Abstract:Exponential random graph models (ERGMs) are one of the most popular statistical methods for analysing relational network structures. ERGMs represent generative statistical network processes that allow researchers to specify sufficient statistics in the form of counts of network configurations associated to potential dependencies between and across particular sets of nodes. In this paper, we review some of the most interesting recent advances for the ERGM framework. In particular, we focus on the modelling extensions for valued, multi-layer and multi-level networks.
{"title":"Recent Advances In Exponential Random Graph Modelling","authors":"A. Caimo, Isabella Gollini","doi":"10.1353/mpr.2023.0000","DOIUrl":"https://doi.org/10.1353/mpr.2023.0000","url":null,"abstract":"Abstract:Exponential random graph models (ERGMs) are one of the most popular statistical methods for analysing relational network structures. ERGMs represent generative statistical network processes that allow researchers to specify sufficient statistics in the form of counts of network configurations associated to potential dependencies between and across particular sets of nodes. In this paper, we review some of the most interesting recent advances for the ERGM framework. In particular, we focus on the modelling extensions for valued, multi-layer and multi-level networks.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121693966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tumour invasion is an essential stage of cancer progression. Its main drivers are diffusion and taxis, a directed movement along the gradient of a stimulus. Here we review models with flux limited diffusion and/or taxis which have applications in modelling of cell migration, particularly in cancer. Flux limitation ensures control upon propagation speeds, precluding unnaturally quick spread which is typical for traditional parabolic equations. We recall the main properties of models with flux limitation effects and discuss ways to construct them, concentrating on multiscale derivations from kinetic transport equations.
{"title":"Flux Limitation Mechanisms Arising in Multiscale Modelling of Cancer Invasion","authors":"A. Zhigun","doi":"10.1353/mpr.2022.0001","DOIUrl":"https://doi.org/10.1353/mpr.2022.0001","url":null,"abstract":"Tumour invasion is an essential stage of cancer progression. Its main drivers are diffusion and taxis, a directed movement along the gradient of a stimulus. Here we review models with flux limited diffusion and/or taxis which have applications in modelling of cell migration, particularly in cancer. Flux limitation ensures control upon propagation speeds, precluding unnaturally quick spread which is typical for traditional parabolic equations. We recall the main properties of models with flux limitation effects and discuss ways to construct them, concentrating on multiscale derivations from kinetic transport equations.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114512205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-31DOI: 10.3318/PRIA.2021.121.03
M. Mackey, P. Mellon
. We show that there are many sets in the boundary of a bounded symmetric domain that determine the values and norm of holomorphic functions on the domain having continuous extensions to the boundary. We provide an analogue of the Bergmann-Shilov boundary for finite rank JB ∗ -triples.
{"title":"The Bergmann-Shilov boundary of a Bounded Symmetric\u0000 Domain","authors":"M. Mackey, P. Mellon","doi":"10.3318/PRIA.2021.121.03","DOIUrl":"https://doi.org/10.3318/PRIA.2021.121.03","url":null,"abstract":". We show that there are many sets in the boundary of a bounded symmetric domain that determine the values and norm of holomorphic functions on the domain having continuous extensions to the boundary. We provide an analogue of the Bergmann-Shilov boundary for finite rank JB ∗ -triples.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"2015 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127685539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let R be an associative ring with 1, which is not commutative. Assume that any non-zero commutator v ∈ R satisfies: v2 is in the centre of R, and v is not a zero divisor.
We prove that R has no zero divisors, and that if char(R) ≠ 2, then the localisation of R at its centre is a quaternion division algebra. Our proof is elementary and self contained.
