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This paper focuses on the dense uniform Li-Yorke chaos for linear operators on a Banach space. Some sufficient conditions and equivalent conditions are established under which the dynamical system is densely uniformly Li-Yorke chaotic. It is shown that there are plenty of densely uniformly Li-Yorke chaotic operators. For unilateral backward weighted shifts and bilateral backward weighted shifts on $\ell^p$, it is shown that Li-Yorke chaos is equivalent to dense uniform Li-Yorke chaos.

Cities are rarely flat, yet urban network analysis usually represents streets as planar graphs. This simplification affects modeled impedance, route choice, and the interpretation of accessibility, particularly where alternative paths differ in grade. This paper introduces Gridnberg ('grid-n-berg', grid and mountain), a topography-aware pedestrian routing dataset for New York City. The dataset enriches the NYCWalks network with vertex-level elevations derived from the New York City Planimetric Database. For each pedestrian-network geometry vertex, the workflow averages selected elevation observations within a 50 m radius, retains segments with complete vertex support, and uses direction-specific cumulative ascent and descent to calculate three routing costs: horizontal distance, a comfort-oriented slope score, and an accessibility-sensitive slope score. The release retains 313184 of 315577 source segments (99.24%). Gridnberg supports reproducible terrain-aware analysis, transparent scenario comparison, and improved pedestrian-network representations in New York and other cities.

We estimate the number of street vendors in New York City. First, we summarize the process by which vendors receive licenses and permits to operate legally in New York City. We then describe a survey that was administered by the Street Vendor Project while distributing coronavirus relief aid to vendors operating in New York City both with and without a license or permit. Finally, we review ratio estimation and develop a theoretical justification based on the theory of point processes. We find approximately 23,000 street vendors operate in New York City: 20,500 mobile food vendors and 2,400 general merchandise vendors. One third are located in just six ZIP Codes: 11368 (16%), 11372 (3%), and 11354 (3%) in North and West Queens and 10036 (5%), 10019 (4%), and 10001 (3%) in the Chelsea and Clinton neighborhoods of Manhattan. Our estimates suggest the American Community Survey misses the majority of New York City street vendors.

We examine the impact of New York City's congestion pricing program through automated analysis of traffic camera data. Our computer vision pipeline processes footage from over 900 cameras distributed throughout Manhattan and New York, comparing traffic patterns from November 2024 through the program's implementation in January 2025 until January 2026. We establish baseline traffic patterns and identify systematic changes in vehicle density across the monitored region.

The Berger-Ebin and York $L^2$-orthogonal decompositions of the vector space of symmetric bilinear differential two-forms are fundamental tools in global Riemannian geometry. In this paper, we investigate the structure of Ricci tensors on compact Riemannian manifolds, with a particular focus on compact Ricci almost solitons, utilizing both the Berger-Ebin and York $L^2$-orthogonal decompositions. In addition, we explore applications of the York $L^2$-orthogonal decomposition to the theory of submanifolds and to the study of harmonic maps between Riemannian manifolds.

The New York Times (NYT) games have found widespread popularity in recent years and reportedly account for an increasing fraction of the newspaper's readership. In this paper, we bring the computational lens to the study of New York Times games and consider four of them not previously studied: Letter Boxed, Pips, Strands and Tiles. We show that these games can be just as hard as they are fun. In particular, we characterize the hardness of several variants of computational problems related to these popular puzzle games. For Letter Boxed, we show that deciding whether an instance is solvable is in general NP-Complete, while in some parameter settings it can be done in polynomial time. Similarly, for Pips we prove that deciding whether a puzzle has a solution is NP-Complete even in some restricted classes of instances. We then show that one natural computational problem arising from Strands is NP-Complete in most parameter settings. Finally, we demonstrate that deciding whether a Tiles puzzle is solvable with a single, uninterrupted combo requires polynomial time.

We prove a \(Γ\)-convergence result for a diffeomorphism-natural discrete MDL-type functional to the Einstein-Hilbert action with the Gibbons-Hawking-York boundary term. On boundary-fitted, shape-regular meshes we establish interior and boundary blow-ups, identify the Carathéodory densities \(f_{\mathrm{in}}=α_0+α_1 R\) and \(f_{\mathrm{bdry}}=β_1 K\), and obtain the \(\liminf/\limsup\) bounds via a recovery sequence based on reflected Fermi smoothing. A boundary first-layer asymptotics shows that boundary cells contribute at order \(h^{d-1}\), yielding a global \(O(h)\) boundary remainder, while the interior remainder is \(O(h^2)\). The paper is foundational; Appendix~E specifies a reproducible protocol for rate checks and calibration of \(α_0,α_1,β_1\).

