34,541 results
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2. A note on the paper 'Best proximity point results for $$p$$-proximal contractions'
- Author
-
M. Gabeleh and J. Markin
- Subjects
Combinatorics ,Metric space ,Class (set theory) ,General Mathematics ,Fixed-point theorem ,Point (geometry) ,Mathematics - Abstract
Very recently, I. Altun, M. Aslantas and H. Sahin [1] introduced the notion of $$p$$ -proximal contractions and surveyed the existence of best proximity points for such class of non-self mappings in metric spaces. In this note we show that this existence result is a straightforward consequence of the same conclusion in fixed point theory.
- Published
- 2021
3. Addendum to the paper 'High‐order symmetric cubature rules for tetrahedra and pyramids'
- Author
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Jan Jaśkowiec and N. Sukumar
- Subjects
Combinatorics ,Numerical Analysis ,Applied Mathematics ,General Engineering ,Tetrahedron ,Addendum ,High order ,Mathematics - 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
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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. Ramsey, Paper, Scissors
- Author
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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
7. Correcting mistakes in the paper 'A mass formula for negacyclic codes of length 2k and some good negacyclic codes over $\mathbb {Z}_{4}+u\mathbb {Z}_{4}$' [Cryptogr. Commun. (2017) 9: 241–272]
- Author
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Rama Krishna Bandi, Yuan Cao, Yonglin Cao, and Fang-Wei Fu
- Subjects
Mass formula ,Combinatorics ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Computer Networks and Communications ,Applied Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Mathematics - Abstract
We correct some mistakes in the paper “A mass formula for negacyclic codes of length 2k and some good negacyclic codes over $\mathbb {Z}_{4}+u\mathbb {Z}_{4}$ ” (Bandi et al. Cryptogr. Commun. 9, 241–272, 2017).
- Published
- 2020
8. On a paper of Dressler and Van de Lune
- Author
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Pablo Andres Panzone
- Subjects
Combinatorics ,Lune ,General Mathematics ,Arithmetic function ,Natural number ,Prime (order theory) ,Mathematics - Abstract
If $$z\in {\mathbb {C}}$$ and $$1\le n$$ is a natural number then $$\begin{aligned} \sum _{d_1 d_2 =n} (1-z^{p_1})\cdots (1-z^{p_m}) z^{q_1 e_{1}+\cdots +q_i e_{i} }=1, \end{aligned}$$ where $$d_1=p_1^{r_1}\dots p_m^{r_m }$$ , $$d_2=q_1^{e_1}\dots q_i^{e_i }$$ are the prime decompositions of $$d_1, d_2$$ . This is one of the identities involving arithmetic functions that we prove using ideas from the paper of Dressler and van de Lune [3].
- Published
- 2020
9. Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski
- Author
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Jie Wu
- Subjects
Combinatorics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Very recently Bordelles, Dai, Heyman, Pan and Shparlinski studied asymptotic behaviour of the quantity $$\begin{aligned} S_f(x) := \sum _{n\leqslant x} f\left( \left[ \frac{x}{n}\right] \right) , \end{aligned}$$and established some asymptotic formulas for $$S_f(x)$$ under three different types of assumptions on f. In this short note we improve some of their results.
- Published
- 2019
10. A Note on a Paper of Aivazidis, Safonova and Skiba
- Author
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M. M. Al-Shomrani, Adolfo Ballester-Bolinches, and A. A. Heliel
- Subjects
Subnormal subgroup ,Combinatorics ,Mathematics::Group Theory ,Finite group ,General Mathematics ,Mathematics - Abstract
The main result of this paper states that if $${\mathcal {F}}$$ is a subgroup-closed saturated formation of full characteristic, then the $${\mathcal {F}}$$ -residual of a K- $${\mathcal {F}}$$ -subnormal subgroup S of a finite group G is a large subgroup of G provided that the $${\mathcal {F}}$$ -hypercentre of every subgroup X of G containing S is contained in the $${\mathcal {F}}$$ -residual of X. This extends a recent result of Aivazidis, Safonova and Skiba.
