74,769 results
Search Results
2. Torn-Paper Coding
- Author
-
Ilan Shomorony and Alireza Vahid
- Subjects
Combinatorics ,Computer science ,Block (permutation group theory) ,Binary number ,Library and Information Sciences ,Dna storage ,Computer Science Applications ,Information Systems - Abstract
We consider the problem of communicating over a channel that randomly “tears” the message block into small pieces of different sizes and shuffles them. For the binary torn-paper channel with block length $n$ and pieces of length ${\mathrm{ Geometric}}(p_{n})$ , we characterize the capacity as $C = e^{-\alpha }$ , where $\alpha = \lim _{n\to \infty } p_{n} \log n$ . Our results show that the case of ${\mathrm{ Geometric}}(p_{n})$ -length fragments and the case of deterministic length- $(1/p_{n})$ fragments are qualitatively different and, surprisingly, the capacity of the former is larger. Intuitively, this is due to the fact that, in the random fragments case, large fragments are sometimes observed, which boosts the capacity.
- Published
- 2021
3. An improved bound on the optimal paper Moebius band
- Author
-
Richard Evan Schwartz
- Subjects
Combinatorics ,Aspect ratio ,Differential geometry ,Hyperbolic geometry ,Geometry and Topology ,Algebraic geometry ,Lambda ,Topology (chemistry) ,Projective geometry ,Mathematics - Abstract
We show that a smooth embedded paper Moebius band must have aspect ratio at least $$\begin{aligned}\lambda _1= \frac{2 \sqrt{4-2 \sqrt{3}}+4}{\root 4 \of {3} \sqrt{2}+2 \sqrt{2 \sqrt{3}-3}} =1.69497\ldots \end{aligned}$$ This bound comes more than 3/4 of the way from the old known bound of $$\pi /2=1.5708\ldots $$ to the conjectured bound of $$\sqrt{3} = 1.732\ldots $$
- Published
- 2021
4. Sendov’s Conjecture: A Note on a Paper of Dégot
- Author
-
T. P. Chalebgwa
- Subjects
Combinatorics ,Conjecture ,General Mathematics ,Sendov's conjecture ,Complex polynomial ,Unit distance ,Unit disk ,Critical point (mathematics) ,Mathematics - Abstract
Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Degot proved that, for each a ∈ (0, 1), there exists an integer N such that for any polynomial P(z) with degree greater than N, if P(a) = 0 and all zeroes lie inside the unit disk, the disk |z − a| ≤ 1 contains a critical point of P(z). Based on this result, we derive an explicit formula N(a) for each a ∈ (0, 1) and, consequently obtain a uniform bound N for all a ∈ [α, β] where 0 < α < β < 1. This (partially) addresses the questions posed in Degot’s paper.
- Published
- 2020
5. Disruptive papers published in Scientometrics: meaningful results by using an improved variant of the disruption index originally proposed by Wu, Wang, and Evans (2019)
- Author
-
George Chacko, Alexander Tekles, Lutz Bornmann, and Sitaram Devarakonda
- Subjects
Combinatorics ,05 social sciences ,General Social Sciences ,Field (mathematics) ,0509 other social sciences ,Library and Information Sciences ,Scientometrics ,050905 science studies ,050904 information & library sciences ,Measure (mathematics) ,Computer Science Applications ,Mathematics - Abstract
Wu et al. (Nature 566:378–382, 2019) introduced a new indicator measuring disruption ($${DI}_{1}$$DI1). Bornmann et al. (Do disruption index indicators measure what they propose to measure? The comparison of several indicator variants with assessments by peers, 2019. https://arxiv.org/abs/1911.08775) compared variants of the disruption index and pointed to $${DI}_{5}$$DI5 as an interesting variant. The calculation of a field-specific version of $${DI}_{5}$$DI5 (focusing on disruptiveness within the same field) for Scientometrics papers in the current study reveals that the variant is possibly able to identify landmark papers in scientometrics. This result is in contrast to the Scientometrics analysis previously published by Bornmann and Tekles (Scientometrics 120(1):331–336, 2019) based on the original disruption index ($${DI}_{1}$$DI1).
- Published
- 2020
6. Partner choice correlates with fine scale kin structuring in the paper wasp Polistes dominula
- Author
-
Paul J. Parsons, Jeremy Field, and Lena Grinsted
- Subjects
0106 biological sciences ,0301 basic medicine ,Topography ,Heredity ,Wasps ,NERC ,Social Sciences ,01 natural sciences ,Nesting Behavior ,Habits ,Nest ,Psychology ,Inbreeding ,Islands ,education.field_of_study ,Multidisciplinary ,Behavior, Animal ,biology ,Agricultural and Biological Sciences(all) ,Eusociality ,Spring ,Databases as Topic ,Physical Sciences ,NE/M003191/1 ,Medicine ,Female ,Seasons ,Research Article ,Statistical Distributions ,Kin recognition ,Permutation ,Science ,Population ,Cuticular Hydrocarbons ,Insect Physiology ,Polistes dominula ,NE/K00655X/1 ,010603 evolutionary biology ,Nesting Habits ,03 medical and health sciences ,Genetics ,Animal Physiology ,Animals ,Social Behavior ,education ,General ,Invertebrate Physiology ,Paper wasp ,Evolutionary Biology ,Behavior ,Landforms ,Population Biology ,Discrete Mathematics ,Biochemistry, Genetics and Molecular Biology(all) ,Biology and Life Sciences ,RCUK ,Geomorphology ,Probability Theory ,biology.organism_classification ,Statistical Dispersion ,030104 developmental biology ,Natal homing ,Combinatorics ,Evolutionary biology ,Earth Sciences ,Philopatry ,Zoology ,Entomology ,Population Genetics ,Mathematics - Abstract
Cooperation among kin is common in animal societies. Kin groups may form by individuals directly discriminating relatives based on kin recognition cues, or form passively through natal philopatry and limited dispersal. We describe the genetic landscape for a primitively eusocial wasp, Polistes dominula, and ask whether individuals choose cooperative partners that are nearby and/or that are genetic relatives. Firstly, we genotyped an entire sub-population of 1361 wasps and found genetic structuring on an extremely fine scale: the probability of finding genetic relatives decreases exponentially within just a few meters of an individual’s nest. At the same time, however, we found a lack of genetic structuring between natural nest aggregations within the population. Secondly, in a separate dataset where ~2000 wasps were genotyped, we show that wasps forced experimentally to make a new nest choice tended to choose new nests near to their original nests, and that these nests tended to contain some full sisters. However, a significant fraction of wasps chose nests that did not contain sisters, despite sisters being present in nearby nests. Although we cannot rule out a role for direct kin recognition or natal nest-mate recognition, our data suggest that kin groups may form via a philopatric rule-of-thumb, whereby wasps simply select groups and nesting sites that are nearby. The result is that most subordinate helpers obtain indirect fitness benefits by breeding cooperatively.
