We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime. Swap cosystolic expansion is defined by considering, for a given complex $X$, its faces complex $F^r X$, whose vertices are $r$-faces of $X$ and where two vertices are connected if their disjoint union is also a face in $X$. The faces complex $F^r X$ is a derandomizetion of the product of $X$ with itself $r$ times. The graph underlying $F^rX$ is the swap walk of $X$, known to have excellent spectral expansion. The swap cosystolic expansion of $X$ is defined to be the cosystolic expansion of $F^r X$. Our main result is a $\exp(-O(\sqrt r))$ lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders w
Urban bike-sharing systems require strategic station expansion to meet growing demand. Traditional allocation approaches rely on explicit demand modelling that may not capture the urban characteristics distinguishing successful stations. This study addresses the need to exploit patterns from existing stations to inform expansion decisions, particularly in data-constrained environments. We present a data-driven framework leveraging existing stations deemed desirable by operational metrics. A hybrid denoising autoencoder (HDAE) learns compressed latent representations from multi-source grid-level features (socio-demographic, built environment, and transport network), with a supervised classification head regularising the embedding space structure. Expansion candidates are selected via greedy allocation with spatial constraints based on latent-space similarity to existing stations. Evaluation on Trondheim's bike-sharing network demonstrates that HDAE embeddings yield more spatially coherent clusters and allocation patterns than raw features. Sensitivity analyses across similarity methods and distance metrics confirm robustness. A consensus-based procedure across multiple parametrisati
This article provides a primer on the spectral representation of random fields via the Karhunen-Loève Expansion (KLE). The goal is to bridge the gap between the theoretical foundations of the KLE and its application in computational modeling under uncertainty. We detail how tools from operator theory and probability are combined to analyze the convergence and optimality of the KLE. We also emphasize the associated computational and mathematical modeling considerations.
The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the smaller side of the cut. We study the Hypergraph Small Set Expansion problem, which, for a parameter $δ\in (0,1/2]$, asks to compute the cut having the least expansion while having at most $δ$ fraction of the vertices on the smaller side of the cut. We present two algorithms. Our first algorithm gives an $\tilde O(δ^{-1} \sqrt{\log n})$ approximation. The second algorithm finds a set with expansion $\tilde O(δ^{-1}(\sqrt{d_{\text{max}}r^{-1}\log r\, φ^*} + φ^*))$ in a $r$--uniform hypergraph with maximum degree $d_{\text{max}}$ (where $φ^*$ is the expansion of the optimal solution). Using these results, we also obtain algorithms for the Small Set Vertex Expansion problem: we get an $\tilde O(δ^{-1} \sqrt{\log n})$ approximation algorithm and an algorithm that finds a set with vertex expansion $O\left(δ^{-1}\sqrt{φ^V \log d_{\text{max}} } + δ^{-1} φ^V\right)$ (where $φ^V$ is the vertex expansion of the optimal solution). For $δ=1/2$, Hypergraph Small Set Expansion is equi
In recent years, high dimensional expanders have been found to have a variety of applications in theoretical computer science, such as efficient CSPs approximations, improved sampling and list-decoding algorithms, and more. Within that, an important high dimensional expansion notion is \emph{cosystolic expansion}, which has found applications in the construction of efficiently decodable quantum codes and in proving lower bounds for CSPs. Cosystolic expansion is considered with systems of equations over a group where the variables and equations correspond to faces of the complex. Previous works that studied cosystolic expansion were tailored to the specific group $\mathbb{F}_2$. In particular, Kaufman, Kazhdan and Lubotzky (FOCS 2014), and Evra and Kaufman (STOC 2016) in their breakthrough works, who solved a famous open question of Gromov, have studied a notion which we term ``parity'' expansion for small sets. They showed that small sets of $k$-faces have proportionally many $(k+1)$-faces that contain \emph{an odd number} of $k$-faces from the set. Parity expansion for small sets could be used to imply cosystolic expansion only over $\mathbb{F}_2$. In this work we introduce a stro
In this work several semantic approaches to concept-based query expansion and reranking schemes are studied and compared with different ontology-based expansion methods in web document search and retrieval. In particular, we focus on concept-based query expansion schemes, where, in order to effectively increase the precision of web document retrieval and to decrease the users browsing time, the main goal is to quickly provide users with the most suitable query expansion. Two key tasks for query expansion in web document retrieval are to find the expansion candidates, as the closest concepts in web document domain, and to rank the expanded queries properly. The approach we propose aims at improving the expansion phase for better web document retrieval and precision. The basic idea is to measure the distance between candidate concepts using the PMING distance, a collaborative semantic proximity measure, i.e. a measure which can be computed by using statistical results from web search engine. Experiments show that the proposed technique can provide users with more satisfying expansion results and improve the quality of web document retrieval.
