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In this paper, we present an error estimate of a second-order linearized finite element (FE) method for the 2D Navier-Stokes equations with variable density. In order to get error estimates, we first introduce an equivalent form of the original system. Later, we propose a general BDF2-FE method for solving this equivalent form, where the Taylor-Hood FE space is used for discretizing the Navier-Stokes equations and conforming FE space is used for discretizing density equation. We show that our scheme ensures discrete energy dissipation. Under the assumption of sufficient smoothness of strong solutions, an error estimate is presented for our numerical scheme for variable density incompressible flow in two dimensions. Finally, some numerical examples are provided to confirm our theoretical results.

Future wireless networks and sensing systems will benefit from access to large chunks of spectrum above 100 GHz, to achieve terabit-per-second data rates in 6th Generation (6G) cellular systems and improve accuracy and reach of Earth exploration and sensing and radio astronomy applications. These are extremely sensitive to interference from artificial signals, thus the spectrum above 100 GHz features several bands which are protected from active transmissions under current spectrum regulations. To provide more agile access to the spectrum for both services, active and passive users will have to coexist without harming passive sensing operations. In this paper, we provide the first, fundamental analysis of Radio Frequency Interference (RFI) that large-scale terrestrial deployments introduce in different satellite sensing systems now orbiting the Earth. We develop a geometry-based analysis and extend it into a data-driven model which accounts for realistic propagation, building obstruction, ground reflection, for network topology with up to $10^5$ nodes in more than $85$ km$^2$. We show that the presence of harmful RFI depends on several factors, including network load, density and topology, satellite orientation, and building density. The results and methodology provide the foundation for the development of coexistence solutions and spectrum policy towards 6G.

In this paper we present the initial screening order problem, a crucial step within candidate screening. It involves a human-like screener with an objective to find the first k suitable candidates rather than the best k suitable candidates in a candidate pool given an initial screening order. The initial screening order represents the way in which the human-like screener arranges the candidate pool prior to screening. The choice of initial screening order has considerable effects on the selected set of k candidates. We prove that under an unbalanced candidate pool (e.g., having more male than female candidates), the human-like screener can suffer from uneven efforts that hinder its decision-making over the protected, under-represented group relative to the non-protected, over-represented group. Other fairness results are proven under the human-like screener. This research is based on a collaboration with a large company to better understand its hiring process for potential automation. Our main contribution is the formalization of the initial screening order problem which, we argue, opens the path for future extensions of the current works on ranking algorithms, fairness, and automation for screening procedures.

In a well-calibrated risk prediction model, the average predicted probability is close to the true event rate for any given subgroup. Such models are reliable across heterogeneous populations and satisfy strong notions of algorithmic fairness. However, the task of auditing a model for strong calibration is well-known to be difficult -- particularly for machine learning (ML) algorithms -- due to the sheer number of potential subgroups. As such, common practice is to only assess calibration with respect to a few predefined subgroups. Recent developments in goodness-of-fit testing offer potential solutions but are not designed for settings with weak signal or where the poorly calibrated subgroup is small, as they either overly subdivide the data or fail to divide the data at all. We introduce a new testing procedure based on the following insight: if we can reorder observations by their expected residuals, there should be a change in the association between the predicted and observed residuals along this sequence if a poorly calibrated subgroup exists. This lets us reframe the problem of calibration testing into one of changepoint detection, for which powerful methods already exist. We begin with introducing a sample-splitting procedure where a portion of the data is used to train a suite of candidate models for predicting the residual, and the remaining data are used to perform a score-based cumulative sum (CUSUM) test. To further improve power, we then extend this adaptive CUSUM test to incorporate cross-validation, while maintaining Type I error control under minimal assumptions. Compared to existing methods, the proposed procedure consistently achieved higher power in simulation studies and more than doubled the power when auditing a mortality risk prediction model.

Because "out-of-the-box" large language models are capable of generating a great deal of objectionable content, recent work has focused on aligning these models in an attempt to prevent undesirable generation. While there has been some success at circumventing these measures -- so-called "jailbreaks" against LLMs -- these attacks have required significant human ingenuity and are brittle in practice. In this paper, we propose a simple and effective attack method that causes aligned language models to generate objectionable behaviors. Specifically, our approach finds a suffix that, when attached to a wide range of queries for an LLM to produce objectionable content, aims to maximize the probability that the model produces an affirmative response (rather than refusing to answer). However, instead of relying on manual engineering, our approach automatically produces these adversarial suffixes by a combination of greedy and gradient-based search techniques, and also improves over past automatic prompt generation methods. Surprisingly, we find that the adversarial prompts generated by our approach are quite transferable, including to black-box, publicly released LLMs. Specifically, we train an adversarial attack suffix on multiple prompts (i.e., queries asking for many different types of objectionable content), as well as multiple models (in our case, Vicuna-7B and 13B). When doing so, the resulting attack suffix is able to induce objectionable content in the public interfaces to ChatGPT, Bard, and Claude, as well as open source LLMs such as LLaMA-2-Chat, Pythia, Falcon, and others. In total, this work significantly advances the state-of-the-art in adversarial attacks against aligned language models, raising important questions about how such systems can be prevented from producing objectionable information. Code is available at github.com/llm-attacks/llm-attacks.