- Published
- 2021
11. Capacity of the Torn Paper Channel with Lost Pieces
- Author
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Aditya Narayan Ravi, Alireza Vahid, and Ilan Shomorony
- Subjects
Combinatorics ,Capacity planning ,Fragment (logic) ,Channel (programming) ,Binary number ,Length distribution ,Nuclear Experiment ,Information theory ,Computer Science::Information Theory ,Block (data storage) ,Mathematics - Abstract
We study the problem of transmitting a message over a channel that randomly breaks the message block into small fragments, deletes a subset of them, and shuffles the remaining fragments. We characterize the capacity of the binary torn-paper channel under arbitrary fragment length distribution and fragment deletion probabilities. We show that, for a message with block length $n$ , discarding fragments shorter than $\log(n)$ does not affect the achievable rates, and that the capacity is given by a simple closed-form expression that can be understood as “coverage minus reordering-cost”.
- Published
- 2021
12. Metapopulation model of rock-scissors-paper game with subpopulation-specific victory rates stabilized by heterogeneity
- Author
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Genki Ichinose, Takashi Nagatani, and Kei-ichi Tainaka
- Subjects
0301 basic medicine ,Statistics and Probability ,General Immunology and Microbiology ,Applied Mathematics ,Victory ,Metapopulation ,General Medicine ,Models, Theoretical ,Random walk ,01 natural sciences ,General Biochemistry, Genetics and Molecular Biology ,Graph ,Combinatorics ,03 medical and health sciences ,030104 developmental biology ,Game Theory ,Homogeneous ,Modeling and Simulation ,0103 physical sciences ,Neutral stability ,Computer Simulation ,010306 general physics ,General Agricultural and Biological Sciences ,Mathematics - Abstract
Recently, metapopulation models for rock-paper-scissors games have been presented. Each subpopulation is represented by a node on a graph. An individual is either rock (R), scissors (S) or paper (P); it randomly migrates among subpopulations. In the present paper, we assume victory rates differ in different subpopulations. To investigate the dynamic state of each subpopulation (node), we numerically obtain the solutions of reaction-diffusion equations on the graphs with two and three nodes. In the case of homogeneous victory rates, we find each subpopulation has a periodic solution with neutral stability. However, when victory rates between subpopulations are heterogeneous, the solution approaches stable focuses. The heterogeneity of victory rates promotes the coexistence of species.
- Published
- 2018
13. A few observations on the paper 'Maximum distance in graphs'
- Author
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K. V. S. Sarma, Venkata Subrahmanyam Sajja, Dasari Jai Ramesh, and I. H. Nagaraja Rao
- Subjects
Combinatorics ,Cycle graph ,Complete graph ,Extension (predicate logic) ,Complete bipartite graph ,Mathematics - Abstract
Thamarai Selvi and Vaidya Nathan [2] discussed maximum distance on graphs. We made an extension of that paper [2]. We discussed about complete graph Kn, star graph Sn, the complete bipartite graph Km,n. In this paper, we studied about the cycle graph Cn (n being any interger ≥ 3) and the multi star graph Bm,n, where m, n are integers, both greater than or equal to 2.
- Published
- 2021
14. Chasing convex bodies with linear competitive ratio (invited paper)
- Author
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Guru Guruganesh, Anupam Gupta, Ziye Tang, and C. J. Argue
- Subjects
Combinatorics ,Sequence ,Competitive analysis ,Convex body ,Steiner point ,State (functional analysis) ,Function (mathematics) ,Online algorithm ,Convex function ,Mathematics - Abstract
The problem of chasing convex functions is easy to state: faced with a sequence of convex functions f t over d-dimensional Euclidean spaces, the goal of the algorithm is to output a point x t at each time, so that the sum of the function costs f t (x t ), plus the movement costs ||x t − x t − 1 || is minimized. This problem generalizes questions in online algorithms such as caching and the k-server problem. In 1994, Friedman and Linial posed the question of getting an algorithm with a competitive ratio that depends only on the dimension d. In this talk we give an O (d)-competitive algorithm, based on the notion of the Steiner point of a convex body.