We find a positive $e_I$-expansion for the chromatic symmetric function of KPKP graphs, which are graphs obtained by connecting a vertex in a complete graph with a vertex in the maximal clique of a lollipop graph by a path. This generalizes the positive $e_I$-expansion for the chromatic symmetric function of lollipops obtained by Tom, for that of KPK graphs obtained by Wang and Zhou, and as well for those of KKP graphs and PKP graphs obtained by Qi, Tang and Wang. As an application, we confirm the $e$-positivity of twinned lollipops. We also discover the first positive $e_I$-expansion for the chromatic symmetric function of kayak paddle graphs which are formed by connecting a vertex on a cycle and a vertex on another cycle with a path. This refines the $e$-positivity of kayak paddle graphs which was obtained by Aliniaeifard, Wang, and van Willigenburg.
In the rapidly evolving field of conversational AI, Ontology Expansion (OnExp) is crucial for enhancing the adaptability and robustness of conversational agents. Traditional models rely on static, predefined ontologies, limiting their ability to handle new and unforeseen user needs. This survey paper provides a comprehensive review of the state-of-the-art techniques in OnExp for conversational understanding. It categorizes the existing literature into three main areas: (1) New Intent Discovery, (2) New Slot-Value Discovery, and (3) Joint OnExp. By examining the methodologies, benchmarks, and challenges associated with these areas, we highlight several emerging frontiers in OnExp to improve agent performance in real-world scenarios and discuss their corresponding challenges. This survey aspires to be a foundational reference for researchers and practitioners, promoting further exploration and innovation in this crucial domain.
We consider first order expansions of convex penalized estimators in high-dimensional regression problems with random designs. Our setting includes linear regression and logistic regression as special cases. For a given penalty function $h$ and the corresponding penalized estimator $\hatβ$, we construct a quantity $η$, the first order expansion of $\hatβ$, such that the distance between $\hatβ$ and $η$ is an order of magnitude smaller than the estimation error $\|\hatβ - β^*\|$. In this sense, the first order expansion $η$ can be thought of as a generalization of influence functions from the mathematical statistics literature to regularized estimators in high-dimensions. Such first order expansion implies that the risk of $\hatβ$ is asymptotically the same as the risk of $η$ which leads to a precise characterization of the MSE of $\hatβ$; this characterization takes a particularly simple form for isotropic design. Such first order expansion also leads to inference results based on $\hatβ$. We provide sufficient conditions for the existence of such first order expansion for three regularizers: the Lasso in its constrained form, the lasso in its penalized form, and the Group-Lasso. T
In this paper, we demonstrate that locally, the $α^{\prime}$ expansion of a string propagating in AdS can be summed into a closed expression, where the $α'$ dependence is manifested. The T-dual of this sum exactly matches the expression controlling all genus expansion in the Goparkumar-Vafa formula, which in turn also matches the loop expansion of the Chern-Simons gauge theory. We therefore find an exact correspondence between the $α^{\prime}$ expansion for a string moving in AdS and the genus expansion of a string propagating in four dimensional flat spacetime. We are then able to give a closed form of the $α'$ expansion for all values of $\sqrt{α'}/R_{AdS}$. Moreover, the correspondence makes it possible to conjecture the exact $g_s$ dependence of the strongly coupled theories.
The generator of time-translations on the solution space of the wave equation on stationary spacetimes specialises to the square root of the Laplacian on Riemannian manifolds when the spacetime is ultrastatic. Its spectral analysis therefore constitutes a generalization of classical spectral geometry. If the spacetime is spatially compact the spectrum is discrete and admits a wave-trace expansion at time zero. A Weyl law for the eigenvalues and a wave-trace formula was shown in a previous paper and related to the geometry of the space of null-geodesics. In this paper we investigate the relation to heat kernel coefficients and residues of zeta functions in this context and compute the second non-zero term in the wave-trace expansion. This second coefficient is an analogue in the category of stationary spacetimes of the second heat kernel coefficient of the Laplace operator. The general formula is quite involved but reduces to the usual term involving the scalar curvature when specialised to ultra-static spacetimes.
To generate genuine random numbers, random number generators based on quantum theory are essential. However, ensuring that the process used to produce randomness meets desired security standards can pose challenges for traditional quantum random number generators. This thesis delves into Device Independent (DI) and Semi-Device Independent (semi-DI) protocols of randomness expansion, based on a minimal set of experimentally verifiable security assumptions. The security in DI protocols relies on the violation of Bell inequalities, which certify the quantum behavior of devices. The semi-DI protocols discussed in this thesis require the characterization of only one device - a power meter. These protocols exploit the fact that quantum states can be prepared such that they cannot be distinguished with certainty, thereby creating a randomness resource. In this study, we introduce enhanced DI and semi-DI protocols that surpass existing ones in terms of output randomness rate, security, or in some instances, both. Our analysis employs the Entropy Accumulation Theorem (EAT) to determine the extractable randomness for finite rounds. A notable contribution is the introduction of randomness exp
In this master's thesis, we introduce expansion systems as a general framework to describe a large variety of approximation algorithms, such as Taylor approximation, decimal expansion and continued fraction. We consider some basic properties of expansion systems, and also study criteria for convergence. Further, we introduce the notion of isomorphisms between expansion systems. In the appendix, we discuss another class of expansion systems, which we call approximation systems. Many claims of convergence in this appendix, remain to be proven.
