An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open neighborhood of $v$. The largest cardinality of odd independent sets of a graph $G$, denoted $\alpha_{od}(G)$, is called the odd independence number of $G$. This new parameter is a natural companion to the recently introduced strong odd chromatic number. A proper vertex coloring of a graph $G$ is a strong odd coloring if, for every vertex $v \in V(G)$, each color used in the neighborhood of $v$ appears an odd number of times in $N(v)$. The minimum number of colors in a strong odd coloring of $G$ is denoted by $\chi_{so}(G)$. A simple relation involving these two parameters and the order $|G|$ of $G$ is $\alpha_{od}(G)\cdot \chi_{so}(G) \geq |G|$, parallel to the same on chromatic number and independence number. We develop several basic inequalities concerning $\alpha_{od}(G)$, and use already existing results on strong odd coloring, to derive lower bounds for odd independence in many families of graphs. We prove that $\alpha_{od}(G) = \alpha(G^2)$ holds for all claw-free graphs $G$, and present many results, using various techniques, concerning the odd independence number of cycles, paths, Moore graphs, Kneser graphs, the complete subdivision $S(K_n)$ of $K_n$, the half graphs $H_{n,n}$, and $K_p \Box K_q$. Further, we consider the odd independence number of the hypercube $Q_d$ and also of the complements of triangle-free graphs. Many open problems for future research are stated.

Large language models are often adapted through parameter efficient fine tuning, but current release practices provide weak assurances about what data were used and how updates were computed. We present Verifiable Fine Tuning, a protocol and system that produces succinct zero knowledge proofs that a released model was obtained from a public initialization under a declared training program and an auditable dataset commitment. The approach combines five elements. First, commitments that bind data sources, preprocessing, licenses, and per epoch quota counters to a manifest. Second, a verifiable sampler that supports public replayable and private index hiding batch selection. Third, update circuits restricted to parameter efficient fine tuning that enforce AdamW style optimizer semantics and proof friendly approximations with explicit error budgets. Fourth, recursive aggregation that folds per step proofs into per epoch and end to end certificates with millisecond verification. Fifth, provenance binding and optional trusted execution property cards that attest code identity and constants. On English and bilingual instruction mixtures, the method maintains utility within tight budgets while achieving practical proof performance. Policy quotas are enforced with zero violations, and private sampling windows show no measurable index leakage. Federated experiments demonstrate that the system composes with probabilistic audits and bandwidth constraints. These results indicate that end to end verifiable fine tuning is feasible today for real parameter efficient pipelines, closing a critical trust gap for regulated and decentralized deployments.

Economic complexity methods have become popular tools in economic development, economic geography, and innovation. Yet, despite their widespread adoption, we lack a mechanistic model that provides these methods with a solid mathematical foundation. Here, we analytically derive the economic complexity eigenvector associated with a mechanistic model where the output of an economy in an activity (e.g. of a country in a product) depends on the combined presence of the factors required by the activity. Using analytical and numerical derivations we show that the economic complexity index (or ECI) is a monotonic function of the probability that an economy is endowed with many factors, validating the idea that it is an agnostic estimate of the presence of multiple factors of production. We then generalize this result to other production functions and to a short-run equilibrium framework with prices, wages, and consumption, showing that the derived wage function is consistent with economies converging to an income that is compatible with their complexity. Finally, we use this model to explain differences in the networks of related activities, such as the product space and the research space, showing that the shape of these networks can be explained by different factor distributions. These findings solve long standing puzzles in the economic complexity literature and validate commonly used metrics of economic complexity as estimates of the combined presence of multiple factors.

Developing and testing modern RDF-based applications often requires access to RDF datasets with certain characteristics. Unfortunately, it is very difficult to publicly find domain-specific knowledge graphs that conform to a particular set of characteristics. Hence, in this paper we propose RDFGraphGen, an open-source RDF graph generator that uses characteristics provided in the form of SHACL (Shapes Constraint Language) shapes to generate synthetic RDF graphs. RDFGraphGen is domain-agnostic, with configurable graph structure, value constraints, and distributions. It also comes with a number of predefined values for popular this http URL classes and properties, for more realistic graphs. Our results show that RDFGraphGen is scalable and can generate small, medium, and large RDF graphs in any domain.

