Text indexing is a fundamental and well-studied problem. Classic solutions either replace the original text with a compressed representation, e.g., the FM-index and its variants, or keep it uncompressed but attach some redundancy - an index - to accelerate matching. The former solutions thus retain excellent compressed space, but are slow in practice. The latter approaches, like the suffix array, instead sacrifice space for speed. We show that efficient text indexing can be achieved using just a small extra space on top of the original text, provided that the query patterns are sufficiently long. More specifically, we develop a new indexing paradigm in which a sketch of a query pattern is first matched against a sketch of the text. Once candidate matches are retrieved, they are verified using the original text. This paradigm is thus universal in the sense that it allows us to use any solution to index the sketched text, like a suffix array, FM-index, or r-index. We explore both the theory and the practice of this universal framework. With an extensive experimental analysis, we show that, surprisingly, universal indexes can be constructed much faster than their unsketched counterpar
We propose a new supersymmetry index for the D1-D5 CFT for $T^4$, relevant to the AdS$_3$/CFT$_2$ correspondence. In a novel formulation of symmetric orbifold CFTs based on the Schur-Weyl duality, we show how this index can be naturally described and its protection is argued based on the detailed nature of exactly marginal operators in these theories. This index is a one-parameter generalization of the standard index and gives more fine-grained information about the structure of microstates than previously available. We demonstrate precise matching of the new index between supergravity and CFT below the black-hole threshold, where the standard index -- the modified elliptic genus -- is trivial. Above the threshold, we uncover a decomposition of black-hole microstates into distinct sectors, invisible to the modified elliptic genus.
Important game-changer economic events and transformations cause uncertainties that may affect investment decisions, capital flows, international trade, and macroeconomic variables. One such major transformation is Brexit, which refers to the multiyear process through which the UK withdrew from the EU. This study develops and uses a new Brexit-Related Uncertainty Index (BRUI). In creating this index, we apply Text Mining, Context Window, Natural Language Processing (NLP), and Large Language Models (LLMs) from Deep Learning techniques to analyse the monthly country reports of the Economist Intelligence Unit from May 2012 to January 2025. Additionally, we employ a standard vector autoregression (VAR) analysis to examine the model-implied responses of various macroeconomic variables to BRUI shocks. While developing the BRUI, we also create a complementary COVID-19 Related Uncertainty Index (CRUI) to distinguish the uncertainties stemming from these distinct events. Empirical findings and comparisons of BRUI with other earlier-developed uncertainty indexes demonstrate the robustness of the new index. This new index can assist British policymakers in measuring and understanding the impa
For a (molecular) graph $G$ and any real number $α e 0$ , the zero-order general Randić index , denote by $^0R_α$, is defined by the following equation: \begin{align*} {^0R_α} (G) =\sum_{v\in G}d_G (v) ^α (α\in \mathbb{R}-\left\{0\right\}) . \end{align*} In this paper, we use this index to give sufficient conditions for a graph $G$ to satisfy the Hamiltonian (or $k$-Hamiltonian) property, and show that none of these conditions can be dropped. Finally we give similar results for the case when $G$ is a balanced bipartite graph.
In this paper, topological indices play a significant role in the analysis of caterpillar trees, especially due to their applications in chemical graph theory. We presented a study on topological indices related to the Sigma index, which we carefully selected on caterpillar trees with multiple levels. Through the election of these topological indices, we provide efficient models of these indices on caterpillar trees, as it is known that the Albertson index is the basis on which most of the topological indices are built and we have shown this through the close correlation between the Albertson's index and the Sigma index.
Portfolio optimization under cardinality constraints transforms the classical Markowitz mean-variance problem from a convex quadratic problem into an NP-hard combinatorial optimization problem. This paper introduces a novel approach using THRML (Thermodynamic HypergRaphical Model Library), a JAX-based library for building and sampling probabilistic graphical models that reformulates index tracking as probabilistic inference on an Ising Hamiltonian. Unlike traditional methods that seek a single optimal solution, THRML samples from the Boltzmann distribution of high-quality portfolios using GPU-accelerated block Gibbs sampling, providing natural regularization against overfitting. We implement three key innovations: (1) dynamic coupling strength that scales inversely with market volatility (VIX), adapting diversification pressure to market regimes; (2) rebalanced bias weights prioritizing tracking quality over momentum for index replication; and (3) sector-aware post-processing ensuring institutional-grade diversification. Backtesting on a 100-stock S and P 500 universe from 2023 to 2025 demonstrates that THRML achieves 4.31 percent annualized tracking error versus 5.66 to 6.30 perce
Index structures are a building block of query processing and computer science in general. Since the dawn of computer technology there have been index structures. And since then, a myriad of index structures are being invented and published each and every year. In this paper we argue that the very idea of "inventing an index" is a misleading concept in the first place. It is the analogue of "inventing a physical query plan". This paper is a paradigm shift in which we propose to drop the idea to handcraft index structures (as done for binary search trees over B-trees to any form of learned index) altogether. We present a new automatic index breeding framework coined Genetic Generic Generation of Index Structures (GENE). It is based on the observation that almost all index structures are assembled along three principal dimensions: (1) structural building blocks, e.g., a B-tree is assembled from two different structural node types (inner and leaf nodes), (2) a couple of invariants, e.g., for a B-tree all paths have the same length, and (3) decisions on the internal layout of nodes (row or column layout, etc.). We propose a generic indexing framework that can mimic many existing index
Automatic speech recognition (ASR) has witnessed remarkable progress in recent years, largely driven by the emergence of LLM-based ASR paradigm. Despite their strong performance on a variety of open-source benchmarks, existing LLM-based ASR systems still suffer from two critical limitations. First, they are prone to hallucination errors, often generating excessively long and repetitive outputs that are not well grounded in the acoustic input. Second, they provide limited support for flexible and fine-grained contextual customization. To address these challenges, we propose Index-ASR, a large-scale LLM-based ASR system designed to simultaneously enhance robustness and support customizable hotword recognition. The core idea of Index-ASR lies in the integration of LLM and large-scale training data enriched with background noise and contextual information. Experimental results show that our Index-ASR achieves strong performance on both open-source benchmarks and in-house test sets, highlighting its robustness and practicality for real-world ASR applications.