- Published
- 2021
15. Observation On The Paper Entitled 'Special Pairs Of Rectangles And Sphenic Number'
- Author
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S. Vidhyalakshmi, Gopalan M A, and S. Aarthy Thangam
- Subjects
Combinatorics ,Sphenic number ,Mathematics - Abstract
This paper aims at presenting pairs of rectangles representing the same sphenic number where, in each pair, the sum of the areas is 2 times sphenic number -1
- Published
- 2019
16. A remark on a paper of P. B. Djakov and M. S. Ramanujan
- Author
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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
17. Fuzzy quotient-3 cordial labeling of star related graphs- Paper I
- Author
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P. Sumathi and J. Suresh Kumar
- Subjects
Combinatorics ,Star (graph theory) ,Fuzzy logic ,Quotient ,Mathematics - Published
- 2019
18. Partner choice correlates with fine scale kin structuring in the paper wasp Polistes dominula
- Author
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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
19. Menon-type identities again: A note on a paper by Li, Kim and Qiao
- Author
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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
20. A Remark on the Paper 'Properties of Intersecting Families of Ordered Sets' by O. Einstein
- Author
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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
21. 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
22. A Note on the Paper 'The Algebraic Structure of the Arbitrary-Order Cone'
- Author
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Yen Chi Roger Lin, Xin-He Miao, and Jein Shan Chen
- Subjects
Pure mathematics ,021103 operations research ,Control and Optimization ,Algebraic structure ,Applied Mathematics ,0211 other engineering and technologies ,Structure (category theory) ,Order (ring theory) ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Cone (formal languages) ,Combinatorics ,Operator (computer programming) ,Product (mathematics) ,Light cone ,0101 mathematics ,Mathematics ,Counterexample - Abstract
In this short paper, we look into a conclusion drawn by Alzalg (J Optim Theory Appl 169:32---49, 2016). We think the conclusion drawn in the paper is incorrect by pointing out three things. First, we provide a counterexample that the proposed inner product does not satisfy bilinearity. Secondly, we offer an argument why a pth-order cone cannot be self-dual under any reasonable inner product structure on $$\mathbb {R}^n$$Rn. Thirdly, even under the assumption that all elements operator commute, the inner product becomes an official inner product and the arbitrary-order cone can be shown as a symmetric cone, we think this condition is still unreasonable and very stringent so that the result can only be applied to very few cases.
- Published
- 2017
23. 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
24. Local Restrictions from the Furst-Saxe-Sipser Paper
- Author
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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
25. Remark on the paper 'On products of Fourier coefficients of cusp forms'
- Author
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Yuk-Kam Lau, Deyu Zhang, and Yingnan Wang
- Subjects
Cusp (singularity) ,Discrete group ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,02 engineering and technology ,01 natural sciences ,Cusp form ,Combinatorics ,Integer ,Product (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,0101 mathematics ,Fourier series ,Mathematics - Abstract
Let a(n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of \(SL_2({\mathbb R})\). This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of \(a(n)a(n+r)\) for a fixed positive integer r.
- Published
- 2016
26. Addendum and corrigenda to the paper 'Infinitary superperfect numbers'
- Author
-
Tomohiro Yamada
- Subjects
Combinatorics ,General Computer Science ,General Mathematics ,Addendum ,Mathematics - Published
- 2018
27. Short Paper: Tight Bounds for Universal and Cautious Self-stabilizing 1-Maximal Matching
- Author
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Michiko Inoue, Sébastien Tixeuil, Nara Institute of Science and Technology - Graduate School of Information Science (NAIST), Nara Institute of Science and Technology, Networks and Performance Analysis (NPA), LIP6, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratory of Information, Network and Communication Sciences (LINCS), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Mines-Télécom [Paris] (IMT)-Sorbonne Université (SU)
- Subjects
Matching (graph theory) ,Spacetime ,Existential quantification ,0102 computer and information sciences ,02 engineering and technology ,Binary logarithm ,Space (mathematics) ,01 natural sciences ,Upper and lower bounds ,Combinatorics ,010201 computation theory & mathematics ,020204 information systems ,0202 electrical engineering, electronic engineering, information engineering ,Graph (abstract data type) ,[INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC] ,ComputingMilieux_MISCELLANEOUS ,Blossom algorithm ,Mathematics - Abstract
We consider the problem of constructing a matching in an n-nodes graph in a distributed and self-stabilizing manner. We prove that there exists a lower bound in space of \(\varOmega (n\log n)\) bits for universal maximal matching algorithms, and a lower bound in time of \(\varOmega (e)\) moves for universal and cautious 1-maximal matching algorithms. A side contribution of our result is the optimality in both time and space of the self-stabilizing 1-maximal matching algorithm of Inoue et al. [8].
