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Inductive link prediction with knowledge hypergraphs is the task of predicting missing hyperedges involving completely novel entities (i.e., nodes unseen during training). Existing methods for inductive link prediction with knowledge hypergraphs assume a fixed relational vocabulary and, as a result, cannot generalize to knowledge hypergraphs with novel relation types (i.e., relations unseen during training). Inspired by knowledge graph foundation models, we propose HYPER as a foundation model for link prediction, which can generalize to any knowledge hypergraph, including novel entities and novel relations. Importantly, HYPER can learn and transfer across different relation types of varying arities, by encoding the entities of each hyperedge along with their respective positions in the hyperedge. To evaluate HYPER, we construct 16 new inductive datasets from existing knowledge hypergraphs, covering a diverse range of relation types of varying arities. Empirically, HYPER consistently outperforms all existing methods in both node-only and node-and-relation inductive settings, showing strong generalization to unseen, higher-arity relational structures.

In a previous paper, we introduced the notion of hyper swap structures, a novel class of hyperalgebras that naturally generalizes swap structures semantics. In this paper we introduce the concept of hyper Boolean algebras based on Morgado hyperlattices, proving some basic properties. From this, we show that several paraconsistent logics in the hierarchy of Logics of Formal Inconsistency (LFIs) can be naturally characterized in terms of hyper swap structures semantics generated by hyper Boolean algebras. Finally, for each of these LFIs we obtain a Kalman-style functor which establishes an equivalence between the category of hyper Boolean algebras and a category of hyper algebras for the corresponding LFI having the hyper swap structures as representative objects.

In this paper, we first introduce the notion of a hyper relative differential operator on a Lie algebra, in which Nijenhuis operators are used to characterize the relative differential operators and their inverse. We then introduce the notions of DN-structures, KN-structures, and KD-structures on Lie algebras and study the relationships between DN-structures, KD-structures, KN-structures, and hyper relative differential operators. Finally, we investigate hyper symplectic structures and hyper Hessian structures from the view point of hyper relative differential operators, and provide equivalent descriptions for both hyper symplectic structures and hyper Hessian structures.

Approximate solutions of partial differential equations (PDEs) obtained by neural networks are highly affected by hyper parameter settings. For instance, the model training strongly depends on loss function design, including the choice of weight factors for different terms in the loss function, and the sampling set related to numerical integration; other hyper parameters, like the network architecture and the optimizer settings, also impact the model performance. On the other hand, suitable hyper parameter settings are known to be different for different model problems and currently no universal rule for the choice of hyper parameters is known. In this paper, for second order elliptic model problems, various hyper parameter settings are tested numerically to provide a practical guide for efficient and accurate neural network approximation. While a full study of all possible hyper parameter settings is not possible, we focus on studying the formulation of the PDE loss as well as the incorporation of the boundary conditions, the choice of collocation points associated with numerical integration schemes, and various approaches for dealing with loss imbalances will be extensively studi

We present Hyper-Py, a fully restructured and extended Python implementation of HYPER (HYbrid Photometry and Extraction Routine, Traficante et al. 2015). HYPER was originally implemented in IDL, aiming to deliver robust and reproducible photometry of compact sources in FIR/sub-mm/mm maps. HYPER combines source detection via high-pass filtering, background estimation through local polynomial fitting, and source modeling with 2D elliptical Gaussians, simultaneously fitting multiple Gaussians to deblend overlapping sources. Hyper-Py preserves the original logic while offering improvements in performance, configurability, and background modeling capabilities, making it a flexible modern tool for source extraction and photometry across diverse datasets. Notably, Hyper-Py enables background estimation and subtraction across individual slices of 3D datacubes, allowing consistent background modeling along the spectral axis for line or continuum studies in spectrally resolved observations.