We discuss a real-valued expansion of any Hermitian operator defined in a Hilbert space of finite dimension N, where N is a prime number, or an integer power of a prime. The expansion has a direct interpretation in terms of the operator expectation values for a set of complementary bases. The expansion can be said to be the complement of the discrete Wigner function. We expect the expansion to be of use in quantum information applications since qubits typically are represented by a discrete, and finite-dimensional physical system of dimension N=2^p, where p is the number of qubits involved. As a particular example we use the expansion to prove that an intermediate measurement basis (a Breidbart basis) cannot be found if the Hilbert space dimension is 3 or 4.
Can one inject new concepts into an already trained generative model, while respecting its existing structure and knowledge? We propose a new task - domain expansion - to address this. Given a pretrained generator and novel (but related) domains, we expand the generator to jointly model all domains, old and new, harmoniously. First, we note the generator contains a meaningful, pretrained latent space. Is it possible to minimally perturb this hard-earned representation, while maximally representing the new domains? Interestingly, we find that the latent space offers unused, "dormant" directions, which do not affect the output. This provides an opportunity: By "repurposing" these directions, we can represent new domains without perturbing the original representation. In fact, we find that pretrained generators have the capacity to add several - even hundreds - of new domains! Using our expansion method, one "expanded" model can supersede numerous domain-specific models, without expanding the model size. Additionally, a single expanded generator natively supports smooth transitions between domains, as well as composition of domains. Code and project page available at https://yotamnitz
We report on new 5-GHz VLA radio observations of the pulsar-powered supernova remnant G21.5-0.9. These observations have allowed us to make a high-quality radio image of this remnant with a resolution of ~0.7". It has a filamentary structure similar to that seen in the Crab Nebula. Radio structure suggestive of the torus seen around the Crab pulsar is tentatively identified. We also compared the new image with one taken ~15 yr earlier at 1.5 GHz, both to find the expansion speed of the remnant and to make a spectral index image. Between 1991 and 2006, we find that the average expansion rate of the remnant is 0.11 +/- 0.02 %/year, corresponding, for a distance of 5 kpc, to a speed of 910 +/- 160 km/s wrt. the centre of the nebula. Assuming undecelerated expansion, this expansion speed implies that the age of G21.5-0.9 is 870 (+200,-150) yr, which makes PSR J1833-1034 one of the youngest, if not the youngest, known pulsars in the Galaxy.
For a free--field flat monodromy defect, a formula for the finite part of the correlator is obtained as a double power series in $(1-x)$ and $(1-\ol x)$ where $x$ and $\ol x$ are lightcone coordinates. It takes the particular form of a series in $(1-x)$ with coefficients finite sums of hypergeometric functions of $1-\ol x$ and is identified with a bulk block expansion. A simple expression for the coefficient of the $(1-x)^n(1-\ol x)^m$ term is thereby found as an explicit function of the flux and dimension. Some typical examples are presented.A transformation allows the bulk block expansion to be written as an Appell $F_3$ function which has simplifying consequences.
In this paper, we provide a systematic methodology for calculating multi-order asymptotic expansion of blow-up solutions near blow-up for autonomous ordinary differential equations (ODEs). Under the specific form of the principal term of blow-up solutions for a class of vector fields, we extract algebraic objects determining all possible orders in the asymptotic expansions. Examples for calculating concrete multi-order asymptotic expansions of blow-up solutions are finally collected.
Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical inte
We modify a Lie algebra expansion method recently introduced for the (2+1)-dimensional kinematical algebras so as to work for higher dimensions. This new improved and geometrical procedure is applied to expanding the (3+1)-dimensional Galilei algebra and leads to its physically meaningful `expanded' neighbours. One expansion gives rise to the Poincare algebra, introducing a curvature $-1/c^2$ in the flat Galilean space of worldlines, while keeping a flat spacetime which changes from absolute to relative time in the process. This formally reverses, at a Lie algebra level, the well known non-relativistic contraction $c\to \infty$ that goes from the Poincare group to the Galilei one; this expansion is done in an explicit constructive way. The other possible expansion leads to the Newton-Hooke algebras, endowing with a non-zero spacetime curvature $\pm 1/τ^2$ the spacetime, while keeping a flat space of worldlines.