1,720,989 research outputs found
Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic
No full textWe study the scaling asymptotics of the eigenspace projection kernels Π[subscript ℏ,E](x,y) of the isotropic Harmonic Oscillator [^ over H][subscript ℏ]=−ℏ[superscript 2]Δ+|x|[superscript 2] of eigenvalue E=ℏ(N+d/2) in the semi-classical limit ℏ→0. The principal result is an explicit formula for the scaling asymptotics of Π[subscript ℏ,E](x,y) for x, y in a ℏ[superscript 2/3] neighborhood of the caustic C[subscript E] as ℏ→0. The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as ℏ→0. In previous work we proved that the density of zeros of Gaussian random eigenfunctions of [^ over H][subscript ℏ] have different orders in the Planck constant ℏ in the allowed and forbidden regions: In the allowed region the density is of order ℏ[superscript −1] while it is ℏ[superscript −1/2] in the forbidden region. Our main result on nodal sets is that the density of zeros is of order ℏ[superscript −2/3] in an ℏ[superscript 2/3] -tube around the caustic. This tube radius is the ‘critical radius’. For annuli of larger inner and outer radii ℏ[superscript α] with 0<α<2/3 we obtain density results that interpolate between this critical radius result and our prior ones in the allowed and forbidden region. We also show that the Hausdorff (d−2)-dimensional measure of the intersection of the nodal set with the caustic is of order ℏ[superscript −2/3] .National Science Foundation (U.S.) (Grant DMS-1400822
C[superscript ∞] Scaling Asymptotics for the Spectral Projector of the Laplacian
No full textThis article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near any non-self-focal point, the scaling limit of the spectral projector of the Laplacian onto frequency windows of constant size is a normalized Bessel function depending only on n. Keywords: Spectral projector, Pointwise Weyl Law, Scaling limits, Laplace eigenfunctions, Non-self-focal point
Fixed frequency eigenfunction immersions and supremum norms of random waves
No full textA compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions
Style Transfer in Poetry
No full textThe goal of this project is to approximate style transfer in poetry, where
style is defined as the particular flavor of a certain author. To make this more
concrete we enforce a model on poems - assume that there are two separate
attributes which make up a poem. The first attribute we call ”content” - this
can be thought of as the attribute which allows us to identify text as a poem
about a certain topic. This is common among all poems - we will sharpen this
notion of ”common” later. The second attribute is ”style” - this is the attribute
which separates poets from one another. By separating content from style we
make it possible to transfer poetry written in one style to another.
To use the parlance of machine learning, we interpret this as the visible
output (the poem) being determined by a ”hidden” layer which can be used to
generate the visible output. If we could learn a function mapping the visible
layer to the hidden layer and vice versa, theoretically we can interpret the attributes of any poem as well as generate a poem given some attributes. In the
autoencoder model the map from visible to hidden is known as the encoder,
and the map from hidden to visible is the decoder. In this particular use case,
because style is known for each poem the style transfer takes the following form:
the encoder maps poem input x and style input y to a hidden layer z, and the
decoder takes hidden layer z and desired style y
′
to generate poem x
′ which has
been transferred from style y to style y
′
.
In learning these mappings, we take advantage of the fact that we should be
able to reconstruct any sample by decoding the samples encoding. Therefore
our loss function will minimize reconstruction error - the difference between the
real and reconstructed samples. In addition to this, we can add constraints - one
important one is the distribution of the hidden layer. In the variational autoencoder (VAE) we assume a prior distribution and add the KL-divergence between
the implied and prior distributions to the optimizer. The KL-divergence term
serves to keep our hidden layer close to the prior while learning enough information to reconstruct the sample.
To make style transfer meaningful, we must ensure that all authors are the
same apart from their stylistic attributes, or that the content is the same subject
to some variation in intent (for example: every author writing about the moon
should write the same poem independent of style, but if one is writing about
the sun there can be variation between the two poems). Formalizing this, we
enforce that the content has the same distribution across styles - they do not
have to be exactly the same but they must be drawn from the same underlying
distribution. One way is to use the variational method described above - simply assuming a prior distribution and forcing content toward that distribution.
The authors in (Shen et al 2017) argue, however, that using this makes the content distribution too simple and most of the complexity will be forced on
the decoder. They propose the cross-aligned autoencoder, which introduces adversarial discriminators to distinguish between the content distributions of real
and transferred samples. In this method we align the two content distributions
by forcing real and transferred samples of the same style together.
