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In this short note, we prove several infinite family of congruences for some restricted partitions introduced by Pushpa and Vasuki (2022) (thereby, also proving a conjecture of Dasappa et. al. (2023)). We also prove some isolated congruences which seem to have been missed by earlier authors. Our proof techniques uses both elementary means as well as the theory of modular forms.
In addition to reproducing discriminatory relationships in the training data, machine learning systems can also introduce or amplify discriminatory effects. We refer to this as introduced unfairness, and investigate the conditions under which it may arise. To this end, we propose introduced total variation as a measure of introduced unfairness, and establish graphical conditions under which it may be incentivised to occur. These criteria imply that adding the sensitive attribute as a feature removes the incentive for introduced variation under well-behaved loss functions. Additionally, taking a causal perspective, introduced path-specific effects shed light on the issue of when specific paths should be considered fair.
The present review of bibliometric counting methods investigates 1) the number of unique counting methods in the bibliometric research literature, 2) to what extent the counting methods can be categorized according to selected characteristics of the counting methods, 3) methods and elements to assess the internal validity of the counting methods, and 4) to what extent and with which characteristics the counting methods are used in research evaluations. The review identifies 32 counting methods introduced during the period 1981 - 2018. Two frameworks categorize these counting methods. Framework 1 describes selected mathematical properties of counting methods, and Framework 2 describes arguments for choosing a counting method. Twenty of the 32 counting methods are rank-dependent, fractionalized, and introduced to measure contribution, participation, etc. of an object of study. Next, three criteria for internal validity are used to identify five methods that test the adequacy of counting methods, two elements that test sensitivity, and three elements that test homogeneity of the counting methods. These methods and elements may be used to assess the internal validity of counting method
Erdös and Nicolas [erdos1976methodes] introduced an arithmetical function $F(n)$ related to divisors of $n$ in short intervals $\left] \frac{t}{2}, t\right]$. The aim of this note is to prove that $F(n)$ is the largest coefficient of polynomial $P_n(q)$ introduced by Kassel and Reutenauer [kassel2015counting]. We deduce that $P_n(q)$ has a coefficient larger than $1$ if and only if $2n$ is the perimeter of a Pythagorean triangle. We improve a result due to Vatne [vatne2017sequence] concerning the coefficients of $P_n(q)$.
A phenomenon that strongly influences the demography of small introduced populations and thereby potentially their genetic diversity is the Allee effect, a reduction in population growth rates at small population sizes. We take a stochastic modeling approach to investigate levels of genetic diversity in populations that successfully overcame a strong demographic Allee effect, a scenario in which populations smaller than a certain critical size are expected to decline. Our results indicate that compared to successful populations without Allee effect, successful Allee-effect populations tend to 1) derive from larger founder population sizes and thus have a higher initial amount of genetic variation, 2) spend fewer generations at small population sizes where genetic drift is particularly strong, and 3) spend more time around the critical population size and thus experience more drift there. Altogether, the Allee effect can either increase or decrease genetic diversity, depending on the average founder population size. In the case of multiple introduction events, there is an additional increase in diversity because Allee-effect populations tend to derive from a larger number of introdu
Conditional Equi-concentration of Types on I-projections (ICET) and Extended Gibbs Conditioning Principle (EGCP) provide an extension of Conditioned Weak Law of Large Numbers and of Gibbs Conditioning Principle to the case of non-unique Relative Entropy Maximizing (REM) distribution (aka I-projection). ICET and EGCP give a probabilistic justification to REM under rather general conditions. mu-projection variants of the results are introduced. They provide a probabilistic justification to Maximum Probability (MaxProb) method. 'REM/MaxEnt or MaxProb?' question is discussed, briefly. Jeffreys Conditioning Principle is mentioned.
RANS simulations with the Spalart-Allmaras turbulence model are improved for cases with flow separation using the Field Inversion and Machine Learning approach. A compensatory discrepancy term is introduced into the turbulence model and optimized using high-fidelity reference data from experiments. Influences on the optimization results with respect to regularization, grid resolution and areas in which the optimization is active are investigated. Finally, a neural network is trained and used to augment simulations on a test case.
We show two results about the Conway potential function which is known as the normalized multivariable Alexander polynomial. We first show that the Conway potential function introduced by Kauffman in "Formal Knot Theory" is indeed a link invariant. Next we show that Kauffman's potential function equals Hartley's potential function. We will prove it by using Murakami's axioms for the multivariable Alexander polynomial.
