Procedurally Rational Volunteers

The mixed-strategy equilibrium applied to the volunteer's dilemma (Diekmann, 1985 Diekmann , A. ( 1985 ). Volunteer's dilemma . Journal of Conflict Resolution , 29 , 605 – 610 .[Crossref], [Web of Science ®] , [Google Scholar]) predicts that the probability that at least 1 bystander volunteers decreases, if the number of bystander increases. I argue that this prediction lacks empirical support and demonstrate that an alternative solution concept from the literature on bounded rationality, the procedurally rational equilibrium (Osborne &; Rubinstein, 1998 Osborne , M. J. &; Rubinstein , A. ( 1998 ). Games with procedurally rational players . American Economic Review , 88 , 834 – 847 .[Web of Science ®] , [Google Scholar]), yields alternative predictions. I supply some empirical evidence that the alternative solution concept fares better in explaining observed behavior in the volunteer's dilemma.     

The Measures of Rank or Status: A Reformulation and Reinterpretation

This article investigates methods in social network analysis to identify the most important or the most prominent actors in a social network by ranking them appropriately. Although much has been done since Seeley's (1949 Seeley , J. R. ( 1949 ). The net of reciprocal influence: A problem in treating sociometric data . Canadian Journal of Psychology , 3 , 234 – 240 .[Crossref], [Web of Science ®] , [Google Scholar]) seminal work, several questions remain: How does scaling an adjacency matrix of a social network affect its eigenvalues and their corresponding eigenvectors? How can the differences between the left and the right eigenvectors be reconciled? Do both depict the two sides of the same coin? Can they be “merged” objectively to yield a single eigenvector, and how? Under what conditions does an adjacency matrix characterize, as a first-order approximation, the dynamics of its corresponding social network? Most importantly, how can a general dynamic model be derived to make clear the interrelationships of existing models?

In answering these questions, we derive a general model for the evolving rank (which we define as a general measure of prominence) of a (fixed) set of actors in a social network from a system of nonlinear equations; show that, under a rank equivalence condition, an adjacency matrix characterizes the dynamics of its corresponding social network as a first-order approximation; put some existing models within a single framework; and give a solution to the vexing problem of specifying the reference values of the rank.     

A General Model of Church Growth and Decline

An earlier model of church growth (Hayward, 1999 Hayward , J. ( 1999 ). Mathematical modeling of church growth . Journal of Mathematical Sociology , 23 ( 4 ): 255 – 292 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) is extended to include long-term effects due to births, deaths and reversion from the church. It is proposed that only a subset of the church, the enthusiasts, is involved in the recruitment process, and only for a limited period of time after their conversion. It is found that the church reaches equilibrium in its proportion of society according to the potential of these enthusiasts to reproduce themselves, and the losses from the church. If this reproduction potential is below a threshold that depends on losses, then extinction occurs. If it is above a higher threshold, then the church sees rapid revival growth. The model is applied to a number of church denominations to examine their prospects for survival or revial growth. Generally, declining churches do so because their reproduction potential is inadequate, rather than due to excessive losses.     

A Wealth and Status-Based Model of Residential Segregation

We extend the classic “Schelling model” (1971 Schelling , T. C. (1971). Dynamic models of segregation. Journal of Mathematical Sociology , 1, 143–186.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] 1978 Schelling , T. C. ( 1978 ). Micromotives and Macrobehavior , New York and London : W.W. Norton and Company . [Google Scholar]) to incorporate the wealth and status of agents and the desirability and affordability of residences. We analyze the effects of 1) the degree of the status-wealth correlation, and 2) the extent to which the wealth of residents shapes the affordability of residences, on levels of status and wealth segregation. Both factors generally exert a positive effect on both forms of segregation and interact to produce higher levels of segregation. The greater the correlation between status and wealth, the more the agents tend to segregate, either due to choice (for the wealthy and high status) or exclusion (for the poor and low status). We also find that housing price endogeneity is a precondition for status segregation.     

Key Player Policies When Contextual Effects Matter

We consider a model where the criminal decision of each individual is affected by not only her own characteristics, but also by the characteristics of her friends (contextual effects). We determine who the key player is, i.e., the criminal who once removed generates the highest reduction in total crime in the network. We propose a new measure, the contextual intercentrality measure, that generalizes the one proposed by Ballester, Calvó-Armengol, and Zenou (2006 Ballester , C. , Calvó-Armengol , A. , &; Zenou , Y. ( 2006 ). Who's who in networks. Wanted: The key player . Econometrica , 74 , 1403 – 1417 .[Crossref], [Web of Science ®] , [Google Scholar]) by taking into account the change in contextual effects following the removal of the key player. We also provide an example showing that the key player can be different whether contextual effects are taken into account or not. This means that the planner may target the wrong person if it ignores the effect of the “context” when removing a criminal from a network.     