- Published
- 2019
7. Ramsey, Paper, Scissors
- Author
-
Jacob Fox, Xiaoyu He, and Yuval Wigderson
- Subjects
Computer Science::Computer Science and Game Theory ,Applied Mathematics ,General Mathematics ,Combinatorial game theory ,0102 computer and information sciences ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Upper and lower bounds ,Combinatorics ,010201 computation theory & mathematics ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,FOS: Mathematics ,Mathematics - Combinatorics ,Graph (abstract data type) ,Combinatorics (math.CO) ,Ramsey's theorem ,Null graph ,Software ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics ,Independence number - Abstract
We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on $n$ vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least $s$. We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants $0B\sqrt{n}\log{n}$. This is a factor of $\Theta(\sqrt{\log{n}})$ larger than the lower bound coming from the off-diagonal Ramsey number $r(3,s)$.
- Published
- 2020
8. Abelian powers in paper-folding words
- Author
-
Holub, Štěpán
- Subjects
- *
ABELIAN groups , *PAPER arts , *VOCABULARY , *ARBITRARY constants , *ALGEBRA , *COMBINATORICS - Abstract
Abstract: We show that paper-folding words contain arbitrarily large abelian powers. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
9. On the Dispersions of the Gel’fand–Pinsker Channel and Dirty Paper Coding
- Author
-
Jonathan Scarlett
- Subjects
Independent and identically distributed random variables ,Gaussian ,Variable-length code ,Data_CODINGANDINFORMATIONTHEORY ,Library and Information Sciences ,Computer Science Applications ,Gel’fand-Pinsker channel ,Combinatorics ,symbols.namesake ,Second-order coding rate ,Shannon–Fano coding ,Dirty paper coding ,symbols ,Channel dispersion ,Channels with state ,Algorithm ,Encoder ,Decoding methods ,Information Systems ,Mathematics ,Coding (social sciences) - Abstract
This paper studies the second-order coding rates for memoryless channels with a state sequence known non-causally at the encoder. In the case of finite alphabets, an achievability result is obtained using constant-composition random coding, and by using a small fraction of the block to transmit the empirical distribution of the state sequence. For error probabilities less than 0.5, it is shown that the second-order rate improves on an existing one based on independent and identically distributed random coding. In the Gaussian case (dirty paper coding) with an almost-sure power constraint, an achievability result is obtained using random coding over the surface of a sphere, and using a small fraction of the block to transmit a quantized description of the state power. It is shown that the second-order asymptotics are identical to the single-user Gaussian channel of the same input power without a state.
- Published
- 2015
10. A remark on a paper of P. B. Djakov and M. S. Ramanujan
- Author
-
Murat Yurdakul and Elif Uyanik
- Subjects
Unbounded operator ,Combinatorics ,symbols.namesake ,Monotone polygon ,Basis (linear algebra) ,General Mathematics ,Bounded function ,Operator (physics) ,symbols ,Sequence space ,Continuous linear operator ,Ramanujan's sum ,Mathematics - Abstract
Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\"{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\"{o}the matrices when every continuous linear operator between l-K\"{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\"{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.
- Published
- 2019
11. Communicating over the Torn-Paper Channel
- Author
-
Alireza Vahid and Ilan Shomorony
- Subjects
FOS: Computer and information sciences ,0301 basic medicine ,Computer science ,Computer Science - Information Theory ,Information Theory (cs.IT) ,Block (permutation group theory) ,020206 networking & telecommunications ,02 engineering and technology ,Combinatorics ,03 medical and health sciences ,030104 developmental biology ,Channel (programming) ,0202 electrical engineering, electronic engineering, information engineering ,Computer Science::Information Theory ,Communication channel - Abstract
We consider the problem of communicating over a channel that randomly "tears" the message block into small pieces of different sizes and shuffles them. For the binary torn-paper channel with block length $n$ and pieces of length ${\rm Geometric}(p_n)$, we characterize the capacity as $C = e^{-\alpha}$, where $\alpha = \lim_{n\to\infty} p_n \log n$. Our results show that the case of ${\rm Geometric}(p_n)$-length fragments and the case of deterministic length-$(1/p_n)$ fragments are qualitatively different and, surprisingly, the capacity of the former is larger. Intuitively, this is due to the fact that, in the random fragments case, large fragments are sometimes observed, which boosts the capacity.