Many exotic species combine low probability of establishment at each introduction with rapid population growth once introduction does succeed. To analyze this phenomenon, we note that invaders often cluster spatially when rare, and consequently an introduced exotic's population dynamics should depend on locally structured interactions. Ecological theory for spatially structured invasion relies on deterministic approximations, and determinism does not address the observed uncertainty of the exotic-introduction process. We take a new approach to the population dynamics of invasion and, by extension, to the general question of invasibility in any spatial ecology. We apply the physical theory for nucleation of spatial systems to a lattice-based model of competition between plant species, a resident and an invader, and the analysis reaches conclusions that differ qualitatively from the standard ecological theories. Nucleation theory distinguishes between dynamics of single-cluster and multi-cluster invasion. Low introduction rates and small system size produce single-cluster dynamics, where success or failure of introduction is inherently stochastic. Single-cluster invasion occurs only
This paper discusses the meaning and scope of biological hypercomputation (BH). The framework here is computational, and from the outset it should be clear that life is not a standard Turing Machine. Living systems hypercompute, but the distinction is made between classical and non-classical hypercomputation. We argue that living processes are non-classical hypercomputation. Yet, BH entails new computational models, for it does not correspond, any longer, to the Turing Machine model of computation. Hence, we introduce BH having a twofold scope, thus: on the one hand, it implies new computational models, while on the other hand we aim at understanding life not by what it is, but rather by what it does. From a computational point of view, life hypercomputes. At the end we sketch out the possibilities, stances and reach of BH. The aim of BH is basically help understanding life from a computational standpoint.
The introduction of non-native organisms into complex microbiome communities holds enormous potential to benefit society. However, microbiome engineering faces several challenges including successful establishment of the organism into the community, its persistence in the microbiome to serve a specified purpose, and constraint of the organism and its activity to the intended environment. A theoretical framework is needed to represent the complex interactions that drive these dynamics. Building on the concept of the community functional landscape, we define the persistence landscape as the metabolic, genetic, and broader functional composition and ecological context of the target microbiome that can be used to predict the environmental fitness of an introduced organism. Here, we discuss critical aspects of persistence landscapes that impact interactions between an introduced organism and the target microbiome, including the communitys genetic and metabolic complementation potential, cellular defense strategies, spatial and temporal dynamics, and the introduced organisms ability to compete for resources to survive. Finally, we highlight important knowledge gaps in the fields of micro
This paper investigates code quality education by analyzing how errors are introduced and corrected in group projects within an embedded systems course. We identify who introduces errors, who fixes them, and when these actions occur. Students learn code quality rules for C and embedded systems. We address three questions: RQ1: What is the impact of group formation on code quality? RQ2: How do students interact to fix code issues? RQ3: When are issues introduced and resolved? We analyzed data from eight individual labs and two group projects involving 34 students. The course provides continuous, automated feedback on code quality. Findings show that the most active contributors often introduce the most issues. Many issues are fixed late in the project. Individual labs tend to have fewer issues due to their structured nature. Most problems are fixed by the original author, while cross-student fixes take longer, especially in shared code. Critical issues are fixed quickly, but non-critical ones may be ignored, showing a focus on functionality over quality.
Developing a systematic view of where quantum computers will outperform classical ones is important for researchers, policy makers and business leaders. But developing such a view is challenging because quantum advantage analyses depend not only on algorithm properties, but also on a host of technical characteristics (error correction, gate speeds, etc.). Because various analyses make different assumptions about these technical characteristics, it can be challenging to make comparisons across them. In this paper, we introduce an open-access web-tool designed to make such comparisons easy. Built on the framework introduced by Choi, Moses, and Thompson (2023), it calculates when quantum systems will outperform classical computers for a given algorithmic problem. These estimates can be easily updated based on various assumptions for error correction, overhead, and connectivity. Different hardware roadmaps can also be used and algorithm running times can be customized to particular applications. It can currently be accessed at https://futuretech.mit.edu/quantum-economic-advantage-calculator. This integrated prediction tool also allows us to explore which technical factors are most impo
The problem of image data generation in computer vision has traditionally been a harder problem to solve, than discriminative problems. Such data generation entails placing relevant objects of appropriate sizes each, at meaningful location in a scene canvas. There have been two classes of popular approaches to such generation: graphics based, and generative models-based. Optimization problems are known to lurk in the background for both these classes of approaches. In this paper, we introduce a novel, practically useful manifestation of the classical Bin Packing problem in the context of generation of synthetic image data. We conjecture that the newly introduced problem, Resizable Anchored Region Packing(RARP) Problem, is NP-hard, and provide detailed arguments about our conjecture. As a first solution, we present a novel heuristic algorithm that is generic enough and therefore scales and packs arbitrary number of arbitrary-shaped regions at arbitrary locations, into an image canvas. The algorithm follows greedy approach to iteratively pack region pairs in a careful way, while obeying the optimization constraints. The algorithm is validated by an implementation that was used to gen
In this paper, we present an extension to Melda (a library which implements a general purpose delta state JSON CRDT) to support move operations. This enhancement relies on minimal changes to the underlying logic of the data structure, has virtually no runtime overhead and zero storage overhead compared to the original version of the library, ensuring simplicity while addressing multiple use cases. Although concurrent reordering of the elements in a list was already supported in the original version of the library, moving objects between different containers lead to undesired outcomes, namely duplicate entries. To address this problem we revisited the original approach and introduced the necessary changes to support for relocating elements within a JSON structure. We detail those changes and provide some examples.