{"title":"A Characterisation of the Quaternions Using Commutators","authors":"E. Kleinfeld, Yoav Segev","doi":"10.1353/mpr.2022.0000","DOIUrl":"https://doi.org/10.1353/mpr.2022.0000","url":null,"abstract":"<p>Abstract:</p><p>Let <i>R</i> be an associative ring with 1, which is not commutative. Assume that any non-zero commutator <i>v</i> ∈ <i>R</i> satisfies: <i>v</i>2 is in the centre of <i>R</i>, and <i>v</i> is not a zero divisor.</p><p>We prove that <i>R</i> has no zero divisors, and that if char(<i>R</i>) ≠ 2, then the localisation of <i>R</i> at its centre is a quaternion division algebra. Our proof is elementary and self contained.</p>","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127282494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-14DOI: 10.3318/pria.2021.121.02
S. C. Power
The first-order flex space of the bar-joint framework $G_P$ of a parallelogram tiling $P$ is determined in terms of an explicit free basis. Applications are given to braced parallelogram frameworks and to quasicrystal frameworks associated with multigrids in the sense of de Bruijn and Beenker. In particular we characterise rigid bracing patterns, identify quasicrystal frameworks with finite dimensional flex spaces, and define a zero mode spectrum.
{"title":"Parallelogram Frameworks and Flexible Quasicrystals","authors":"S. C. Power","doi":"10.3318/pria.2021.121.02","DOIUrl":"https://doi.org/10.3318/pria.2021.121.02","url":null,"abstract":"The first-order flex space of the bar-joint framework $G_P$ of a parallelogram tiling $P$ is determined in terms of an explicit free basis. Applications are given to braced parallelogram frameworks and to quasicrystal frameworks associated with multigrids in the sense of de Bruijn and Beenker. In particular we characterise rigid bracing patterns, identify quasicrystal frameworks with finite dimensional flex spaces, and define a zero mode spectrum.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123314928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-19DOI: 10.3318/pria.2021.121.01
J. McCarron
A finite group is said to have "perfect order classes" if the number of elements of any given order is either zero or a divisor of the order of the group. The purpose of this note is to describe explicitly the finite Hamiltonian groups with perfect order classes. We show that a finite Hamiltonian group has perfect order classes if, and only if, it is isomorphic to the direct product of the quaternion group of order $8$, a non-trivial cyclic $3$-group and a group of order at most $2$. Theorem. A finite Hamiltonian group has perfect order classes if, and only if, it is isomorphic either to $Qtimes C_{3^k}$ or to $Qtimes C_{2}times C_{3^k}$, for some positive integer $k$.
{"title":"Hamiltonian Groups with Perfect Order Classes","authors":"J. McCarron","doi":"10.3318/pria.2021.121.01","DOIUrl":"https://doi.org/10.3318/pria.2021.121.01","url":null,"abstract":"A finite group is said to have \"perfect order classes\" if the number of elements of any given order is either zero or a divisor of the order of the group. \u0000The purpose of this note is to describe explicitly the finite Hamiltonian groups with perfect order classes. We show that a finite Hamiltonian group has perfect order classes if, and only if, it is isomorphic to the direct product of the quaternion group of order $8$, a non-trivial cyclic $3$-group and a group of order at most $2$. \u0000Theorem. A finite Hamiltonian group has perfect order classes if, and only if, it is isomorphic either to $Qtimes C_{3^k}$ or to $Qtimes C_{2}times C_{3^k}$, for some positive integer $k$.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133266379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider formal maps in any finite dimension $d$ with coefficients in an integral domain $K$ with identity. Those invertible under formal composition form a group $G$. We consider the centraliser $C_g$ of an element $ginG$ which is tangent to the identity of $G$. Elements of finite order always have a large centraliser. If $g$ has infinite order our main result is that $C_g$ is uncountable, and in fact contains an uncountable abelian subgroup. This holds regardless of the characteristic of $K$, but the proof is quite different in finite characteristic than in characteritic zero.