Supercharacter Theory Constructions Corresponding to Schur Ring Products

Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group [7 Diaconis , P. , Isaacs , I. M. ( 2008 ). Supercharacters and superclasses for algebra groups . Trans. Amer. Math. Soc. 360 : 2359 – 2392 .[Crossref], [Web of Science ®] , [Google Scholar]]. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12 Leung , K. H. , Man , S. H. ( 1996 ). On Schur rings over cyclic groups, II . J. Algebra 183 : 273 – 285 .[Crossref], [Web of Science ®] , [Google Scholar]], Hirasaka and Muzychuk [10 Hirasaka , M. , Muzychuk , M. ( 2001 ). An elementary abelian group of rank 4 is a CI-group . J. Combin. Theory Ser. A 94 : 339 – 362 .[Crossref], [Web of Science ®] , [Google Scholar]], and Tamaschke [20 Tamaschke , O. ( 1970 ). On Schur-rings which define a proper character theory on finite groups . Math. Z. 117 : 340 – 360 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Corrigendum to: “Clifford Theory of a Finite Group that Contains a Defect 0 p-Block of a Normal Subgroup”

At some point, after publication, the author realized that the proof of [3 Harris, M. E. (2013). Clifford theory of a finite group that contains a defect 0 p-block of a normal subgroup. Comm. in Alg. 41:3509–3540.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 5.2] is incorrect. This proof incorrectly adapts the proof of [1 Broué, M. (1990). Isométries parfaites, types de blocs, cégories dérivees. Aérisque 181–182:61–92. [Google Scholar], Theorem 4.8] since [3 Harris, M. E. (2013). Clifford theory of a finite group that contains a defect 0 p-block of a normal subgroup. Comm. in Alg. 41:3509–3540.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], (5.5)] is incorrect. Using the same proof outline, we correct the proof of [3 Harris, M. E. (2013). Clifford theory of a finite group that contains a defect 0 p-block of a normal subgroup. Comm. in Alg. 41:3509–3540.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 5.2].     

On the Associated Primes of Matlis Duals of Local Cohomology Modules II

In continuation of [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] we study associated primes of Matlis duals of local cohomology modules (MDLCM). We combine ideas from Helmut Zöschinger on coassociated primes of arbitrary modules with results from [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] 4-6 Hellus , M. , Stückrad , J. ( 2008 ). On endomorphism rings of local cohomology modules . Proceedings of the American Mathematical Society 136 : 2333 – 2341 . Hellus , M. , Stückrad , J. ( 2008 ). Matlis duals of top local cohomology modules . Proceedings of the American Mathematical Society 136 : 489 – 498 . Hellus , M. , Stückrad , J. ( 2009 ). Artinianness of local cohomology . Journal of Commutative Algebra 1 : 269 – 274 . ], and obtain partial answers to questions which were left open in [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. These partial answers give further support for conjecture (*) from [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] on the set of associated primes of MDLCMs. In addition, and also inspired by ideas from Zöschinger, we prove some non-finiteness results of local cohomology.     

Global Existence and Uniform Boundedness of Smooth Solutions to a Cross-Diffusion System with Equal Diffusion Rates

We consider the Shigesada-Kawasaki-Teramoto cross-diffusion model for two competing species. If both species have the same random diffusion coefficients and the space dimension is less than or equal to three, we establish the global existence and uniform boundedness of smooth solutions to the model in convex domains. This extends some previous works of Kim [12 Kim, J.U. (1984). Smooth solutions to a quasilinear system of diffusion equations for a certain population model. Nonlinear Anal. 8:1121–1144.[Crossref], [Web of Science ®] , [Google Scholar]] and Shim [21 Shim, S.-A. (2002). Uniform boundedness and convergence of solutions to cross-diffusion systems. J. Diff. Eqs. 185:281–305.[Crossref], [Web of Science ®] , [Google Scholar]] in one dimensional space.     