Evolution Strategy (ES) is a promising alternative to gradient-based fine-tuning for resource-constrained Large Language Model (LLM) reasoning. However, directly applying ES to billion-parameter LLMs is highly ineffective. In such high-dimensional parameter spaces, most random perturbations are nearly orthogonal to useful update directions, leading to unstable optimization. We propose Hyper-ES, a subspace-based ES framework that avoids the weakness of ES in full-parameter search while exploiting its strength in low-dimensional optimization. Instead of asking ES to discover useful directions from random perturbations in the LLM parameter space, Hyper-ES first performs a small number of inexpensive gradient-based fine-tuning runs to obtain descent directions. Although each direction may provide only a limited improvement on its own, their span forms a compact adaptation subspace that captures useful reasoning updates. Hyper-ES then applies CMA-ES to optimize layer-wise DARE-TIES merging coefficients within this subspace, allowing ES to search over combinations of meaningful descent directions rather than over arbitrary full-model perturbations. We evaluate Hyper-ES on three Qwen2.5-I

Motivated by the recent work of Xiao and Zhong [AIMS Math. 9 (2024), 35125--35150: MR4840882], we propose a generalized inverse for a hyper-dual matrix called hyper-dual group generalized inverse (HDGGI). Firstly, we characterize the existence of the HDGGI of a hyper-dual matrix and we show that it is unique, whenever exists. The HDGGI is then used to solve a linear hyper-dual system. We discuss the minimal $P$-norm least-squares properties of hyper-dual group inverse. We also exploit some sufficient conditions under which the reverse and forward-order laws for a particular form of the HDGGI and hyper-dual Moore-Penrose generalized inverse (HDMPGI) hold. Using the definition of dual matrix of order $n$, we finally establish necessary and sufficient condition for the existence of the group inverse of a dual matrix of order $n$.

Network representation learning has aroused widespread interests in recent years. While most of the existing methods deal with edges as pairwise relationships, only a few studies have been proposed for hyper-networks to capture more complicated tuplewise relationships among multiple nodes. A hyper-network is a network where each edge, called hyperedge, connects an arbitrary number of nodes. Different from conventional networks, hyper-networks have certain degrees of indecomposability such that the nodes in a subset of a hyperedge may not possess a strong relationship. That is the main reason why traditional algorithms fail in learning representations in hyper-networks by simply decomposing hyperedges into pairwise relationships. In this paper, we firstly define a metric to depict the degrees of indecomposability for hyper-networks. Then we propose a new concept called hyper-path and design hyper-path-based random walks to preserve the structural information of hyper-networks according to the analysis of the indecomposability. Then a carefully designed algorithm, Hyper-gram, utilizes these random walks to capture both pairwise relationships and tuplewise relationships in the whole h

Binary mixtures of the ferronematic compound DIO with recently reported non-ferroelectric material WJ-16 which shows Colossal Permittivity (CP) ~5000 and superparaelectricity (SPE) were studied by POM, electrical switching studies, and dielectric spectroscopy. Three mixtures with different contents of WJ-16 as 10, 25 and 50% in DIO as host were prepared. Our original expectation was the development of new nematic materials with both ferroelectric nematic (NF) and non-ferroelectric CP phases. The non-ferroelectric phase in mixtures exhibits a CP mode originally observed in pure WJ-16 and was termed as superparaelectric. However, the dielectric spectroscopy of mixtures shows two distinct relaxation processes: the typical paraelectric response and the CP mode. Therefore, this CP mode cannot be called superparaelectric and is redefined it as Hyper Dielectric mode. This is the first direct demonstration of materials with both ferroelectric and Hyper-dielectric phases in liquid crystalline materials. The hyper dielectric phase has a good potential as a working media for supercapacitors industry.

In runtime verification, pattern matching, which searches for occurrences of a specific pattern within a word, provides more information than a simple violation detection of the monitored property, by locating concrete evidence of the violation. However, witnessing violations of some properties, particularly hyperproperties, requires evidence across multiple input words or different parts of the same word, which goes beyond the scope of conventional pattern matching. We propose here hyper pattern matching, a generalization of pattern matching over a set of words. Properties of interest include robustness and (non-)interference. As a formalism for patterns, we use nondeterministic asynchronous finite automata (NAAs). We first provide a naive algorithm for hyper pattern matching and then devise several heuristics for better efficiency. Although we prove the NP-completeness of the problem, our implementation HypPAu is able to address several case studies scalable in the length, number of words (or logs) and number of dimensions, suggesting the practical relevance of our approach.