- Published
- 2020
12. Fuzzy quotient-3 cordial labeling of star related graphs- Paper I
- Author
-
P. Sumathi and J. Suresh Kumar
- Subjects
Combinatorics ,Star (graph theory) ,Fuzzy logic ,Quotient ,Mathematics - Published
- 2019
13. The Analysis of Type and Expression Method for Chinese Paper-cut Sign
- Author
-
NamHojung
- Subjects
Combinatorics ,Paper cut ,medicine ,Semiotics ,General Medicine ,Type (model theory) ,Expression (computer science) ,medicine.disease ,Sign (mathematics) ,Mathematics - Published
- 2013
14. Menon-type identities again: A note on a paper by Li, Kim and Qiao
- Author
-
László Tóth, Pentti Haukkanen, Informaatioteknologian ja viestinnän tiedekunta - Faculty of Information Technology and Communication Sciences, and Tampere University
- Subjects
Combinatorics ,Identity (mathematics) ,Character (mathematics) ,Mathematics - Number Theory ,Simple (abstract algebra) ,General Mathematics ,Matematiikka - Mathematics ,Arithmetic function ,Function (mathematics) ,11A07, 11A25 ,Type (model theory) ,Mathematics - Group Theory ,Mathematics - Abstract
We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function. We present a simple character free approach for the proof., Comment: 14 pages
- Published
- 2019
15. A Remark on the Paper 'Properties of Intersecting Families of Ordered Sets' by O. Einstein
- Author
-
Sounggun Wee and Sang-il Oum
- Subjects
010102 general mathematics ,Mistake ,0102 computer and information sciences ,01 natural sciences ,Linear subspace ,Combinatorics ,Computational Mathematics ,symbols.namesake ,010201 computation theory & mathematics ,Ordered set ,FOS: Mathematics ,symbols ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Einstein ,05C35 ,Finite set ,Mathematics - Abstract
O. Einstein (2008) proved Bollob\'as-type theorems on intersecting families of ordered sets of finite sets and subspaces. Unfortunately, we report that the proof of a theorem on ordered sets of subspaces had a mistake. We prove two weaker variants., Comment: 6 pages. Improved bound for Theorem 4
- Published
- 2018
16. A close-to-capacity dirty paper coding scheme
- Author
-
S. ten Brink and Uri Erez
- Subjects
Theoretical computer science ,Quantization (signal processing) ,Detector ,Transmitter ,Vector quantization ,Data_CODINGANDINFORMATIONTHEORY ,Library and Information Sciences ,Interference (wave propagation) ,EXIT chart ,Precoding ,Computer Science Applications ,Combinatorics ,Channel capacity ,Single antenna interference cancellation ,Code (cryptography) ,Dirty paper coding ,Algorithm ,Decoding methods ,Computer Science::Information Theory ,Information Systems ,Mathematics ,Data compression - Abstract
The "writing on dirty paper"-channel model offers an information-theoretic framework for precoding techniques for canceling arbitrary interference known at the transmitter. It indicates that lossless precoding is theoretically possible at any signal-to-noise ratio (SNR), and thus dirty-paper coding may serve as a basic building block in both single-user and multiuser communication systems. We design an end-to-end coding realization of a system materializing a significant portion of the promised gains. We employ multidimensional quantization based on trellis shaping at the transmitter. Coset decoding is implemented at the receiver using "virtual bits." Combined with iterative decoding of capacity-approaching codes we achieve an improvement of 2dB over the best scalar quantization scheme. Code design is done using the EXIT chart technique.
- Published
- 2005
17. Addendum to Paper Entitled 'Do Prime Numbers Obey a Three Dimensional Double Helix?'
- Author
-
Peter Bissonnet
- Subjects
Combinatorics ,Almost prime ,Prime factor ,Twin prime ,Calculus ,Prime triplet ,Prime power ,Probable prime ,Prime k-tuple ,Mathematics ,Sphenic number - Abstract
This paper again specifies the major points of the article “Do Prime Numbers Obey a Three-Dimensional Double Helix?” [1] which was received on February 16, 2006 by Hadronic Journal. New information has been added and elucidated upon, such as why the numbers 2 and 3 are not considered true prime numbers, and why s in the following formulas for 6s - 1 and for 6s + 1 is really a composite number equal to the sum of two other numbers, suggesting that s is always to be considered as an integer. Other new information is added as well, such as how an engineer in a matter of seconds decomposed a large prime product into its constituent primes using basic software and won a contract for his firm.
- Published
- 2017
18. Local Restrictions from the Furst-Saxe-Sipser Paper
- Author
-
Osamu Watanabe and Suguru Tamaki
- Subjects
Discrete mathematics ,Computational complexity theory ,Parity function ,True quantified Boolean formula ,Boolean circuit ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Bounded function ,Theory of computation ,Isomorphism ,0101 mathematics ,Boolean satisfiability problem ,Mathematics - Abstract
In their celebrated paper (Furst et al., Math. Syst. Theory 17(1), 13---27 (12)), Furst, Saxe, and Sipser used random restrictions to reveal the weakness of Boolean circuits of bounded depth, establishing that constant-depth and polynomial-size circuits cannot compute the parity function. Such local restrictions have played important roles and have found many applications in complexity analysis and algorithm design over the past three decades. In this article, we give a brief overview of two intriguing applications of local restrictions: the first one is for the Isomorphism Conjecture and the second one is for moderately exponential time algorithms for the Boolean formula satisfiability problem.