Uniqueness Criteria for the Adjoint Equation in State-Constrained Elliptic Optimal Control

The article considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control problems. By a counter example of Serrin [18 J. Serrin ( 1964 ). Pathological solutions of elliptic differential equations . Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 : 385 – 388 . [Google Scholar]], it is known that, in the presence of non-smooth data, a standard weak formulation does not ensure uniqueness for such equations. Therefore several notions of solution have been developed that guarantee uniqueness. In this note, we compare different definitions of solutions, namely the ones of Stampacchia [19 G. Stampacchia ( 1965 ). Le probléme de Dirichlet pour les équations elliptiques du second ordre à coéffcients discontinus . Ann. Inst. Fourier 15 : 189 – 258 .[Crossref] , [Google Scholar]] and Boccardo-Galouët [4 L. Boccardo and T. Gallouët ( 1989 ). Nonlinear elliptic and parabolic equations involving measure data . J. Func. Anal. 87 : 149 – 169 .[Crossref], [Web of Science ®] , [Google Scholar]] and the two notions of solutions of [2 J.-J. Alibert and J.-P. Raymond ( 1997 ). Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls . Numer. Func. Anal. Optim. 18 : 235 – 250 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 7 E. Casas (1993). Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31:993–1006.[Crossref], [Web of Science ®] , [Google Scholar]], and show that they are equivalent. As side results, we reformulate the solution in the sense of [19 G. Stampacchia ( 1965 ). Le probléme de Dirichlet pour les équations elliptiques du second ordre à coéffcients discontinus . Ann. Inst. Fourier 15 : 189 – 258 .[Crossref] , [Google Scholar]], and prove the existence of solutions in the sense of [2 J.-J. Alibert and J.-P. Raymond ( 1997 ). Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls . Numer. Func. Anal. Optim. 18 : 235 – 250 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 4 L. Boccardo and T. Gallouët ( 1989 ). Nonlinear elliptic and parabolic equations involving measure data . J. Func. Anal. 87 : 149 – 169 .[Crossref], [Web of Science ®] , [Google Scholar], 7 E. Casas (1993). Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31:993–1006.[Crossref], [Web of Science ®] , [Google Scholar]] in case of mixed boundary conditions.     

Equivalences for Weak Crossed Products

In this article, we give a criterion that characterizes equivalent weak crossed products. By duality, we obtain a similar result for weak crossed coproducts and, as a consequence, we find the conditions that assures the equivalence between two weak crossed biproducts. As an application, we show that the main results proved by Panaite in [12 Panaite, F. (2014). Equivalent crossed products and cross product bialgebras. Commun. Algebr. 42:1937–1952.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] (see also [11 Panaite, F. (2012). Invariance under twisting for crossed products. Proc. Am. Math. Soc. 140:755–763.[Crossref], [Web of Science ®] , [Google Scholar]]), for Brzeziński's crossed products, admits a substantial reduction in the imposed conditions.     

A new family of set-theoretic solutions of the Yang–Baxter equation

A new family of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation is constructed. Two subfamilies, consisting of irretractable square-free solutions, are new counterexamples to Gateva-Ivanova’s Strong Conjecture [7 Gateva-Ivanova, T. (2004). A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation. J. Math. Phys. 45(10):3828–3858.[Crossref], [Web of Science ®] , [Google Scholar]]. They are in addition to those obtained by Vendramin [15 Vendramin, L. (2016). Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova. J. Pure Appl. Algebra 220:2064–2076.[Crossref], [Web of Science ®] , [Google Scholar]] and [1 Bachiller, D., Cedó, F., Jespers, E., Okniński, J. (2017). A family of irretractable square-free solutions of the Yang-Baxter equation. Forum Math. (to appear). [Google Scholar]].     

Remarks on the Global Solutions of 3-D Navier-Stokes System with One Slow Variable

By applying Wiegner's method in [16 Wiegner , M. ( 1987 ). Decay results for weak solutions to the Navier-Stokes equations on ? ⁿ . J. London Math. Soc. 35 : 303 – 313 .[Crossref], [Web of Science ®] , [Google Scholar]], we first prove the large time decay estimate for the global solutions of a 2.5 dimensional Navier-Stokes system, which is a sort of singular perturbed 2-D Navier-Stokes system in three space dimension. As an application of this decay estimate, we give a simplified proof for the global wellposedness result in [6 Chemin , J.-Y. , Gallagher , I. ( 2010 ). Large, global solutions to the Navier-Stokes equations, slowly varying in one direction . Transactions of the American Mathematical Society 362 : 2859 – 2873 .[Crossref], [Web of Science ®] , [Google Scholar]] for 3-D Navier-Stokes system with one slow variable. Let us also mention that compared with the assumptions for the initial data in [6 Chemin , J.-Y. , Gallagher , I. ( 2010 ). Large, global solutions to the Navier-Stokes equations, slowly varying in one direction . Transactions of the American Mathematical Society 362 : 2859 – 2873 .[Crossref], [Web of Science ®] , [Google Scholar]], here the assumptions in Theorem 1.3 are weaker.     