In contrast to regular (simple) networks, hyper networks possess the ability to depict more complex relationships among nodes and store extensive information. Such networks are commonly found in real-world applications, such as in social interactions. Learning embedded representations for nodes involves a process that translates network structures into more simplified spaces, thereby enabling the application of machine learning approaches designed for vector data to be extended to network data. Nevertheless, there remains a need to delve into methods for learning embedded representations that prioritize structural aspects. This research introduces HyperS2V, a node embedding approach that centers on the structural similarity within hyper networks. Initially, we establish the concept of hyper-degrees to capture the structural properties of nodes within hyper networks. Subsequently, a novel function is formulated to measure the structural similarity between different hyper-degree values. Lastly, we generate structural embeddings utilizing a multi-scale random walk framework. Moreover, a series of experiments, both intrinsic and extrinsic, are performed on both toy and real networks. T

We introduce Hyper-Trees as a novel framework for modeling time series data using gradient boosted trees. Unlike conventional tree-based approaches that forecast time series directly, Hyper-Trees learn the parameters of a target time series model, such as ARIMA or Exponential Smoothing, as functions of features. These parameters are then used by the target model to generate the final forecasts. Our framework combines the effectiveness of decision trees on tabular data with classical forecasting models, thereby inducing a time series inductive bias into tree-based models. To resolve the scaling limitations of boosted trees when estimating a high-dimensional set of target model parameters, we combine decision trees and neural networks within a unified framework. In this hybrid approach, the trees generate informative representations from the input features, which a shallow network then uses as input to learn the parameters of a time series model. With our research, we explore the effectiveness of Hyper-Trees across a range of forecasting tasks and extend tree-based modeling beyond its conventional use in time series analysis.

Amodal object completion is a complex task that involves predicting the invisible parts of an object based on visible segments and background information. Learning shape priors is crucial for effective amodal completion, but traditional methods often rely on two-stage processes or additional information, leading to inefficiencies and potential error accumulation. To address these shortcomings, we introduce a novel framework named the Hyper-Transformer Amodal Network (H-TAN). This framework utilizes a hyper transformer equipped with a dynamic convolution head to directly learn shape priors and accurately predict amodal masks. Specifically, H-TAN uses a dual-branch structure to extract multi-scale features from both images and masks. The multi-scale features from the image branch guide the hyper transformer in learning shape priors and in generating the weights for dynamic convolution tailored to each instance. The dynamic convolution head then uses the features from the mask branch to predict precise amodal masks. We extensively evaluate our model on three benchmark datasets: KINS, COCOA-cls, and D2SA, where H-TAN demonstrated superior performance compared to existing methods. Addit

Let $\kl_n(a,b;m)$ be the hyper-Kloosterman sum. Fix integers $n\geqslant2,a eq0$, $b eq0$ and $k\geqslant2$. For any $0 eqη\in\mathbb{C}$ and multiplicative function $f: \mathbb{N} \rightarrow \mathbb{C}$, we prove that $\kl_n(a,b;m) eqηf(m)$ holds for $100\%$ square-free $k$-almost prime numbers $m$ and $100\%$ square-free numbers $m$. Counterintuitively, if $\kl_n(a,b;p)=ηf(p)$ holds for all but finitely many primes $p$, we further show that \begin{align*} \ab|\{m\leqslant X:\kl_n(a,b;m)=ηf(m), m \text{ square-free }k\text{-almost prime}\}|= O(X^{1-\frac{1}{k+1}}). \end{align*} These results overturn the general belief that $\kl_n(a,b;m)$ is nearly multiplicative in $m$, and that its distribution at almost prime moduli $m$ closely approximates that at primes. Moreover, we prove that these results also hold for general algebraic exponential sums satisfying some natural conditions.