We establish complete characterizations of the notion of Li-Yorke chaos for weighted composition operators on $C_0(X)$ spaces and on $L^p(μ)$ spaces. As a consequence, we obtain simple characterizations of the Li-Yorke chaotic weighted shifts on $c_0$ and on $\ell^p$ ($1 \leq p < \infty$) that complement previously known results. We also investigate the notion of Li-Yorke chaos for weighted shifts on Fréchet sequence spaces. As applications, we obtain characterizations of the Li-Yorke chaotic weighted shifts on Köthe sequence spaces depending only on the entries of the Köthe matrix and the weights of the shift.

Fashion is a perpetual topic in human social life, and the mass has the penchant to emulate what large city residents and celebrities wear. Undeniably, New York City is such a bellwether large city with all kinds of fashion leadership. Consequently, to study what the fashion trends are during this year, it is very helpful to learn the fashion trends of New York City. Discovering fashion trends in New York City could boost many applications such as clothing recommendation and advertising. Does the fashion trend in the New York Fashion Show actually influence the clothing styles on the public? To answer this question, we design a novel system that consists of three major components: (1) constructing a large dataset from the New York Fashion Shows and New York street chic in order to understand the likely clothing fashion trends in New York, (2) utilizing a learning-based approach to discover fashion attributes as the representative characteristics of fashion trends, and (3) comparing the analysis results from the New York Fashion Shows and street-chic images to verify whether the fashion shows have actual influence on the people in New York City. Through the preliminary experiments o

It is an open problem whether a homeomorphism on a compact metric space satisfying that each proper pair is either positively or negatively Li--Yorke, called completely Li--Yorke chaotic, can have positive entropy. In the present paper, an affirmative answer to this question is given. In fact, for each homeomorphism $T$ with positive entropy such that each proper pair is not two-sided asymptotic, a completely Li--Yorke chaotic homeomorphism with positive entropy associated with the given homeomorphism can be constructed.

We introduce the definition of Li-Yorke chaos for the map f on G-spaces, and show G-Li-Yorke chaos is iterable for f. Li-Yorke chaos implies G-Li-Yorke chaos, while the converse is not true. Then we give a sufficient condition for f to be chaotic in the sense of G-Li-Yorke. Also, we prove that if f is G-transitive and there exists a common fixed point for f and all of the maps in G, then f is chaotic in the sense of G-Li-Yorke.

We investigate the notion of mean Li-Yorke chaos for operators on Banach spaces. We show that it differs from the notion of distributional chaos of type 2, contrary to what happens in the context of topological dynamics on compact metric spaces. We prove that an operator is mean Li-Yorke chaotic if and only if it has an absolutely mean irregular vector. As a consequence, absolutely Cesàro bounded operators are never mean Li-Yorke chaotic. Dense mean Li-Yorke chaos is shown to be equivalent to the existence of a dense (or residual) set of absolutely mean irregular vectors. As a consequence, every mean Li-Yorke chaotic operator is densely mean Li-Yorke chaotic on some infinite-dimensional closed invariant subspace. A (Dense) Mean Li-Yorke Chaos Criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained. Moreover, we construct an example of an invertible hypercyclic operator $T$ such that every nonzero vector is absolutely mean irregular for both $T$ and $T^{-1}$. Several other examples are also presented. Finally, mean Li-Yorke chaos is also investigated for $C_0$-semigroups of operators on Banach spaces.

Akin and Kolyada in 2003 [E. Akin, S. Kolyada, Li-Yorke sensitivity, Nonlinearity 16 (2003) 1421 - 1433] introduced the notion of Li-Yorke sensitivity. They proved that every weak mixing system $(X, T)$, where $X$ is a compact metric space and $T$ a continuous map of $X$ is Li-Yorke sensitive. An example of Li-Yorke sensitive system without weak mixing factors was given in [M. Čiklová, Li-Yorke sensitive minimal maps, Nonlinearity 19 (2006) 517 - 529] (see also [M. Čiklová-Mlíchová, Li-Yorke sensitive minimal maps II, Nonlinearity 22 (2009) 1569 -1573]). In their paper, Akin and Kolyada conjectured that every minimal system with a weak mixing factor, is Li-Yorke sensitive. We provide arguments supporting this conjecture though the proof seems to be difficult.