- Published
- 2019
28. 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
29. Paper or and digital : A study on combinatorics in preschool class
- Author
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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
30. On a paper of Beltrán and Shao about coprime action
- Author
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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
31. Polynomial Silent Self-Stabilizing p-Star Decomposition (Short Paper)
- Author
-
Mohammed Haddad, Colette Johnen, Sven Köhler, Graphes, AlgOrithmes et AppLications (GOAL), Laboratoire d'InfoRmatique en Image et Systèmes d'information (LIRIS), Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École Centrale de Lyon (ECL), Université de Lyon-Université Lumière - Lyon 2 (UL2)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Université Lumière - Lyon 2 (UL2), Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), University of Freiburg, Sustainability Center Freiburg, Germany, and ANR-10-IDEX-0003,IDEX BORDEAUX,Initiative d'excellence de l'Université de Bordeaux(2010)
- Subjects
Polynomial (hyperelastic model) ,020203 distributed computing ,Degree (graph theory) ,Round complexity ,Star (game theory) ,Short paper ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Combinatorics ,010201 computation theory & mathematics ,Distributed algorithm ,0202 electrical engineering, electronic engineering, information engineering ,[INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC] ,Algorithm ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
We present a silent self-stabilizing distributed algorithm computing a maximal p-star decomposition of the underlying communication network. Under the unfair distributed scheduler, the most general scheduler model, the algorithm converges in at most \(12\varDelta m + \mathcal {O}(m+n)\) moves, where m is the number of edges, n is the number of nodes, and \(\varDelta \) is the maximum node degree. Regarding the move complexity, our algorithm outperforms the previously known best algorithm by a factor of \(\varDelta \). While the round complexity for the previous algorithm was unknown, we show a \(5\left\lfloor \frac{n}{p+1} \right\rfloor +5\) bound for our algorithm.
- Published
- 2016
32. Leader Election in Rings with Bounded Multiplicity (Short Paper)
- Author
-
Lawrence L. Larmore, Karine Altisen, Stéphane Devismes, Ajoy K. Datta, and Anaïs Durand
- Subjects
Discrete mathematics ,020203 distributed computing ,Leader election ,Short paper ,Multiplicity (mathematics) ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Computer Science::Multiagent Systems ,Combinatorics ,010201 computation theory & mathematics ,Bounded function ,0202 electrical engineering, electronic engineering, information engineering ,Mathematics - Abstract
We study leader election in unidirectional rings of homonym processes that have no a priori knowledge on the number of processes. We show that message-terminating leader election is impossible for any class of rings \(\mathcal K_k\) with bounded multiplicity \(k \ge 2\). However, we show that process-terminating leader election is possible in the sub-class \(\mathcal U^* \cap \mathcal K_k\), where \(\mathcal U^*\) is the class of rings which contain a process with a unique label.
- Published
- 2016
33. 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
34. 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
35. An Addendum to a Paper by Li and Zhang
- Author
-
Conrad Mädler and Lutz Klotz
- Subjects
Combinatorics ,Trace norm ,Zhàng ,Addendum ,Block matrix ,Positive-definite matrix ,Main diagonal ,Schur polynomial ,Mathematics - Abstract
Recently C.-K. Li and F. Zhang gave a short proof of S.W. Drury’s result that the trace norm preserves positive semidefiniteness of 3 ×3 partitioned positive semidefinite matrices. Slightly generalizing Li and Zhang’s proof we obtain a more general result under the assumption that all matrices on the principal diagonal are of the same size. Also, we answer a question asked by Marcus and Watkins in 1971.
- Published
- 2018
36. 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
37. Corrigendum to the paper 'On the K^2 of degenerations of surfaces and the multiple point formula'
- Author
-
Alberto Calabri, Ciro Ciliberto, Richard Miranda, and Flaminio Flamini
- Subjects
Pure mathematics ,deformations, degenerations, enumerative problems, singularities ,deformations ,010102 general mathematics ,Minor (linear algebra) ,Socio-culturale ,01 natural sciences ,degenerations ,Combinatorics ,Multiple point ,Section (fiber bundle) ,Mathematics (miscellaneous) ,0103 physical sciences ,enumerative problems ,Gravitational singularity ,010307 mathematical physics ,Settore MAT/03 - Geometria ,0101 mathematics ,Statistics, Probability and Uncertainty ,singularities ,Mathematics - Abstract
We correct an error in the Multiple Point Formula (7.3) in the paper mentioned in the title. This correction propagates to formulas (7.5), (7.6), (7.23) and (8.18), and it affects minor results in Section 8, where few statements require an extra assumption, but it does not affect the main results of Section 8.