Dosimetry audits are carried out to determine how well radiotherapy is delivered to the patient. It is also used to understand the uncertainty introduced into the measurement result when using different computational models. As measurement procedures are becoming increasingly complex with technological advancements, it is harder to establish sources of variability in measurements and understand if they stem from true differences in measurands or in the measurement pipelines themselves. The gamma index calculation is a widely accepted metric used for the comparison of measured and predicted doses in radiotherapy. However, various steps in the measurement pipeline can introduce variation in the measurement result. In this paper, we perform a sensitivity and correlation analysis to investigate the influence of various input factors (i.e. setting) in gamma index calculations on the uncertainty introduced in dosimetry audits. We identify a number of factors where standardization will improve measurements by reducing variability in outputs. Furthermore, we also compare gamma index metrics and similarities across audit sites.
Drawing from engineering systems and control theory, we introduce a framework to understand repository stability, which is a repository activity capacity to return to equilibrium following disturbances - such as a sudden influx of bug reports, key contributor departures, or a spike in feature requests. The framework quantifies stability through four indicators: commit patterns, issue resolution, pull request processing, and community engagement, measuring development consistency, problem-solving efficiency, integration effectiveness, and sustainable participation, respectively. These indicators are synthesized into a Composite Stability Index (CSI) that provides a normalized measure of repository health proxied by its stability. Finally, the framework introduces several important theoretical properties that validate its usefulness as a measure of repository health and stability. At a conceptual phase and open to debate, our work establishes mathematical criteria for evaluating repository stability and proposes new ways to understand sustainable development practices. The framework bridges control theory concepts with modern collaborative software development, providing a foundation
Quantum computing is an emerging field with growing implications across science and industry, making early educational exposure increasingly important. This paper examines how quantum computing concepts can be introduced into high-school STEM curricula within existing structures to enhance foundational learning in mathematics, computer science, and physics. We outline a modular integration strategy introducing key quantum ideas into standard courses, leveraging open-source educational resources to ensure global accessibility. Emphasis is placed on educational opportunity and equity: the approach is designed to be inclusive and to bridge current curricular gaps so that students worldwide can develop basic quantum literacy. Our analysis demonstrates that integrating quantum topics at the secondary level is feasible and can enrich STEM learning.
Applying a machine learning model for decision-making in the real world requires to distinguish what the model knows from what it does not. A critical factor in assessing the knowledge of a model is to quantify its predictive uncertainty. Predictive uncertainty is commonly measured by the entropy of the Bayesian model average (BMA) predictive distribution. Yet, the properness of this current measure of predictive uncertainty was recently questioned. We provide new insights regarding those limitations. Our analyses show that the current measure erroneously assumes that the BMA predictive distribution is equivalent to the predictive distribution of the true model that generated the dataset. Consequently, we introduce a theoretically grounded measure to overcome these limitations. We experimentally verify the benefits of our introduced measure of predictive uncertainty. We find that our introduced measure behaves more reasonably in controlled synthetic tasks. Moreover, our evaluations on ImageNet demonstrate that our introduced measure is advantageous in real-world applications utilizing predictive uncertainty.
In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes using the evaluation at non-rational places of a function field. In this paper, we show that one can obtain a locality parameter $r$ in such codes by using only non-rational places of degrees at most $r$. This is, up to the author's knowledge, a new way to construct locally recoverable codes (LRCs). We give an example of such a code reaching the Singleton-like bound for LRCs, and show the parameters obtained for some longer codes over $\mathbb F_3$. We then investigate similarities with certain concatenated codes. Contrary to previous methods, our construction allows one to obtain directly codes whose dimension is not a multiple of the locality. Finally, we give an asymptotic study using the Garcia-Stichtenoth tower of function fields, for both our construction and a construction of concatenated codes. We give explicit infinite families of LRCs with locality 2 over any finite field of cardinality greater than 3 following our new approach.