The eraser produces any union of white open unit disks. You may switch tools as many times as desired. Our main result is that drawability cannot be characterized by local obstructions: A bounded set can be locally drawable, while not being drawable. We also show that if drawable sets are defined using closed unit disks the cardinality of the collection of drawable sets is strictly larger compared with the definition involving open unit disks.
With current technology, a number of entities have access to user mobility traces at different levels of spatio-temporal granularity. At the same time, users frequently reveal their location through different means, including geo-tagged social media posts and mobile app usage. Such leaks are often bound to a pseudonym or a fake identity in an attempt to preserve ones privacy. In this work, we investigate how large-scale mobility traces can de-anonymize anonymous location leaks. By mining the country-wide mobility traces of tens of millions of users, we aim to understand how many location leaks are required to uniquely match a trace, how spatio-temporal obfuscation decreases the matching quality, and how the location popularity and time of the leak influence de-anonymization. We also study the mobility characteristics of those individuals whose anonymous leaks are more prone to identification. Finally, by extending our matching methodology to full traces, we show how large-scale human mobility is highly unique. Our quantitative results have implications for the privacy of users traces, and may serve as a guideline for future policies regarding the management and publication of mobility data.
Reddits WallStreetBets (WSB) community has come to prominence in light of its notable role in affecting the stock prices of what are now referred to as meme stocks. Yet very little is known about the reliability of the highly speculative investment advice disseminated on WSB. This paper analyses WSB data spanning from January 2019 to April 2021 in order to assess how successful an investment strategy relying on the communitys recommendations could have been. We detect buy and sell advice and identify the communitys most popular stocks, based on which we define a WSB portfolio. Our evaluation shows that this portfolio has grown approx. 200% over the last three years and approx. 480% over the last year, significantly outperforming the S&P500. The average short-term accuracy of buy and sell signals, in contrast, is not found to be significantly better than randomly or equally distributed buy decisions within the same time frame. However, we present a technique for estimating whether posts are proactive as opposed to reactive and show that by focusing on a subset of more promising buy signals, a trader could have made investments yielding higher returns than the broader market or the strategy of trusting all posted buy signals. Lastly, the analysis is also conducted specifically for the period before 2021 in order to factor out the effects of the GameStop hype of January 2021 - the results confirm the conclusions and suggest that the 2021 hype merely amplified pre-existing characteristics.
We show that the classical algebra of quaternions is a commutative $Z_2timesZ_2timesZ_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible.
Cryptography with quantum states exhibits a number of surprising and counterintuitive features. In a 2002 work, Barnum et al. argue that these features imply that digital signatures for quantum states are impossible (Barnum et al., FOCS 2002). In this work, we ask: can all forms of signing quantum data, even in a possibly weak sense, be completely ruled out? We give two results which shed significant light on this basic question. First, we prove an impossibility result for digital signatures for quantum data, which extends the result of Barnum et al. Specifically, we show that no nontrivial combination of correctness and security requirements can be fulfilled, beyond what is achievable simply by measuring the quantum message and then signing the outcome. In other words, only classical signature schemes exist. We then show a positive result: a quantum state can be signed with the same security guarantees as classically, provided that it is also encrypted with the public key of the intended recipient. Following classical nomenclature, we call this notion quantum signcryption. Classically, signcryption is only interesting if it provides superior performance to encypt-then-sign. Quantumly, it is far more interesting: it is the only signing method available. We develop as-strong-as-classical security definitions for quantum signcryption and give secure constructions based on post-quantum public-key primitives. Along the way, we show that a natural hybrid method of combining classical and quantum schemes can be used to upgrade a secure classical scheme to the fully-quantum setting, in a wide range of cryptographic settings including signcryption, authenticated encryption, and CCA security.
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