{"title":"Centralisers of Formal Maps","authors":"A. O’Farrell","doi":"10.1353/mpr.2022.0005","DOIUrl":"https://doi.org/10.1353/mpr.2022.0005","url":null,"abstract":"We consider formal maps in any finite dimension $d$ with coefficients in an integral domain $K$ with identity. Those invertible under formal composition form a group $G$. We consider the centraliser $C_g$ of an element $ginG$ which is tangent to the identity of $G$. Elements of finite order always have a large centraliser. If $g$ has infinite order our main result is that $C_g$ is uncountable, and in fact contains an uncountable abelian subgroup. This holds regardless of the characteristic of $K$, but the proof is quite different in finite characteristic than in characteritic zero.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"216 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132586930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-07DOI: 10.3318/pria.2020.120.06
S. Power
Necessary and sufficient conditions are obtained for the infinitesimal rigidity of braced grids in the plane with respect to non-Euclidean norms. Component rectangles of the grid may carry 0, 1 or 2 diagonal braces, and the combinatorial part of the conditions is given in terms of a matroid for the bicoloured bipartite multigraph defined by the braces.
{"title":"Non-Euclidean braced grids","authors":"S. Power","doi":"10.3318/pria.2020.120.06","DOIUrl":"https://doi.org/10.3318/pria.2020.120.06","url":null,"abstract":"Necessary and sufficient conditions are obtained for the infinitesimal rigidity of braced grids in the plane with respect to non-Euclidean norms. Component rectangles of the grid may carry 0, 1 or 2 diagonal braces, and the combinatorial part of the conditions is given in terms of a matroid for the bicoloured bipartite multigraph defined by the braces.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124651946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-17DOI: 10.3318/pria.2019.119.07
E. Kastis, D. Kitson, S. Power
We consider, from the point of view of operator theory, a class of infinite matrices in which the matrix entries are determined by an underlying graph structure with accompanying geometric data. This class includes the rigidity matrices of infinite bar-joint frameworks as well as the incidence matrices of infinite directed graphs. We consider the following questions: When do these matrices give rise to bounded operators? Can we compute the operator norm? When are these operators compact? And when are they bounded below?
{"title":"Coboundary operators for infinite frameworks","authors":"E. Kastis, D. Kitson, S. Power","doi":"10.3318/pria.2019.119.07","DOIUrl":"https://doi.org/10.3318/pria.2019.119.07","url":null,"abstract":"We consider, from the point of view of operator theory, a class of infinite matrices in which the matrix entries are determined by an underlying graph structure with accompanying geometric data. This class includes the rigidity matrices of infinite bar-joint frameworks as well as the incidence matrices of infinite directed graphs. We consider the following questions: When do these matrices give rise to bounded operators? Can we compute the operator norm? When are these operators compact? And when are they bounded below?","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127076847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-07-29DOI: 10.3318/pria.2019.119.08
A. Defant, I. Schoolmann
A theorem of Henry Helson shows that for every ordinary Dirichlet series $sum a_n n^{-s}$ with a square summable sequence $(a_n)$ of coefficients, almost all vertical limits $sum a_n chi(n) n^{-s}$, where $chi: mathbb{N} to mathbb{T}$ is a completely multiplicative arithmetic function, converge on the right half-plane. We survey on recent improvements and extensions of this result within Hardy spaces of Dirichlet series -- relating it with some classical work of Bohr, Banach, Carleson-Hunt, Cesaro, Hardy-Littlewood, Hardy-Riesz, Menchoff-Rademacher, and Riemann.
{"title":"Henry Helson meets other big shots — a brief survey","authors":"A. Defant, I. Schoolmann","doi":"10.3318/pria.2019.119.08","DOIUrl":"https://doi.org/10.3318/pria.2019.119.08","url":null,"abstract":"A theorem of Henry Helson shows that for every ordinary Dirichlet series $sum a_n n^{-s}$ with a square summable sequence $(a_n)$ of coefficients, almost all vertical limits $sum a_n chi(n) n^{-s}$, where $chi: mathbb{N} to mathbb{T}$ is a completely multiplicative arithmetic function, converge on the right half-plane. We survey on recent improvements and extensions of this result within Hardy spaces of Dirichlet series -- relating it with some classical work of Bohr, Banach, Carleson-Hunt, Cesaro, Hardy-Littlewood, Hardy-Riesz, Menchoff-Rademacher, and Riemann.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115051080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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