To reduce waste and improve public health and sanitation in New York City, innovative policies tailored to the city's unique urban landscape are necessary. The first program we propose is the Dumpster and Compost Accessibility Program. This program is affordable and utilizes dumpsters placed near fire hydrants to keep waste off the street without eliminating parking spaces. It also includes legal changes and the provision of compost bins to single/two-family households, which together will increase composting rates. The second program is the Pay-As-You-Throw Program. This requires New Yorkers living in single/two-family households to purchase stickers for each refuse bag they have collected by the city, incentivizing them to sort out compostable waste and recyclables. We conduct a weighted multi-objective optimization to determine the optimal sticker price based on the City's priorities. Roughly in proportion to the price, this program will increase diversion rates and decrease the net costs to New York City's Department of Sanitation. In conjunction, these two programs will improve NYC's diversion rates, eliminate garbage bags from the streets, and potentially save New York City m

Copyright infringement in frontier LLMs has received much attention recently due to the New York Times v. OpenAI lawsuit, filed in December 2023. The New York Times claims that GPT-4 has infringed its copyrights by reproducing articles for use in LLM training and by memorizing the inputs, thereby publicly displaying them in LLM outputs. Our work aims to measure the propensity of OpenAI's LLMs to exhibit verbatim memorization in its outputs relative to other LLMs, specifically focusing on news articles. We discover that both GPT and Claude models use refusal training and output filters to prevent verbatim output of the memorized articles. We apply a basic prompt template to bypass the refusal training and show that OpenAI models are currently less prone to memorization elicitation than models from Meta, Mistral, and Anthropic. We find that as models increase in size, especially beyond 100 billion parameters, they demonstrate significantly greater capacity for memorization. Our findings have practical implications for training: more attention must be placed on preventing verbatim memorization in very large models. Our findings also have legal significance: in assessing the relative m

One usually defines the Brown-York energy for a 2-surface embedded in a spacelike 3-slice as an integration of the mean curvature of the 2-surface isometrically embedded into the 3-slice, with a proper reference 3-space. We demonstrate that this naive definition is ill for stationary spacetimes. As an example, we investigate the Kerr-Newman spacetime in detail. We show that the naive definition of the Brown-York energy is not a component of the Brown-York boundary stress tensor, and thus deviates from the original idea of Brown and York. Furthermore, we present the exact form of the Brown-York energy for the Kerr-Newman spacetime with the proper reference.

The notion of Li-Yorke sensitivity has been studied extensively in the case of topological dynamical systems. We introduce a measurable version of Li-Yorke sensitivity, for nonsingular (and measure-preserving) dynamical systems, and compare it with various mixing notions. It is known that in the case of nonsingular dynamical systems, ergodic Cartesian square implies double ergodicity, which in turn implies weak mixing, but the converses do not hold in general, though they are all equivalent in the finite measure-preserving case. We show that for nonsingular systems, ergodic Cartesian square implies Li-Yorke measurable sensitivity, which in turn implies weak mixing. As a consequence we obtain that, in the finite measure-preserving case, Li-Yorke measurable sensitivity is equivalent to weak mixing. We also show that with respect to totally bounded metrics, double ergodicity implies Li-Yorke measurable sensitivity.

In this paper we characterize Li-Yorke chaotic composition operators on Orlicz spaces. Indeed some necessary and suffcient conditions are provided for Li-Yorke chaotic composition operator C' on the Orlicz space Lp. In some cases we have equivalent conditions for composition operators on Orlicz spaces to be Li-Yorke chaotic. The results of this paper extend similar results in Lp-spaces. Our results are essentially based on the results of [4].

We construct an infinite-dimensional compact metric space $X$, which is a closed subset of $\mathbb{S}\times\mathbb{H}$, where $\mathbb{S}$ is the unit circle and $\mathbb{H}$ is the Hilbert cube, and a skew-product map $F$ acting on $X$ such that $(X,F)$ is Li-Yorke sensitive but possesses at most countable scrambled sets. This disproves the conjecture of Akin and Kolyada that Li-Yorke sensitivity implies Li-Yorke chaos from the article [Akin E., Kolyada S., Li-Yorke sensitivity, Nonlinearity 16, (2003), 1421-1433].