Large language models are often adapted through parameter efficient fine tuning, but current release practices provide weak assurances about what data were used and how updates were computed. We present Verifiable Fine Tuning, a protocol and system that produces succinct zero knowledge proofs that a released model was obtained from a public initialization under a declared training program and an auditable dataset commitment. The approach combines five elements. First, commitments that bind data sources, preprocessing, licenses, and per epoch quota counters to a manifest. Second, a verifiable sampler that supports public replayable and private index hiding batch selection. Third, update circuits restricted to parameter efficient fine tuning that enforce AdamW style optimizer semantics and proof friendly approximations with explicit error budgets. Fourth, recursive aggregation that folds per step proofs into per epoch and end to end certificates with millisecond verification. Fifth, provenance binding and optional trusted execution property cards that attest code identity and constants. On English and bilingual instruction mixtures, the method maintains utility within tight budgets while achieving practical proof performance. Policy quotas are enforced with zero violations, and private sampling windows show no measurable index leakage. Federated experiments demonstrate that the system composes with probabilistic audits and bandwidth constraints. These results indicate that end to end verifiable fine tuning is feasible today for real parameter efficient pipelines, closing a critical trust gap for regulated and decentralized deployments.

Efforts to apply economic complexity to identify diversification opportunities often rely on diagrams comparing the relatedness and complexity of products, technologies, or industries. Yet, the use of these diagrams, is not based on empirical or theoretical evidence supporting some notion of optimality. Here, we introduce a method to identify diversification opportunities based on the minimization of a cost function that captures the constraints imposed by an economy's pattern of specialization and show that this ECI optimization algorithm produces recommendations that are substantially different from those obtained using relatedness-complexity diagrams. This method advances the use of economic complexity methods to explore questions of strategic diversification.

Music transcription is the process of transcribing music audio into music notation. It is a field in which the machines still cannot beat human performance. The main motivation for automatic music transcription is to make it possible for anyone playing a musical instrument, to be able to generate the music notes for a piece of music quickly and accurately. It does not matter if the person is a beginner and simply struggles to find the music score by searching, or an expert who heard a live jazz improvisation and would like to reproduce it without losing time doing manual transcription. We propose Scorpiano -- a system that can automatically generate a music score for simple monophonic piano melody tracks using digital signal processing. The system integrates multiple digital audio processing methods: notes onset detection, tempo estimation, beat detection, pitch detection and finally generation of the music score. The system has proven to give good results for simple piano melodies, comparable to commercially available neural network based systems.

A strong odd coloring of a simple graph $G$ is a proper coloring of the vertices of $G$ such that for every vertex $v$ and every color $c$, either $c$ is used an odd number of times in the open neighborhood $N_G(v)$ or no neighbor of $v$ is colored by $c$. The smallest integer $k$ for which $G$ admits a strong odd coloring with $k$ colors is the strong odd chromatic number, $\chi_{soc}(G)$. These coloring notion and graph parameter were recently defined in [H. Kwon and B. Park, Strong odd coloring of sparse graphs, arXiv:2401.11653v2]. We answer a question raised by the originators concerning the existence of a constant bound for the strong odd chromatic number of all planar graphs. We also consider strong odd colorings of trees, unicyclic graphs and graph products.

The heterogeneity in cellular networks that comprise multiple base stations types imposes new challenges in network planning and deployment of future generation of cellular networks. The Radio Resource Management (RRM) techniques, such as dynamic sharing of the available resources and advanced user association strategies, determine the overall network capacity and efficiency. This paper evaluates the downlink performance of a two tier heterogeneous network (consisting of macro and femto tiers) in terms of rate distribution, i.e. the percentage of users that achieve certain rate in the system. The paper specifically addresses the femto tier RRM by randomization of the allocated resources and the user association process by introducing a modified SINR association strategy with bias factor for load balancing. Also, the paper introduces hybrid access control mechanism at the femto tier that allows the authorized users of the femtocell, which are part of the Closed Subscriber Group (CSG) on the femtocell, to achieve higher data rates up to 10 times compared to the other regular users associated in the access.

The aim of the present paper is to give a formula for the $k$-th covariant derivative of tensor field along a given curve. In order to do that, first the symbols $P^{i}_{j}$ and $Q^{i}_{j}$ which depend on the Christoffel symbols are introduced. Some properties of them are also given. The main result is given by (3.1) and further it is generalized for $k\in R$.

Training large language models requires extensive processing, made possible by many high-performance computing resources. This study compares multi-node and multi-GPU environments for training large language models of electrocardiograms. It provides a detailed mapping of current frameworks for distributed deep learning in multinode and multi-GPU settings, including Horovod from Uber, DeepSpeed from Microsoft, and the built-in distributed capabilities of PyTorch and TensorFlow. We compare various multi-GPU setups for different dataset configurations, utilizing multiple HPC nodes independently and focusing on scalability, speedup, efficiency, and overhead. The analysis leverages HPC infrastructure with SLURM, Apptainer (Singularity) containers, CUDA, PyTorch, and shell scripts to support training workflows and automation. We achieved a sub-linear speedup when scaling the number of GPUs, with values of 1.6x for two and 1.9x for four.