- Published
- 2016
19. Modelling the folding of paper into three dimensions using affine transformations
- Author
-
Thomas C. Hull and sarah-marie belcastro
- Subjects
Paper ,Transformations ,Numerical Analysis ,Algebra and Number Theory ,Non-flat folding ,Affine ,Identity matrix ,Folding ,Folding (DSP implementation) ,Computer Science::Computational Geometry ,Vertex (geometry) ,Combinatorics ,Homogeneous ,Product (mathematics) ,Piecewise ,Discrete Mathematics and Combinatorics ,Order (group theory) ,Geometry and Topology ,Affine transformation ,Mathematics - Abstract
We model the folding of ordinary paper via piecewise isometries R 2 → R 3 . The collection of crease lines and vertices in the unfolded paper is called the crease pattern. Our results generalize the previously known necessity conditions from the more restrictive case of folding paper flat (into R 2 ); if the crease pattern is foldable, then the product (in a non-intuitive order) of the associated rotational matrices is the identity matrix. This condition holds locally in a multiple vertex crease pattern and can be adapted to a global condition. Sufficiency conditions are significantly harder, and are not known except in the two-dimensional single-vertex case.
- Published
- 2002
20. The Byline: Thoughts on the distribution of author ranks in multiauthored papers
- Author
-
Liming Liang, Leo Egghe, and Ronald Rousseau
- Subjects
Combinatorics ,Matrix (mathematics) ,Multiauthorship matrix ,Byline ,Author productivity distribution ,Authors per paper ,Rank distribution ,Predicted rank ,Seed ,Ranking ,Distribution (number theory) ,Modeling and Simulation ,Modelling and Simulation ,Calculus ,Probability distribution ,Rank (graph theory) ,Alphabet ,Mathematics ,Computer Science Applications - Abstract
We analyze the multiauthorship matrix M, defined as the matrix where a cell M (j, @k) denotes the number of times authors with j publications are ranked as the @k^t^h author of an article. We prove that if the distribution of the number of authors per paper follows a power law, then the author rank distribution is approximately equal to this power law (more precisely, equal in Landau's big O sense). We further determine the author rank distribution in the case where authors can be characterized through a seed number; this is the probability of preceding a fixed author in the byline of an article. Such a seed is determined for alphabetical ranking of authors using the standard western alphabet.
- Published
- 2003
- Full Text
- View/download PDF
21. Addendum and corrigenda to the paper 'Infinitary superperfect numbers'
- Author
-
Tomohiro Yamada
- Subjects
Combinatorics ,General Computer Science ,General Mathematics ,Addendum ,Mathematics - Published
- 2018
22. Distribution functions of ratio sequences. An expository paper
- Author
-
Oto Strauch
- Subjects
Combinatorics ,Discrete mathematics ,Distribution function ,General Mathematics ,Mathematics - Abstract
This expository paper presents known results on distribution functions g(x) of the sequence of blocks where xn is an increasing sequence of positive integers. Also presents results of the set G(Xn) of all distribution functions g(x). Specially: - continuity of g(x); - connectivity of G(Xn); - singleton of G(Xn); - one-step g(x); - uniform distribution of Xn, n = 1, 2, . . . ; - lower and upper bounds of g(x); - applications to bounds of ; - many examples, e.g., , where pn is the nth prime, is uniformly distributed. The present results have been published by 25 papers of several authors between 2001-2013.
- Published
- 2015
23. A letter concerning Leonetti's paper 'Continuous Projections onto Ideal Convergent Sequences'
- Author
-
Tomasz Kania
- Subjects
Mathematics::Functional Analysis ,Applied Mathematics ,010102 general mathematics ,46B20, 46B26 (primary), and 40A35 (secondary) ,Space (mathematics) ,Quotient space (linear algebra) ,01 natural sciences ,Functional Analysis (math.FA) ,010101 applied mathematics ,Combinatorics ,Mathematics - Functional Analysis ,Mathematics (miscellaneous) ,FOS: Mathematics ,Uncountable set ,Ideal (ring theory) ,Family of sets ,0101 mathematics ,Mathematics - Abstract
Leonetti proved that whenever $${\mathcal {I}}$$ is an ideal on $${\mathbb {N}}$$ such that there exists an uncountable family of sets that are not in $${\mathcal {I}}$$ with the property that the intersection of any two distinct members of that family is in $${\mathcal {I}}$$ , then the space $$c_{0,{\mathcal {I}}}$$ of sequences in $$\ell _\infty $$ that converge to 0 along $${\mathcal {I}}$$ is not complemented. We provide a shorter proof of a more general fact that the quotient space $$\ell _\infty / c_{0,{\mathcal {I}}}$$ does not even embed into $$\ell _\infty $$ .
- Published
- 2018
24. Call for Papers: Crossover
- Author
-
Cecilia Obiol
- Subjects
Pulmonary and Respiratory Medicine ,Combinatorics ,Computer science ,Pediatrics, Perinatology and Child Health ,Crossover ,education ,Arquitectura [Àrees temàtiques de la UPC] ,Architecture ,Arquitectura - Published