- Published
- 2017
38. 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
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. 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
41. 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
42. 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
43. On the quotients between the eccentric connectivity index and the eccentric distance sum of graphs with diameter 2
- Author
-
Hongbo Hua
- Subjects
Vertex (graph theory) ,Applied Mathematics ,Short paper ,0211 other engineering and technologies ,021107 urban & regional planning ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Upper and lower bounds ,Graph ,Combinatorics ,010201 computation theory & mathematics ,Topological index ,Discrete Mathematics and Combinatorics ,Eccentric ,Quotient ,Connectivity ,Mathematics - Abstract
For a connected graph G and a vertex v in G , let d G ( v ) , e G ( v ) and D G ( v ) be the degree, eccentricity and distance sum of v , respectively. The eccentric connectivity index of G , denoted by ξ c ( G ) , is defined to be ξ c ( G ) = ∑ v ∈ V ( G ) d G ( v ) e G ( v ) , and the eccentric distance sum of G , denoted by ξ d ( G ) , is defined to be ξ d ( G ) = ∑ v ∈ V ( G ) e G ( v ) D G ( v ) . Denote by G n 2 the set of connected graphs of order n and diameter two. More recently, Zhang et al. (2019) investigated the relationship between the eccentric connectivity index and eccentric distance sum, and posed the problem to determine sharp upper and lower bounds on ξ c ( G ) ξ d ( G ) for graph in G n 2 . In this short paper, we solve this problem. Sharp upper and lower bounds on ξ c ( G ) ξ d ( G ) for graph in G n 2 are determined, and the corresponding extremal graphs are characterized as well.
- Published
- 2020
44. Paper and Block Printing—From China to Europe, Thomas F. Carter
- Author
-
Paul Heyer and David Crowley
- Subjects
Combinatorics ,Block (telecommunications) ,China ,Mathematics - Published
- 2015
45. Coordination sequences of crystals are of quasi-polynomial type
- Author
-
Junichi Nakagawa, Takafumi Mase, Ryotaro Sakamoto, and Yusuke Nakamura
- Subjects
Generalization ,graph theory ,Hilbert polynomial ,Coordination number ,monoid theory ,Type (model theory) ,010402 general chemistry ,Quasi-polynomial ,01 natural sciences ,Biochemistry ,Inorganic Chemistry ,Combinatorics ,Structural Biology ,General Materials Science ,Physical and Theoretical Chemistry ,Mathematics ,Sequence ,010405 organic chemistry ,Graph theory ,Condensed Matter Physics ,Research Papers ,coordination sequences ,0104 chemical sciences ,Vertex (geometry) ,Graph (abstract data type) ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
It is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889] in their study of coordination sequences of zeolites., The coordination sequence of a graph measures how many vertices the graph has at each distance from a fixed vertex and is a generalization of the coordination number. Here it is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889].
- Published
- 2021
46. Small one-dimensional Euclidean preference profiles
- Author
-
Jiehua Chen and Sven Grottke
- Subjects
FOS: Computer and information sciences ,Original Paper ,Economics and Econometrics ,Small number ,05 social sciences ,Preference ,Combinatorics ,Computer Science - Computer Science and Game Theory ,0502 economics and business ,Euclidean geometry ,050206 economic theory ,050207 economics ,Social choice theory ,Social Sciences (miscellaneous) ,Computer Science and Game Theory (cs.GT) ,Mathematics - Abstract
We characterize one-dimensional Euclidean preference profiles with a small number of alternatives and voters. We show that every single-peaked preference profile withtwovoters is one-dimensional Euclidean, and that every preference profile with up to five alternatives is one-dimensional Euclidean if and only if it is both single-peaked and single-crossing. By the work of Chen et al. (Social Choice and Welfare 48(2):409–432, 2017), we thus obtain that the smallest single-peaked and single-crossing preference profiles that arenotone-dimensional Euclidean consist of three voters and six alternatives.