In this paper are examined general classes of linear and non-linear analytical systems of partial differential equations. Indeed the integrability conditions are found and if they are satisfied, the solutions are given as functional series in a neighborhood of a given point (x=0).

An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open neighborhood of $v$. The largest cardinality of odd independent sets of a graph $G$, denoted $\alpha_{od}(G)$, is called the odd independence number of $G$. This new parameter is a natural companion to the recently introduced strong odd chromatic number. A proper vertex coloring of a graph $G$ is a strong odd coloring if, for every vertex $v \in V(G)$, each color used in the neighborhood of $v$ appears an odd number of times in $N(v)$. The minimum number of colors in a strong odd coloring of $G$ is denoted by $\chi_{so}(G)$. A simple relation involving these two parameters and the order $|G|$ of $G$ is $\alpha_{od}(G)\cdot\chi_{so}(G) \geq |G|$, parallel to the same on chromatic number and independence number. In the present work, which is a companion to our first paper on the subject [The odd independence number of graphs, I: Foundations and classical classes], we focus on grid-like and chessboard-like graphs and compute or estimate their odd independence number and their strong odd chromatic number. Among the many results obtained, the following give the flavour of this paper: (1) $0.375 \leq \varrho_{od}(P_\infty \Box P_\infty) \leq 0.384615...$, where $\varrho_{od}(P_\infty \Box P_\infty)$ is the odd independence ratio. (2) $\chi_{so}(G_d) = 3$ for all $d \geq 1$, where $G_d$ is the infinite $d$-dimensional grid. As a consequence, $\varrho_{od}(G_d) \geq 1/3$. (3) The $r$-King graph $G$ on $n^2$ vertices has $\alpha_{od}(G) = \lceil n/(2r+1) \rceil^2$. Moreover, $\chi_{so}(G) = (2r + 1)^2$ if $n \geq 2r + 1$, and $\chi_{so}(G) = n^2$ if $n \leq 2r$. Many open problems are given for future research.

Complex energy transfer processes in the intracluster medium (ICM) can revive fossil (with spectral ages $\gg100$ Myr) plasma initially generated by radio galaxies. This leads to the re-ignition of faint radio sources with irregular and filamentary morphologies, and ultra-steep ($\alpha \gtrsim 1.5$) synchrotron spectra, which can be more easily detected at low frequencies ($\sim 100$ MHz). These sources offer the opportunity to investigate the microphysics of the ICM and its interplay with radio galaxies, the origin of seed relativistic electrons, the merging history of the host cluster, and the phenomenology of radio filaments. The study of revived sources has so far been hampered by the requirement of sensitive and high-resolution multi-frequency radio data at low frequencies to characterise their spatial properties and provide a proper classification. We aim to perform the analysis of a sample of candidate revived sources identified among nearby ($z\leq0.35$) and low-mass ($M_{500}\leq5\times 10^{14} M_\odot$) \textit{Planck} clusters in the footprint of LoTSS-DR2. By inspecting LoTSS-DR2 images at 144 MHz, we identified 7 targets with patchy and filamentary morphologies, which have been followed-up with the uGMRT at 400 MHz. By combining LOFAR and uGMRT data, we obtained high-resolution images and spectral index maps, which we used to interpret the nature of the sources. All targets show regions with very steep spectra, confirming the effectiveness of our morphology-based selection in identifying fossil plasma. Based on their morphology, spectral properties, and optical associations, we investigated the origin of the targets. We found a variety of promising revived fossil sources, while also showing that apparently intricate structures can be easily misclassified in the absence of high-resolution and multi-band data.

Regularization is a widely recognized technique in mathematical optimization. It can be used to smooth out objective functions, refine the feasible solution set, or prevent overfitting in machine learning models. Due to its simplicity and robustness, the gradient descent (GD) method is one of the primary methods used for numerical optimization of differentiable objective functions. However, GD is not well-suited for solving $\ell^1$ regularized optimization problems since these problems are non-differentiable at zero, causing iteration updates to oscillate or fail to converge. Instead, a more effective version of GD, called the proximal gradient descent employs a technique known as soft-thresholding to shrink the iteration updates toward zero, thus enabling sparsity in the solution. Motivated by the widespread applications of proximal GD in sparse and low-rank recovery across various engineering disciplines, we provide an overview of the GD and proximal GD methods for solving regularized optimization problems. Furthermore, this paper proposes a novel algorithm for the proximal GD method that incorporates a variable step size. Unlike conventional proximal GD, which uses a fixed step size based on the global Lipschitz constant, our method estimates the Lipschitz constant locally at each iteration and uses its reciprocal as the step size. This eliminates the need for a global Lipschitz constant, which can be impractical to compute. Numerical experiments we performed on synthetic and real-data sets show notable performance improvement of the proposed method compared to the conventional proximal GD with constant step size, both in terms of number of iterations and in time requirements.