The Weighted Path Order of Yamada is a powerful technique for proving termination. It is also supported by CeTA, a certifier for checking untrusted termination proofs. To be more precise, CeTA contains a verified function that computes for two terms whether one of them is larger than the other for a given WPO, i.e., where all parameters of the WPO have been fixed. The problem of this verified function is its exponential runtime in the worst case. Therefore, in this work we develop a polynomial time implementation of WPO that is based on memoization. It also improves upon an earlier verified implementation of the Recursive Path Order: the RPO-implementation uses full terms as keys for the memory, a design which simplified the soundness proofs, but has some runtime overhead. In this work, keys are just numbers, so that the lookup in the memory is faster. Although trivial on paper, this change introduces some challenges for the verification task.

We study the reverse shortest path problem on disk graphs in the plane. In this problem we consider the proximity graph of a set of $n$ disks in the plane of arbitrary radii: In this graph two disks are connected if the distance between them is at most some threshold parameter $r$. The case of intersection graphs is a special case with $r=0$. We give an algorithm that, given a target length $k$, computes the smallest value of $r$ for which there is a path of length at most $k$ between some given pair of disks in the proximity graph. Our algorithm runs in $O^*(n^{5/4})$ randomized expected time, which improves to $O^*(n^{6/5})$ for unit disk graphs, where all the disks have the same radius. Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and $k$ is replaced by a target weight $w$; that is, we seek a path whose length is at most $w$. In other variants, we want to optimize a parameter different from $r$, such as a scale factor of the radii of the disks. The main technique for the decision version of the problem (determining whether the graph with a given $r$ has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra's algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique of [4].

Money transfer is an abstraction that realizes the core of cryptocurrencies. It has been shown that, contrary to common belief, money transfer in the presence of Byzantine faults can be implemented in asynchronous networks and does not require consensus. Nonetheless, existing implementations of money transfer still require a quadratic message complexity per payment, making attempts to scale hard. In common blockchains, such as Bitcoin and Ethereum, this cost is mitigated by payment channels implemented as a second layer on top of the blockchain allowing to make many off-chain payments between two users who share a channel. Such channels only require on-chain transactions for channel opening and closing, while the intermediate payments are done off-chain with constant message complexity. But payment channels in-use today require synchrony, therefore they are inadequate for asynchronous money transfer systems. In this paper, we provide a series of possibility and impossibility results for payment channels in asynchronous money transfer systems. We first prove a quadratic lower bound on the message complexity of on-chain transfers. Then, we explore two types of payment channels, unidirectional and bidirectional. We define them as shared memory abstractions and prove that in certain cases they can be implemented as a second layer on top of an asynchronous money transfer system whereas in other cases it is impossible.

The utilization of finite field multipliers is pervasive in contemporary digital systems, with hardware implementation for bit parallel operation often necessitating millions of logic gates. However, various digital design issues, whether inherent or stemming from soft errors, can result in gate malfunction, ultimately can cause gates to malfunction, which in turn results in incorrect multiplier outputs. Thus, to prevent susceptibility to error, it is imperative to employ a reliable finite field multiplier implementation that boasts a robust fault detection capability. In order to achieve the best fault detection performance for finite field detection performance for finite field multipliers while maintaining a low-complexity implementation, this study proposes a novel fault detection scheme for a recent bit-parallel polynomial basis over GF(2m). The primary concept behind the proposed approach is centered on the implementation of an efficient BCH decoder that utilize Berlekamp-Rumsey-Solomon (BRS) algorithm and Chien-search method to effectively locate errors with minimal delay. The results of our synthesis indicate that our proposed error detection and correction architecture for a 45-bit multiplier with 5-bit errors achieves a 37% and 49% reduction in critical path delay compared to existing designs. Furthermore, a 45-bit multiplicand with five errors has hardware complexity that is only 80%, which is significantly less complex than the most advanced BCH-based fault recognition techniques, such as TMR, Hamming's single error correction, and LDPC-based methods for finite field multiplication which is desirable in constrained applications, such as smart cards, IoT devices, and implantable medical devices.

Graph Neural Networks (GNNs) have been successfully used in many problems involving graph-structured data, achieving state-of-the-art performance. GNNs typically employ a message-passing scheme, in which every node aggregates information from its neighbors using a permutation-invariant aggregation function. Standard well-examined choices such as the mean or sum aggregation functions have limited capabilities, as they are not able to capture interactions among neighbors. In this work, we formalize these interactions using an information-theoretic framework that notably includes synergistic information. Driven by this definition, we introduce the Graph Ordering Attention (GOAT) layer, a novel GNN component that captures interactions between nodes in a neighborhood. This is achieved by learning local node orderings via an attention mechanism and processing the ordered representations using a recurrent neural network aggregator. This design allows us to make use of a permutation-sensitive aggregator while maintaining the permutation-equivariance of the proposed GOAT layer. The GOAT model demonstrates its increased performance in modeling graph metrics that capture complex information, such as the betweenness centrality and the effective size of a node. In practical use-cases, its superior modeling capability is confirmed through its success in several real-world node classification benchmarks.