- 2018
25. Partner choice correlates with fine scale kin structuring in the paper wasp Polistes dominula.
- Author
-
Parsons, Paul John, Grinsted, Lena, and Field, Jeremy
- Subjects
- *
PHILOPATRY , *WASPS , *ANIMAL societies , *PHYSIOLOGY , *NESTS , *BIRD breeding - Abstract
Cooperation among kin is common in animal societies. Kin groups may form by individuals directly discriminating relatives based on kin recognition cues, or form passively through natal philopatry and limited dispersal. We describe the genetic landscape for a primitively eusocial wasp, Polistes dominula, and ask whether individuals choose cooperative partners that are nearby and/or that are genetic relatives. Firstly, we genotyped an entire sub-population of 1361 wasps and found genetic structuring on an extremely fine scale: the probability of finding genetic relatives decreases exponentially within just a few meters of an individual’s nest. At the same time, however, we found a lack of genetic structuring between natural nest aggregations within the population. Secondly, in a separate dataset where ~2000 wasps were genotyped, we show that wasps forced experimentally to make a new nest choice tended to choose new nests near to their original nests, and that these nests tended to contain some full sisters. However, a significant fraction of wasps chose nests that did not contain sisters, despite sisters being present in nearby nests. Although we cannot rule out a role for direct kin recognition or natal nest-mate recognition, our data suggest that kin groups may form via a philopatric rule-of-thumb, whereby wasps simply select groups and nesting sites that are nearby. The result is that most subordinate helpers obtain indirect fitness benefits by breeding cooperatively. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
26. Paper or and digital : A study on combinatorics in preschool class
- Author
-
van Bommel, Jorryt and Palmér, Hanna
- Subjects
Matematik ,Combinatorics ,semi-concrete ,children’s documentation ,systematization ,digital application ,Educational Sciences ,Utbildningsvetenskap ,Mathematics - Abstract
In a design research study conducted in Swedish preschool class, children were asked: In how many ways can three toy bears sit in a sofa? In the initial design cycle the children who used pictographic representations seemed to make more systematic solutions with less duplications than children who used iconic representations. A later design cycle involved a digital application that was designed to provide an opportunity to work with pictographic representations on a semi-concrete level. In this paper, we make a comparison of paper and pencil documentation made by children who have used, respective not have used; the digital application. The results indicate that working with the digital application resulted in more systematic paper and pencil documentation with fewer duplications. Titelns "or" är i konferensbidraget genomstruket.
- Published
- 2018
27. On a paper of Beltrán and Shao about coprime action
- Author
-
H. Meng and Adolfo Ballester-Bolinches
- Subjects
Algebra and Number Theory ,Coprime integers ,Mathematics::Number Theory ,010102 general mathematics ,Structure (category theory) ,Automorphism ,01 natural sciences ,Prime (order theory) ,Action (physics) ,Combinatorics ,Mathematics::Group Theory ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Assume that A and G are finite groups of coprime orders such that A acts on G via automorphisms. Let p be a prime. The following coprime action version of a well-known theorem of Ito about the structure of a minimal non-p-nilpotent groups is proved: if every maximal A-invariant subgroup of G is p-nilpotent, then G is p-soluble. If, moreover, G is not p-nilpotent, then G must be soluble. Some earlier results about coprime action are consequences of this theorem.
- Published
- 2020
28. Discussion of the paper: 'Sampling schemes for generalized linear Dirichlet process random effects models' by M. Kyung, J. Gill, and G. Casella
- Author
-
Antonietta Mira, Mark Girolami, and Maurizio Filippone
- Subjects
Statistics and Probability ,Combinatorics ,Hierarchical Dirichlet process ,Dirichlet process ,Applied mathematics ,Sampling (statistics) ,Statistics, Probability and Uncertainty ,Random effects model ,Mathematics - Published
- 2018
29. On D.Y. Gao and X. Lu paper 'On the extrema of a nonconvex functional with double-well potential in 1D'
- Author
-
Constantin Zălinescu
- Subjects
021103 operations research ,Applied Mathematics ,General Mathematics ,0211 other engineering and technologies ,General Physics and Astronomy ,Double-well potential ,02 engineering and technology ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Maxima and minima ,35J20, 35J60, 74G65, 74S30 ,Optimization and Control (math.OC) ,FOS: Mathematics ,Preprint ,0101 mathematics ,Constant (mathematics) ,Mathematics - Optimization and Control ,Subspace topology ,Mathematics - Abstract
The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3
- Published
- 2017
30. CLASSIFICATION OF GENERALIZED PAPER FOLDING SEQUENCES
- Author
-
Nahmwoo Hahm, Junhwi Lim, and Jung-Hee Yun
- Subjects
Combinatorics ,Mathematics - Abstract
Generalized paper folding sequences X n and (XpYq) n where X;Y 2 fR;L;U;Dg, and n;p;q2 N with p;q 2 are clas- sied in this paper. We show that all generalized paper folding sequences X n p are classied into one type if we classify generalized paper folding sequences along with the numbers of downwards and upwards. In addition, we investigate the numbers of downwards and upwards in (XpYq) n and prove that all generalized paper fold- ing sequences (XpYq) n are classied into two types.
- Published
- 2013
31. COUNTING PROBLEMS IN GENERALIZED PAPER FOLDING SEQUENCES
- Author
-
Jung-Hee Yun and Nahm-Woo Hahm
- Subjects
Discrete mathematics ,Combinatorics ,Counting problem ,Natural number ,Folding (DSP implementation) ,Mathematics - Abstract
In this paper, we discuss numbers of downwards and upwards in generalized paper folding sequences. We compute the exact number of downwards and upwards in R n p and (RpRq) n by using the properties of recursive sequences where n;p and q are natural numbers with p 2 and q 2.