- Published
- 2021
47. A quantitative Lovász criterion for Property B
- Author
-
Asaf Shapira and Asaf Ferber
- Subjects
Statistics and Probability ,Vertex (graph theory) ,Hypergraph ,Mathematics::Combinatorics ,Computer Science::Information Retrieval ,Applied Mathematics ,The Intersect ,Short paper ,0102 computer and information sciences ,01 natural sciences ,Theoretical Computer Science ,05D05 ,Combinatorics ,Computational Theory and Mathematics ,Computer Science::Discrete Mathematics ,010201 computation theory & mathematics ,Mathematics - Combinatorics ,Property B ,Mathematics - Abstract
A well known observation of Lov\'asz is that if a hypergraph is not $2$-colorable, then at least one pair of its edges intersect at a single vertex. %This very simple criterion turned out to be extremly useful . In this short paper we consider the quantitative version of Lov\'asz's criterion. That is, we ask how many pairs of edges intersecting at a single vertex, should belong to a non $2$-colorable $n$-uniform hypergraph? Our main result is an {\em exact} answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollob\'as's two families theorem with Pluhar's randomized coloring algorithm., Comment: A note on Property B
- Published
- 2020
48. Short-Time Heat Content Asymptotics via the Wave and Eikonal Equations
- Author
-
Nathanael Schilling
- Subjects
Eikonal equation ,010102 general mathematics ,Short paper ,Boundary (topology) ,Function (mathematics) ,01 natural sciences ,ddc ,Combinatorics ,Mathematics - Analysis of PDEs ,Differential geometry ,0103 physical sciences ,Content (measure theory) ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this short paper, we derive an alternative proof for some known (van den Berg & Gilkey 2015) short-time asymptotics of the heat content in a compact full-dimensional submanifolds S with smooth boundary. This includes formulae like $$\begin{aligned} \int _{S} \exp (t\Delta ) (f \mathbb {1}_{S}) \,\mathrm {d}V= \int _S f \,\mathrm {d}V- \sqrt{\frac{t}{\pi }} \int _{\partial S} f \,\mathrm {d}A+ o(\sqrt{t}),\quad t \rightarrow 0^+, \end{aligned}$$ ∫ S exp ( t Δ ) ( f 1 S ) d V = ∫ S f d V - t π ∫ ∂ S f d A + o ( t ) , t → 0 + , and explicit expressions for similar expansions involving other powers of $$\sqrt{t}$$ t . By the same method, we also obtain short-time asymptotics of $$\int _S \exp (t^m\Delta ^m)(f \mathbb {1}_S)\,\mathrm {d}V$$ ∫ S exp ( t m Δ m ) ( f 1 S ) d V , $$m \in \mathbb N$$ m ∈ N , and more generally for one-parameter families of operators $$t \mapsto k(\sqrt{-t\Delta })$$ t ↦ k ( - t Δ ) defined by an even Schwartz function k.
- Published
- 2020
49. Fractional Factorials and Prime Numbers (A Remark on the Paper 'On Prime Values of Some Quadratic Polynomials')
- Author
-
A. N. Andrianov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Prime element ,01 natural sciences ,Prime k-tuple ,Prime (order theory) ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Prime factor ,Unique prime ,0101 mathematics ,Fibonacci prime ,Prime power ,Sphenic number ,Mathematics - Abstract
Congruences mod p for a prime p and partial products of the numbers 1,…, p − 1 are obtained. Bibliography: 2 titles.
- Published
- 2016
50. Correction of the paper 'Bicyclic graphs with extremal values of PI index'
- Author
-
Qiuju Bian and Gang Ma
- Subjects
Discrete mathematics ,Bicyclic molecule ,Applied Mathematics ,010102 general mathematics ,Bicyclic graphs ,0102 computer and information sciences ,01 natural sciences ,Graph ,Combinatorics ,010201 computation theory & mathematics ,Pi ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Mathematics - Abstract
In Vukicevic and Stevanovic (2013), Theorem 2 is erroneous when m = 3 k + 1 for some integer k . The correct one is given now. That is, P I ( G ) ≥ m 2 − 3 m + 2 when m = 3 k + 1 for some integer k , where G is a connected bicyclic graph with m edges.
- Published
- 2016
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