A systematic study on the probability for the emission of 4^He and 14^C cluster from hyper ${207-234}^$Ac and non-strange normal ${207-234}^$Ac nuclei are performed for the first time using our fission model, the Coulomb and proximity potential model (CPPM). The predicted half lives show that hyper ${207-234}^$Ac nuclei are unstable against 4^He emission and 14^C emission from hyper ${217-228}^$Ac are favorable for measurement. Our study also show that hyper ${207-234}^$Ac are stable against hyper 4^He and 14^C emission. The role of neutron shell closure (N=126) in hyper 214^Fr daughter and role of proton/ neutron shell closure (Z =82, N =126) in hyper 210^Bi daughter are also revealed. As hyper-nuclei decays to normal nuclei by mesonic/non-mesonic decay and since most of the predicted half lives for 4^He and 14^C emission from normal Ac nuclei are favourable for measurement, we presume that alpha and 14^C cluster emission from hyper Ac nuclei can be detected in laboratory in a cascade (two-step) process.

The transition towards the sixth-generation (6G) wireless telecommunications networks introduces significant challenges for researchers and industry stakeholders. The 6G technology aims to enhance existing usage scenarios through supporting innovative applications that require stringent key performance indicators (KPIs). In some critical use cases of 6G, multiple KPIs, including immersive throughput, with an envisioned peak data rate of $1$ Tbps, hyper-reliability, in the range of $10^{-5}$ to $10^{-7}$, and hyper low-latency, between $0.1$ and $1$ ms, must be achieved simultaneously to deliver the expected service experience. However, this is challenging due to the conflicting nature of these KPIs. This article proposes a new service class of 6G as immersive, hyper reliable, and low-latency communication (IHRLLC), and introduces a potential network architecture to achieve the associated KPIs. Specifically, enhanced technologies, such as ultra-massive multiple-input multiple-output (umMIMO)-aided terahertz (THz) communications, reconfigurable intelligent surfaces (RIS), and non-terrestrial networks (NTN), are viewed as the key enablers for achieving immersive data rates and hyper r

Hoare logics are proof systems that allow one to formally establish properties of computer programs. Traditional Hoare logics prove properties of individual program executions (such as functional correctness). Hoare logic has been generalized to prove also properties of multiple executions of a program (so-called hyperproperties, such as determinism or non-interference). These program logics prove the absence of (bad combinations of) executions. On the other hand, program logics similar to Hoare logic have been proposed to disprove program properties (e.g., Incorrectness Logic), by proving the existence of (bad combinations of) executions. All of these logics have in common that they specify program properties using assertions over a fixed number of states, for instance, a single pre- and post-state for functional properties or pairs of pre- and post-states for non-interference. In this paper, we present Hyper Hoare Logic, a generalization of Hoare logic that lifts assertions to properties of arbitrary sets of states. The resulting logic is simple yet expressive: its judgments can express arbitrary program hyperproperties, a particular class of hyperproperties over the set of termi

This work demonstrates that applying a fixed-effect multiple linear regression (MLR) model to an overparameterized dataset is mathematically equivalent to fitting a hyper-curve parameterized by a single scalar. This reformulation shifts the focus from global coefficients to individual predictors, allowing each to be modeled as a function of a common parameter. We prove that this overparameterized linear framework can yield exact predictions even when the underlying data contains nonlinear dependencies that violate classical linear assumptions. By employing parameterization in terms of the dependent variable and a monomial basis, we validate this approach on both synthetic and experimental datasets. Our results show that the hyper-curve perspective provides a robust framework for regularizing problems with noisy predictors and offers a systematic method for identifying and removing 'improper' predictors that degrade model generalizability.

Recently, a series of diffusion-aware distillation algorithms have emerged to alleviate the computational overhead associated with the multi-step inference process of Diffusion Models (DMs). Current distillation techniques often dichotomize into two distinct aspects: i) ODE Trajectory Preservation; and ii) ODE Trajectory Reformulation. However, these approaches suffer from severe performance degradation or domain shifts. To address these limitations, we propose Hyper-SD, a novel framework that synergistically amalgamates the advantages of ODE Trajectory Preservation and Reformulation, while maintaining near-lossless performance during step compression. Firstly, we introduce Trajectory Segmented Consistency Distillation to progressively perform consistent distillation within pre-defined time-step segments, which facilitates the preservation of the original ODE trajectory from a higher-order perspective. Secondly, we incorporate human feedback learning to boost the performance of the model in a low-step regime and mitigate the performance loss incurred by the distillation process. Thirdly, we integrate score distillation to further improve the low-step generation capability of the mo