We use the well established duality of topological gravity to a double scaled matrix model with the Airy spectral curve to define what we refer to as topological gravity with arbitrary Dyson index $\upbeta$ ($\upbeta$ topological gravity). On the matrix model side this is an interpolation in the Dyson index between the Wigner-Dyson universality classes, on the gravity side it can be thought of as interpolating between orientable and unorientable manifolds in the gravitational path integral, opening up the possibility to study moduli space volumes of manifolds ``in between''. Using the perturbative loop equations we study correlation functions of this theory and prove several structural properties, having clear implications for the generalised moduli space volumes. Additionally we give a geometric interpretation of these properties using the generalisation to arbitrary Dyson index of the recently found Mirzakhani-like recursion for unorientable surfaces. Using these properties, we investigate whether $\upbeta$-topological gravity is quantum chaotic in the sense of the Bohigas-Giannoni-Schmit conjecture. Along the way we answer this question for the symplectic Wigner-Dyson class, not
Let $G$ be a connected, real semisimple Lie group. Let $K<G$ be maximal compact, and let $Γ< G$ be discrete and such that $Γ\backslash G$ has finite volume. If the real rank of $G$ is $1$ and $Γ$ is torsion-free, then Barbasch and Moscovici obtained an index theorem for Dirac operators on the locally symmetric space $Γ\backslash G/K$. We obtain a higher version of this, using an index of Dirac operators on $G/K$ in the $K$-theory of an algebra on which the conjugation-invariant terms in Barbasch and Moscovici's index theorem define continuous traces. The resulting index theorems also apply when $Γ$ has torsion. The cases of these index theorems for traces defined by semisimple orbital integrals extend to Song and Tang's higher orbital integrals, and yield nonzero and computable results even when $\operatorname{rank}(G)> \operatorname{rank}(K)$, or the real rank of $G$ is larger than $1$.
The paper is devoted to an analogue of Atiyah-Bott-Singer index theorem for families of self-adjoint elliptic (i.e. satisfying the Shapiro-Lopatinskii condition) local boundary problems of order 1. The proofs are based on classical topological and pseudo-differential methods, but in the self-adjoint case one encounters some new phenomena. The topological index is defined following Atiyah-Bott, but in the self-adjoint case one encounters an obstruction not present in the classical situation. The analytical index is defined with the help of author's approach arXiv:2111.15081, which generalized the one of Atiyah-Singer. On the analytic index side one encounters an obstruction to the realization of symbols by self-adjoint boundary problems, similar to the obstruction to defining the topological index. As an application, we generalize results of Gorokhovsky and Lesch arXiv:1310.0210. In the first version of this paper the index theorem was proved only under an additional technical assumption. A theory of multiplicative properties of symbols and operators, developed in the second version, allows to remove this assumption.
In 2006, Budur, Mustaţǎ and Saito introduced the notion of Bernstein-Sato polynomial of an arbitrary scheme of finite type over fields of characteristic zero. Because of the strong monodromy conjecture, it should have a corresponding picture on the arithmetic side of ideals in polynomial rings. In this paper, we try to address this problem. Motivated by the Hardy-Littlewood circle method, we introduce the notions of abstract exponential sums modulo $p^m$ and motivic oscillation index of an arbitrary ideal in polynomial rings over number fields. In the arithmetic picture, the abstract exponential sums modulo $p^m$ and the motivic oscillation index of an ideal should play the role of the Bernstein-Sato polynomial and its maximal non-trivial root of the corresponding scheme. We will provide some properties of the motivic oscillation index of ideals in this paper. On the other hand, based on Igusa's conjecture for exponential sums, we propose the averaged Igusa conjecture for exponential sums of ideals. In particular, this conjecture and the motivic oscillation index of ideals will have many interesting applications. We will introduce these applications and prove some variant version o
In the setting of several commuting operators on a Hilbert space one defines the notions of invertibility and Fredholmness in terms of the associated Koszul complex. The index problem then consists of computing the Euler characteristic of such a special type of Fredholm complex. In this paper we investigate transformation rules for the index under the holomorphic functional calculus. We distinguish between two different types of index results: 1) A global index theorem which expresses the index in terms of the degree function of the "symbol" and the locally constant index function of the "coordinates". 2) A local index theorem which computes the Euler characteristic of a localized Koszul complex near a common zero of the "symbol". Our results apply to the example of Toeplitz operators acting on both Bergman spaces over pseudoconvex domains and the Hardy space over the polydisc. The local index theorem is fundamental for future investigations of determinants and torsion of Koszul complexes.
Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problems in geometries that are genuinely noncommutative, particularly in the nonunital situation, was a primary motivation for this study and we illustrate this point with two examples in the text. In order to understand what is new in our approach in the commutative setting we prove an analogue of the Gromov-Lawson relative index formula (for Dirac type operators) for even dimensional manifolds with bounded geometry, without invoking compact supports. For odd dimensional manifolds our index formula appears to be completely new. As we prove our local index formula in the framework of semifinite noncommutative geometry we are also able to prove, for manifolds of bounded geometry, a version of Atiyah's L^2-index Theorem for covering spaces. We also explain how to interp
The coupled evolution of pulsar rotation and inclination angle in the wind braking model is calculated. The oblique pulsar tends to align. The pulsar alignment will affect its spin-down behavior. As a pulsar evolves from the magneto-dipole radiation dominated case to the particle wind dominated case, the braking index will first increase and then decrease. In the early time, the braking index may be larger than 3. And during the following long time, the braking index will be always smaller than 3. The minimum braking index is about one. This can explain the existence of high braking index larger than 3, and low braking index of pulsars simultaneously. The pulsar braking index is expected to evolve from larger than three to about one. A general trend is that the pulsar braking index will evolve from the Crab-like case to the Vela-like case.
We prove that the generalized Randic index over graphs following the Erdős-Renyi model, for both the sparse and dense regimes, is concentrated around its mean when the number of vertices tends to infinity.
We show that existing upsampling operators can be unified using the notion of the index function. This notion is inspired by an observation in the decoding process of deep image matting where indices-guided unpooling can often recover boundary details considerably better than other upsampling operators such as bilinear interpolation. By viewing the indices as a function of the feature map, we introduce the concept of "learning to index", and present a novel index-guided encoder-decoder framework where indices are self-learned adaptively from data and are used to guide the downsampling and upsampling stages, without extra training supervision. At the core of this framework is a new learnable module, termed Index Network (IndexNet), which dynamically generates indices conditioned on the feature map itself. IndexNet can be used as a plug-in applying to almost all off-the-shelf convolutional networks that have coupled downsampling and upsampling stages, giving the networks the ability to dynamically capture variations of local patterns. In particular, we instantiate and investigate five families of IndexNet and demonstrate their effectiveness on four dense prediction tasks, including i
Human hands possess remarkable dexterity and have long served as a source of inspiration for robotic manipulation. In this work, we propose a human $\textbf{H}$and$\textbf{-In}$formed visual representation learning framework to solve difficult $\textbf{Dex}$terous manipulation tasks ($\textbf{H-InDex}$) with reinforcement learning. Our framework consists of three stages: (i) pre-training representations with 3D human hand pose estimation, (ii) offline adapting representations with self-supervised keypoint detection, and (iii) reinforcement learning with exponential moving average BatchNorm. The last two stages only modify $0.36\%$ parameters of the pre-trained representation in total, ensuring the knowledge from pre-training is maintained to the full extent. We empirically study 12 challenging dexterous manipulation tasks and find that H-InDex largely surpasses strong baseline methods and the recent visual foundation models for motor control. Code is available at https://yanjieze.com/H-InDex .
We introduce an invariant of a pair of commuting invertible matrices that we call the rotation index. We apply this invariant, together with the Milnor--Munkres--Novikov pairing, to the study of some questions about group actions of $\mathbb{Z}^2$, specifically the Nielsen realization problem, higher-rank Anosov actions, and extending actions from the sphere $S^{d-1}$ to the disk $D^d$.
The eccentric connectivity index $ξ^c$ is a novel distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as $ξ^c (G) = \sum_{v \in V (G)} deg (v) \cdot ε(v)$\,, where $deg (v)$ and $ε(v)$ denote the vertex degree and eccentricity of $v$\,, respectively. We survey some mathematical properties of this index and furthermore support the use of eccentric connectivity index as topological structure descriptor. We present the extremal trees and unicyclic graphs with maximum and minimum eccentric connectivity index subject to the certain graph constraints. Sharp lower and asymptotic upper bound for all graphs are given and various connections with other important graph invariants are established. In addition, we present explicit formulae for the values of eccentric connectivity index for several families of composite graphs and designed a linear algorithm for calculating the eccentric connectivity index of trees. Some open problems and related indices for further study are also listed.