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Complementary decompositions of monomial ideals and involutive bases
(2022-06-25)
Complementary decompositions of monomial ideals - also known as Stanley decompositions - play an important role in many places in commutative algebra. In this article, we discuss and compare several algorithms for their computation. This includes a classical recursive one, an algorithm already proposed by Janet and a construction proposed by Hironaka in his work on idealistic exponents. We relate Janet’s algorithm to the Janet tree of the Janet basis and extend this idea to Janet-like bases to obtain an optimised ...
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On Non-linear Characterizations of Classical Orthogonal Polynomials
(2022-12-03)
Classical orthogonal polynomials are known to satisfy seven equivalent properties, namely the Pearson equation for the linear functional, the second-order differential/difference/q-differential/ divided-difference equation, the orthogonality of the derivatives, the Rodrigues formula, two types of structure relations, and the Riccati equation for the formal Stieltjes function. In this work, following previous work by Kil et al., we state and prove a non-linear characterization result for classical orthogonal polynomials ...
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How to Improve Performance in Bayesian Inference Tasks: A Comparison of Five Visualizations
(2019-02-20)
Bayes’ formula is a fundamental statistical method for inference judgments in uncertain situations used by both laymen and professionals. However, since people often fail in situations where Bayes’ formula can be applied, how to improve their performance in Bayesian situations is a crucial question. We based our research on a widely accepted beneficial strategy in Bayesian situations, representing the statistical information in the form of natural frequencies. In addition to this numerical format, we used five ...
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The Impact of Visualizing Nested Sets. An Empirical Study on Tree Diagrams and Unit Squares
(2017-01-06)
It is an ongoing debate, what properties of visualizations increase people’s performance when solving Bayesian reasoning tasks. In the discussion of the properties of two visualizations, i.e., the tree diagram and the unit square, we emphasize how both visualizations make relevant subset relations transparent. Actually, the unit square with natural frequencies reveals the subset relation that is essential for the Bayes’ rule in a numerical and geometrical way whereas the tree diagram with natural frequencies does it ...
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A New Identity for Generalized Hypergeometric Functions and Applications
(2019-01-18)
We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized hypergeometric functions, Axioms, 7 (2018), Article 38).
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Improving Prospective Teachers’ Beliefs About a Double Discontinuity Between School Mathematics and University Mathematics
(2022-05-12)
Prospective teachers often perceive a “double discontinuity” between school mathematics and university mathematics. The first discontinuity can be described as the belief that there is no coherence between school mathematics and university mathematics, which forms part of the notoriously problematic transition from school to university. The second discontinuity can be described as a belief about the lack of relevance of university mathematics for the later professional practice of prospective teachers. Beliefs about ...
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An extendable key space integer image-cipher using 4-bit piece-wise linear cat map
(2022-09-30)
This paper presents a multiplierless image-cipher, with extendable 2048-bit key-space, based on a 4-dimensional (4D) quantized piece-wise linear cat map (PWLCM). The quantized PWLCM exhibits limit-cycles of 4-bit encoded integers with periods greater than 10⁷. The synthesis of the PWLCM in a finite state space allows to eliminate the undesirable finite precision effect due to the hardware realization. The proposed image-cipher combines chaos, modular arithmetic, and lattice-based cryptography to encrypt a color image ...
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Data Driven prediction of forced nonlinear vibrations using stabilised Autoregressive Neural Networks
(2023-03-24)
In this work, we propose a novel approach to the data-driven prediction of vibration responses of nonlinear systems. The main idea is based on Autoregressive Neural Networks (ARNN) to model the nonlinear transfer behaviour between an external excitation and the system response. We propose an autoregressive network architecture with embedded symmetry using bias-free tanh activation and guarantee Input-to-State-Stability (ISS) by enforcing a special penalty term to the weights. The resulting training procedure is ...
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An Investigation on Limiting Potentials for Damage Prediction of Viscoelastic Adhesives
(2023-03-24)
In his recent work VOLOKH introduced the energy limiting method as an alternative to continuum damage mechanics developed and envolved by KACHANOV, RABOTNOV, LEMAITRE and many more. In contrast to continuum damage mechanics, the energy limiting method is physically grounded by the process of atomistic debonding and, therefore, motivated on an energy basis. In order to use the energy limiting method within the framework of continuum mechanics, a limiting potential needs to be defined, which can be seen as a macroscopic ...
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Consideration of nonlinear multipoint constraints in finite element analyses based on a master-slave elimination scheme operating at the global level
(2023-03-24)
A method to consider nonlinear multipoint constraints in finite element analysis based on a master-slave elimination scheme operating at the global level is presented. We focus on systems with nonlinear constraints in which the number of constraints is of the same order of magnitude as the number of degrees of freedom. Therefore, we rely on the master-slave elimination which gives us the huge benefit of drastically reducing the problem size. In the literature the presented master-slave elimination schemes for linear ...