To evaluate, we will use both qualitative and quantitative analysis. Quantitatively we have two goals: to evaluate the generative power of the text and the
style transfer. For the former we look at reconstruction error as well as some
other text similarity measures. For the latter, we can check if transferred samples classify in the desired style, or at least show movement toward the desired
style (this concept is formalized later)
The Road to Zcash: A Survey of Zero-Knowledge Technology
No full textZcash is the first widespread application of zero-knowledge succinct non-interactive arguments of knowledge, or zk-SNARKs for short. While research on zero-knowledge proofs began in the 1980s, we noticed a lack of accessible literature to help catch up in a rapidly-expanding field. This work aims to fill in that gap by introducing zero-knowledge and ultimately reviewing the Zcash protocol. We discuss the security and anonymity benefits of Zcash as well as its shortcomings in practice
Market Making and Speculative Trading Feasibility in Play-to-Earn Crypto Games
No full textAs memes get sold for millions of dollar NFTs and governments seek to gain tighter control over decentralized finance platforms, we are witnessing a race between new and increasingly fantastic ways to make money and regulation from governments. Play-to-earn cryptocurrency games are another step in the cycle, with players trading NFT tokens and participating in gameplay for cryptocurrency, often a game’s unique cryptocurrency that can then be sold on external exchanges for more popular coins, such as Bitcoin or Ethereum. More and more, these games are becoming a way to supplement or replace incomes in third world countries, yet there is a significant lack of research regarding the stability and sustainability of these games. Furthermore, interest and hype over NFT products and play-to-earn crypto games has persisted but with little academic analysis on the underlying markets. In this paper, we seek to analyze NFT marketplaces in play-to-earn crypto games by proposing in-game trading strategies
Maximizing REITurns: A Multi-Armed Bandit Approach to Optimizing Trading Strategies on Real Estate Investment Trusts
No full textThe overarching goal of this thesis was to research and develop an effective trading strategy
specific to REIT stocks. Real Estate Investment Trusts, commonly referred to as REITs, are
companies that own incoming generating real estate properties. REIT stocks are common
investments for investors that are seeking larger dividends and therefore do not expect high
growth. The topic of REIT stock movement and the application of trading strategies in
REIT stocks has been relatively under explored in prior academic research. These topics
have been traditionally studied in common stocks, however, the REIT market provides interesting qualities that may have been overlooked in the past. In this paper, we will experiment
with applying reinforcement learning techniques to optimize the use of traditional trading
algorithms on REIT stocks. To do this, we research and apply methods from Multi-Armed
Bandit problems. We set a Multi-Armed Bandit problem where the arms exist as trading
strategies and the rewards of each arm are defined as the returns from that trading strategy. The most complex issue in transforming this into a Multi-Armed Bandit problem was
addressing the non-stationarity of returns in stocks. After a thorough analysis, we focus on
two algorithms that address the seasonality of returns in each trading strategy. We develop
multiple automated trading models and we apply these models to sets of historical data to
determine if the Multi-Armed Bandit algorithms carry any efficacy in REIT stock trading
Retrieval-Based Systems for Open-domain Question Answering
No full textQuestion answering is a popular natural language processing task in which a question answering (QA) system is responsible for automatically producing relevant and factually accurate natural language answers to questions formulated in natural language. Open-domain QA refers to the general setting in which the QA system is not provided the evidence required to answer the questions and in which the questions are not restricted to any particular domain. This is a challenging task as it requires open-domain QA systems to have access to a vast amount of factual knowledge pertaining to a diverse range of topics and real-world entities. We first motivate the need for retrieval in open-domain QA, identifying several distinct advantages it has over competing retrieval-free methods. We then survey the most prominent and influential approaches for retrieval-based open-domain QA, focusing on the popular retriever-reader framework. In particular, we describe and compare the various designs proposed for retriever and reader models and detail the methods to train and evaluate such open-domain QA systems. We finally discuss some interesting novel directions for future research in retrieval-based open-domain QA
Gamma Squeezing: An analysis of the market impact of institutional equity options market makers on the underlying market
No full textWith the increase in retail investors expressing opinions on companies through equity options during the peak of COVID-19 activity, as perhaps best illustrated by increasing activity on forums such as the Reddit community wallstreetbets, derivatives exchanges have increased the frequency with which equity options expire. For instance, Tesla options have weekly expiries, the Russell and Nasdaq indices have expiries on Mondays, Wednesdays, and Fridays, and the S&P has daily expiries. Consequently, it is of great interest to people who participate in equities markets to understand how option expiry affects equity price movement. This thesis aims to demonstrate that as a result of institutional options market makers hedging out their risk to the price of the underlying asset, equity prices with options either experience large moves or virtually no move in the last few hours of the afternoon on the day of expiry. One well-studied specific instance of this phenomenon is pinning, where on the day of expiry, the equity price tends to close very near some option strike price. Outside of being interesting for those who wish to participate in equities trading, such a result offers one possible refutation to the efficient market hypothesis, since such a price movement in equities is not reflective of any information or sentiment about the company, but rather an expression of the risk aversion of certain market participants in a correlated product
Has Social Media given stock markets a soft side? An analysis of the influence of Twitter Sentiment on stock markets prices and volatility
No full textThis thesis aims to better understand social media’s influence on the stock market.
Social media has transformed almost every aspect of the world we know today, and
rests in the hands of the very consumers that drive markets. Social media is in a
unique position to influence consumer markets and has given rise to the idea that
the stock market can be ’emotional’ and therefore, moved by consumer opinion. The
world saw this firsthand when Reddit, a social media company, raised a stock by nearly
two hundred percent in 24 hours.Ultimately, this thesis will consist of a literature
review of existing studies, a discussion of gaps in the literature, and an attempt to
fill some of these gaps.
To accomplish this, I will implement and test the efficiency of natural language
processing (NLP) in collecting qualitative data and predicting future stock prices. I
will use a bank of ’positive’ and ’negative’ words to comb through financial articles,
tweets, and Reddit posts and compare them to stock market indicators. Given the
limited research into the efficacy of NLP on stock market prediction, this code will be
an attempt to 1) make advancements in this area of research, and 2) better quantify
how impactful the words of the consumer are on the market
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