- Published
- 2012
32. Averages of Ramanujan sums: note on two papers by E. Alkan
- Author
-
László Tóth
- Subjects
Algebra and Number Theory ,Mathematics - Number Theory ,Logarithm ,Generalization ,Mathematics::Number Theory ,16. Peace & justice ,Ramanujan's sum ,Bernoulli polynomials ,Combinatorics ,symbols.namesake ,Identity (mathematics) ,11A25, 11B68, 33B15 ,symbols ,Arithmetic function ,Gamma function ,Binomial coefficient ,Mathematics - Abstract
We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd's, the Gamma function, the Bernoulli polynomials, and binomial coefficients., Comment: 8 pages
- Published
- 2014
33. On a paper of S S Pillai
- Author
-
Ravindranathan Thangadurai and M. Ram Murty
- Subjects
Combinatorics ,Discrete mathematics ,Correctness ,Argument ,General Mathematics ,Natural number ,Asymptotic formula ,Term (logic) ,Prime (order theory) ,Square (algebra) ,Mathematics - Abstract
In 1935, Erdos proved that all natural numbers can be written as a sum of a square of a prime and a square-free number. In 1939, Pillai derived an asymptotic formula for the number of such representations. The mathematical review of Pillai’s paper stated that the proof of the above result contained inaccuracies, thus casting a doubt on the correctness of the paper. In this paper, we re-examine Pillai’s paper and show that his argument was essentially correct. Afterwards, we improve the error term in Pillai’s theorem using the Bombieri–Vinogradov theorem.
- Published
- 2012
34. A note on O. Frolkina’s paper 'pairwise disjoint moebius bands in space'
- Author
-
Sergey A. Melikhov
- Subjects
Algebra and Number Theory ,010102 general mathematics ,Disjoint sets ,Space (mathematics) ,01 natural sciences ,Combinatorics ,symbols.namesake ,Simple (abstract algebra) ,0103 physical sciences ,symbols ,010307 mathematical physics ,Möbius strip ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
We note a simple algebraic proof of Frolkina’s result that [Formula: see text] does not contain uncountably many pairwise disjoint copies of the Möbius band, and of a similar result in higher dimensions.
- Published
- 2019
35. REMARKS ON THE PAPER: ORTHOGONALLY ADDITIVE AND ORTHOGONALLY QUADRATIC FUNCTIONAL EQUATION
- Author
-
Hark-Mahn Kim, Kil-Woung Jun, and Ahyoung Kim
- Subjects
Combinatorics ,Stability result ,Quadratic functional ,Mathematics - Abstract
The main goal of this paper is to present the additional stability results of the following orthogonally additive and orthogonally quadratic functional equation $$f(\frac{x}{2}+y)+f(\frac{x}{2}-y)+f(\frac{x}{2}+z)+f(\frac{x}{2}-z)
- Published
- 2013
36. A CORRECTION TO A PAPER ON ROMAN κ-DOMINATION IN GRAPHS
- Author
-
Seyed Mehdi Hosseini Moghaddam and Doost Ali Mojdeh
- Subjects
Combinatorics ,New digraph reconstruction conjecture ,Dominating set ,General Mathematics ,Bound graph ,Graph ,Mathematics ,Vertex (geometry) - Abstract
Let G = (V,E) be a graph and k be a positive integer. A k-dominating set of G is a subset S ⊆ V such that each vertex in V \S has atleast k neighbors in S. A Roman k-dominating function on G is a functionf : V → {0,1,2} such that every vertex v with f(v) = 0 is adjacent toat least k vertices v 1 ,v 2 ,...,v k with f(v i ) = 2 for i = 1,2,...,k. In thepaper titled “Roman k-domination in graphs” (J. Korean Math. Soc. 46(2009), no. 6, 1309–1318) K. Kammerling and L. Volkmann showed thatfor any graph G with n vertices, γ kR (G) + γ kR (G) ≥ min {2n,4k + 1},and the equality holds if and only if n ≤ 2k or k ≥ 2 and n = 2k + 1or k = 1 and G or G has a vertex of degree n − 1 and its complementhas a vertex of degree n − 2. In this paper we find a counterexampleof Kammerling and Volkmann’s result and then give a correction to theresult. 1. IntroductionLet G = (V,E) be agraphwith vertexset V = V (G) andedge set E = E(G).A k-dominatingsetof G is a subset S ⊆ V such that every vertex in V \S has atleast k neighbors in S. The k-domination numberγ
- Published
- 2013
37. Constructing the Cubus simus and the Dodecaedron simum via paper folding
- Author
-
Klaudia Kwickert and Urs Hartl
- Subjects
Snub dodecahedron ,Hyperbolic geometry ,Metric Geometry (math.MG) ,Folding (DSP implementation) ,Snub cube ,Archimedean solid ,Combinatorics ,symbols.namesake ,Dodecahedron ,Mathematics - Metric Geometry ,Differential geometry ,FOS: Mathematics ,symbols ,Geometry and Topology ,51M15 (Primary) 51M20 (Secondary) ,Projective geometry ,Mathematics - Abstract
The archimedean solids Cubus simus (snub cube) and Dodecaedron simum (snub dodecahedron) cannot be constructed by ruler and compass. We explain that for general reasons their vertices can be constructed via paper folding on the faces of a cube, respectively dodecahedron, and we present explicit folding constructions. The construction of the Cubus simus is particularly elegant. We also review and prove the construction rules of the other Archimedean solids., Comment: 13 pages, 12 figures, v2: as published in Geometriae Dedicata
- Published
- 2012
38. Dynamical Systems and Irrational Angle Construction by Paper-Folding
- Author
-
Cayanne McFarlane and Wm. Douglas Withers
- Subjects
Combinatorics ,Number theory ,Dynamical systems theory ,Aperiodic graph ,General Mathematics ,Acute angle ,Irrational number ,Regular polygon ,Periodic sequence ,Mathematics ,Irrational rotation - Abstract