The labelling of speech corpora is a laborious and time-consuming process. The ProsoBeast Annotation Tool seeks to ease and accelerate this process by providing an interactive 2D representation of the prosodic landscape of the data, in which contours are distributed based on their similarity. This interactive map allows the user to inspect and label the utterances. The tool integrates several state-of-the-art methods for dimensionality reduction and feature embedding, including variational autoencoders. The user can use these to find a good representation for their data. In addition, as most of these methods are stochastic, each can be used to generate an unlimited number of different prosodic maps. The web app then allows the user to seamlessly switch between these alternative representations in the annotation process. Experiments with a sample prosodically rich dataset have shown that the tool manages to find good representations of varied data and is helpful both for annotation and label correction. The tool is released as free software for use by the community.

In this study, we identify the relative standard deviation volatility (RSD volatility) in the individual target time fulfilment of the complete set of comparables (e.g., all individuals in the same organisational structure) as a possible key performance indicator (KPI) for predicting employee job performance. KPIs are a well-established, measurable benchmark of an organisation's critical success metrics; thus, in this paper, we attempt to identify employees experiencing a transition in their RSD towards a higher per cent deviation, indicating emerging inadequate work conditions. We believe RSD volatility can be utilised as an additional assessment factor, particularly in profiling.

This paper surveys the empirical literature of inflation targeting. The main findings from our review are the following: there is robust empirical evidence that larger and more developed countries are more likely to adopt the IT regime; the introduction of this regime is conditional on previous disinflation, greater exchange rate flexibility, central bank independence, and higher level of financial development; the empirical evidence has failed to provide convincing evidence that IT itself may serve as an effective tool for stabilizing inflation expectations and for reducing inflation persistence; the empirical research focused on advanced economies has failed to provide convincing evidence on the beneficial effects of IT on inflation performance, while there is some evidence that the gains from the IT regime may have been more prevalent in the emerging market economies; there is not convincing evidence that IT is associated with either higher output growth or lower output variability; the empirical research suggests that IT may have differential effects on exchange-rate volatility in advanced economies versus EMEs; although the empirical evidence on the impact of IT on fiscal policy is quite limited, it supports the idea that IT indeed improves fiscal discipline; the empirical support to the proposition that IT is associated with lower disinflation costs seems to be rather weak. Therefore, the accumulated empirical literature implies that IT does not produce superior macroeconomic benefits in comparison with the alternative monetary strategies or, at most, they are quite modest.

Maintaining a balance between returns and volatility is a common strategy for portfolio diversification, whether investing in traditional equities or digital assets like cryptocurrencies. One approach for diversification is the application of community detection or clustering, using a network representing the relationships between assets. We examine two network representations, one based on a standard distance matrix based on correlation, and another based on mutual information. The Louvain and Affinity propagation algorithms were employed for finding the network communities (clusters) based on annual data. Furthermore, we examine building assets' co-occurrence networks, where communities are detected for each month throughout a whole year and then the links represent how often assets belong to the same community. Portfolios are then constructed by selecting several assets from each community based on local properties (degree centrality), global properties (closeness centrality), or explained variance (Principal component analysis), with three value ranges (max, med, min), calculated on a maximal spanning tree or a fully connected community sub-graph. We explored these various strategies on data from the S\&P 500 and the Top 203 cryptocurrencies with a market cap above 2M USD in the period from Jan 2019 to Sep 2022. Moreover, we study into more details the periods of the beginning of the COVID-19 outbreak and the start of the war in Ukraine. The results confirm some of the previous findings already known for traditional stock markets and provide some further insights, while they reveal an opposing trend in the crypto-assets market.

We investigate searching efficiency of different kinds of random walk on complex networks which rely on local information and one-step memory. For the studied navigation strategies we obtained theoretical and numerical values for the graph mean first passage times as an indicator for the searching efficiency. The experiments with generated and real networks show that biasing based on inverse degree, persistence and local two-hop paths can lead to smaller searching times. Moreover, these biasing approaches can be combined to achieve a more robust random search strategy. Our findings can be applied in the modeling and solution of various real-world problems.