In Variations on a theme in paper folding, Burkard Polster [6] discusses an itera tive construction due to Hilton and Pedersen [3] for approximating any acute rational angle (rational multiple of n radians) to any desired accuracy by folding a strip of paper. Hilton and Pedersen, motivated by the construction of regular polygons and star-polygons, focus on the construction of rational angles. The construction employs a periodic sequence composed of two types of folding moves, guided by number theoretic properties of the angle's fractional representation (in fact, analysis of the construction yields new results in number theory?see [4], [1], and [2]). This note generalizes this construction to any acute angle, rational or irrational. For irrational angles, the number-theoretic prescription must obviously be replaced by something new; we show how to derive the appropriate aperiodic sequence of folding moves, via a connection between the folding construction and a simple dynamical system. Hilton and Pedersen's construction recursively generates a sequence of angles a0, otx, ... by a pair of folding operations on a strip of paper which we term Folds and Switchfolds. The angles ak appear between an edge of the strip and a crease created by folding. The Fold operation generates ak+x by bisecting ak (Figure 1)
- Published
- 2008
39. Comments on Y. O. Hamidoune's Paper 'Adding Distinct Congruence Classes'
- Author
-
Béla Bajnok
- Subjects
Statistics and Probability ,11B75 ,Lemma (mathematics) ,Mathematics - Number Theory ,Applied Mathematics ,Mistake ,Cyclic group ,Theoretical Computer Science ,Combinatorics ,Computational Theory and Mathematics ,Congruence (geometry) ,Order (group theory) ,Abelian group ,Argument (linguistics) ,Mathematics - Abstract
The main result in Y. O. Hamidoune's paper ‘Adding distinct congruence classes' (Combin. Probab. Comput.7 (1998) 81–87) is as follows. If S is a generating subset of a cyclic group G such that 0 ∉ S and |S| ⩾ 5, then the number of sums of the subsets of S is at least min(|G|, 2|S|). Unfortunately, the argument of the author, who, sadly, passed away in 2011, relies on a lemma whose proof is incorrect; in fact, the lemma is false for all cyclic groups of even order. In this short note we point out this mistake, correct the proof, and discuss why the main result is actually true for all finite abelian groups.
- Published
- 2015
40. An Upper Bound on the Capacity of the Causal Dirty-Paper Channel
- Author
-
M. Kesal and Uri Erez
- Subjects
Library and Information Sciences ,Precoding ,Noise (electronics) ,Upper and lower bounds ,Computer Science Applications ,Combinatorics ,symbols.namesake ,Channel capacity ,Interference (communication) ,Gaussian noise ,Bounded function ,symbols ,Computer Science::Information Theory ,Information Systems ,Mathematics ,Communication channel - Abstract
A bound on the capacity of the causal dirty-paper channel with arbitrary interference and general independent identically distributed additive noise is derived. In particular, it is shown that for the case of Gaussian noise, the capacity is upper bounded by log2(1+SNR/e) bits per real dimension. This bound is useful for SNR ≤ e(e-2) .
- Published
- 2011
41. A Question from a Famous Paper of Erdős
- Author
-
Edgardo Roldán-Pensado and Imre Bárány
- Subjects
010102 general mathematics ,Convex curve ,Regular polygon ,01 natural sciences ,Upper and lower bounds ,Theoretical Computer Science ,Combinatorics ,Computational Theory and Mathematics ,Discrete Mathematics and Combinatorics ,Convex body ,Point (geometry) ,Geometry and Topology ,0101 mathematics ,Mathematics - Abstract
Given a convex body $$K$$K, consider the smallest number $$N$$N so that there is a point $$P\in \partial K$$PźźK such that every circle centred at $$P$$P intersects $$\partial K$$źK in at most $$N$$N points. In 1946 Erdźs conjectured that $$N=2$$N=2 for all $$K$$K, but there are convex bodies for which this is not the case. As far as we know there is no known global upper bound. We show that no convex body has $$N=\infty $$N=ź and that there are convex bodies for which $$N = 6$$N=6.
- Published
- 2013
42. On an open question of V. Colao and G. Marino presented in the paper 'Krasnoselskii-Mann method for non-self mappings'
- Author
-
Meifang Guo, Xia Li, and Yongfu Su
- Subjects
Nonexpansive mapping ,Fixed-point theorem ,Lambda ,01 natural sciences ,Combinatorics ,symbols.namesake ,Strong convergence ,Countable set ,0101 mathematics ,Weak convergence ,Physics ,47H09 ,Multidisciplinary ,47H05 ,Research ,010102 general mathematics ,Regular polygon ,Hilbert space ,010101 applied mathematics ,Mann iterative scheme ,symbols ,Inward condition ,Non-self mapping ,Convex function ,47H10 - Abstract
Let H be a Hilbert space and let C be a closed convex nonempty subset of H and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T : C\rightarrow H$$\end{document}T:C→H a non-self nonexpansive mapping. A map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h : C\rightarrow R$$\end{document}h:C→R defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x) := \inf \{\lambda \ge 0 : \lambda x+(1-\lambda )Tx \in C \}$$\end{document}h(x):=inf{λ≥0:λx+(1-λ)Tx∈C}. Then, for a fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in C$$\end{document}x0∈C and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{\alpha _0} = \max \left\{ {\frac{1}{2},h({x_0})} \right\}\end{aligned}$$\end{document}α0=max12,h(x0), Krasnoselskii–Mann algorithm is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n+1}=\alpha _n+(1-\alpha _n)Tx_n,$$\end{document}xn+1=αn+(1-αn)Txn, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{n+1}=\max \{\alpha _n, h(x_{x_{n+1}})\}$$\end{document}αn+1=max{αn,h(xxn+1)}. Recently, Colao and Marino (Fixed Point Theory Appl 2015:39, 2015) have proved both weak and strong convergence theorems when C is a strictly convex set and T is an inward mapping. Meanwhile, they proposed a open question for a countable family of non-self nonexpansive mappings. In this article, authors will give an answer and will prove the further generalized results with the examples to support them.
- Published
- 2016
43. Complete Bernstein functions and subordinators with nested ranges. A note on a paper by P. Marchal
- Author
-
Chang-Song Deng and René L. Schilling
- Subjects
Statistics and Probability ,Measurable function ,Bernstein function ,Probability (math.PR) ,complete Bernstein function ,subordinator ,Lambda ,30E20 ,Combinatorics ,Probability space ,30H15 ,Exponent ,FOS: Mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,60G51 ,Mathematics - Abstract
Let $\alpha:[0,1]\to [0,1]$ be a measurable function. It was proved by P. Marchal \cite{Mar15} that the function $$ \phi^{(\alpha)}(\lambda):=\exp\left[ \int_0^1\frac{\lambda-1}{1+(\lambda-1)x}\,\alpha(x)\,d x \right],\quad \lambda>0 $$ is a special Bernstein function. Marchal used this to construct, on a single probability space, a family of regenerative sets $\mathcal R^{(\alpha)}$ such that $\mathcal{R}^{(\alpha)} \stackrel{\text{law}}{=} \overline{\{S^{(\alpha)}_t:t\geq 0\}}$ ($S^{(\alpha)}$ is the subordinator with Laplace exponent $\phi^{(\alpha)}$) and $\mathcal R^{(\alpha)}\subset \mathcal R^{(\beta)}$ whenever $\alpha\leq\beta$. We give two simple proofs showing that $\phi^{(\alpha)}$ is a complete Bernstein function and extend Marchal's construction to all complete Bernstein functions., Comment: to appear in Electron. Comm. Probab
- Published
- 2016
44. Review Papers Related to k-cordial of Research Labeling of Graphs
- Author
-
M.V. Modha
- Subjects
Combinatorics - Published
- 2017
45. Addendum to the paper: 'Artin prime producing quadratics', by P. Moree
- Author
-
Pieter Moree and Yves Gallot
- Subjects
Combinatorics ,Algebra ,Number theory ,General Mathematics ,Addendum ,Algebra over a field ,Primitive root modulo n ,Prime (order theory) ,Mathematics - Abstract
A record mentioned in the paper by Moree (Abh Math Sem Univ Hamburg 77:109–127, 2007) was recently improved on by Akbary and Scholten. However, the record mentioned was not the then record. The then record, due to Gallot (2004), actually slightly improves on that obtained recently by Akbary and Scholten.
- Published
- 2015
46. A note on the paper by Bugeaud and Laurent 'Minoration effective de la distance p-adique entre puissances de nombres algébriques'
- Author
-
Tomohiro Yamada
- Subjects
Fermat quotient ,Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,Upper and lower bounds ,Algebra ,Combinatorics ,11A05, 11A07, 11J86 ,FOS: Mathematics ,Exponent ,Number Theory (math.NT) ,p-Adic logarithm ,Mathematics - Abstract
We shall make a slight improvement to a result of p-adic logarithms which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient x^{p-1}-1., 9 pages, submitted to J. Number Theory
- Published
- 2010
47. Corrigendum to the paper 'Ovoidal packings of PG(3,q) for even q'
- Author
-
Bhaskar Bagchi and N. S. Narasimha Sastry
- Subjects
Discrete mathematics ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,Corollary ,010201 computation theory & mathematics ,Retract ,0202 electrical engineering, electronic engineering, information engineering ,Discrete Mathematics and Combinatorics ,Point (geometry) ,Mathematics - Abstract
We point out that the proof of Theorem 3.3 of Bagchi and Sastry (2013) contains a serious flaw. Accordingly, this theorem needs to be modified. In consequence, we also have to retract Corollary 3.4, Corollary 3.6 and Theorem 3.8 of Bagchi and Sastry (2013).
- Published
- 2018
48. Mod(k) Vertex Magic Labeling in Generalized 2-complement of some Graphs- Paper II
- Author
-
B. Fathima and P. Sumathi
- Subjects
Vertex (graph theory) ,Combinatorics ,Two's complement ,Mod ,Magic (programming) ,Mathematics - Published
- 2017
49. Addendum to the paper: 'On the number of terms of a composite polynomial' (Acta Arith. 127 (2007), 157–167)
- Author
-
Umberto Zannier
- Subjects
Combinatorics ,Polynomial ,Algebra and Number Theory ,Composite number ,Addendum ,Mathematics - Published
- 2009
50. Note on a paper of J. Llibre and G. Rodríguez concerning algebraic limit cycles
- Author
-
Daniel Peralta-Salas
- Subjects
Combinatorics ,Discrete mathematics ,Polynomial ,Degree (graph theory) ,Plane (geometry) ,Applied Mathematics ,Bounded function ,Vector field ,Limit (mathematics) ,Algebraic number ,Homeomorphism ,Analysis ,Mathematics - Abstract
In a recent paper of Llibre and Rodriguez (J. Differential Equations 198 (2004) 374–380) it is proved that every configuration of cycles in the plane is realizable (up to homeomorphism) by a polynomial vector field of degree at most 2 ( n + r ) - 1 , where n is the number of cycles and r the number of primary cycles (a cycle C is primary if there are no other cycles contained in the bounded region limited by C ). In this letter we prove the same theorem by using an easier construction but with a greater polynomial bound (the vector field we construct has degree at most 4 n - 1 ). By using the same technique we also construct R 3 polynomial vector fields realizing (up to homeomorphism) any configuration of limit cycles which can be linked and knotted in R 3 . This answers a question of R. Sverdlove.
- Published
- 2005
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