Note on Conic Bundles over Henselian Discrete Valued Fields with Real Closed Residue Field

ABSTRACT

Let ? be a complete set of Sylow subgroups of a finite group G, that is, ? contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup H of a finite group G is said to be ?-permutable if H permutes with every member of ?. The purpose of this article is to study the influence of ?-permutability of all maximal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of a finite group G on the structure of G. Our results improve and extend the main results of Asaad (1998 Asaad , M. ( 1998 ). On maximal subgroups of Sylow subgroups of finite groups . Comm. Algebra 26 ( 11 ): 3647 – 3652 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Asaad and Heliel (2003 Asaad , M. , Heliel , A. A. ( 2003 ). On permutable subgroups of finite groups . Arch. Math. 80 : 113 – 118 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Asaad et al. (1991 Asaad , M. , Ramadan , M. , Shaalan , A. ( 1991 ). Influence of π-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group . Arch. Math. 56 : 521 – 527 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Li et al. (2003 Li , Y. , Wang , Y. , Wei , H. ( 2003 ). The influence of π-quasinormality of maximal subgroups of Sylow subgroups of a finite group . Arch. Math. 81 ( 3 ): 245 – 252 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Ramadan (1992 Ramadan , M. ( 1992 ). Influence of normality on maximal subgroups of Sylow subgroups of a finite group . Acta Math. Hungar. 59 ( 1–2 ): 107 – 110 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), and Srinivasan (1980 Srinivasan , S. ( 1980 ). Two sufficient conditions for supersolvability of finite groups . Israel J. Math. 35 : 210 – 214 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]).     

A Note on Maximal Properties of Some Subsemigroups of Finite Order-Preserving Transformation Semigroups

Both maximal idempotent-generated subsemigroups and maximal idempotent-generated regular subsemigroups of O _n were studied by Yang [10 Yang , X. , Lu , C. ( 2000 ). Maximal properties of some subsemigroups in finite order-preserving transformation semigroups . Communications in Algebra 28 : 3125 – 3135 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The purpose of this article is to simplify the results of Yang [10 Yang , X. , Lu , C. ( 2000 ). Maximal properties of some subsemigroups in finite order-preserving transformation semigroups . Communications in Algebra 28 : 3125 – 3135 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

Fuzzy Optimality Conditions for Fractional Multiobjective Bilevel Problems Under Fractional Constraints

In this article, we are concerned with fractional multiobjective optimization problems. In order to derive optimality conditions, we consider a new single level problem [12 J.J. Ye ( 2006 ). Constraint qualification and KKT conditions for bilevel programming problems . Mathematics of Operations Research 31 : 811 – 824 .[Crossref], [Web of Science ®] , [Google Scholar]], which is locally equivalent to the bilevel fractional multiobjective problem (P) at the optimal solution. Our approach consists of using another approach initiated by Mordukhovich [7 B.S. Mordukhovich ( 1976 ). Maximum principle in problems of time optimal control with nonsmooth constraints . J. Appl. Math. Mech. 40 : 960 – 969 .[Crossref], [Web of Science ®] , [Google Scholar], 8 B.S. Mordukhovich ( 1980 ). Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems . Soviet. Math. Dokl. 22 : 526 – 530 . [Google Scholar]], which does not involve any convex approximations and convex separation arguments, called the extremal principle [5 B.S. Mordukhovich ( 2006 ). Variational Analysis and Generalized Differentiation, I: Basic Theory . Grundlehren Series (Fundamental Principles of Mathematical Sciences) , Vol. 330 , Springer , Berlin . [Google Scholar], 6 B.S. Mordukhovich ( 2006 ). Variational Analysis and Generalized Differentiation, II: Applications , Grundlehren Series (Fundamental Principles of Mathematical Sciences) , Vol. 331 , Springer , Berlin . [Google Scholar], 9 B.S. Mordukhovich ( 1994 ). Generalized differential calculus for nonsmooth and set-valued mappings . J. Math. Anal. Appl. 183 : 250 – 288 .[Crossref], [Web of Science ®] , [Google Scholar]], for the study of necessary optimality conditions in fractional vector optimization.     

Derived Representation Type of Schur Superalgebras

In this note we extend the results of Bekkert and Futorny in [2 Bekkert , V. , Futorny , V. ( 2003 ). Derived categories of Schur algebras . Comm. Alg. 31 : 1799 – 1822 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and Hemmer, Kujawa and Nakano in [10 Hemmer , D. J. , Kujawa , J. , Nakano , D. K. ( 2006 ). Representation types of Schur superalgebras . J. Group Theory 9 : 283 – 306 .[Crossref], [Web of Science ®] , [Google Scholar]] and determine the derived representation type of Schur superalgebras.     

Regularity of Analytic Algebra of Prime Characteristic

Let A be an analytic algebra over a field k of characteristic p > 0. In this article, for an analytic k-algebra we introduce the concept of analytic pⁿ-basis which generalizes the pⁿ-basis defined in [1 Furuya , M. , Niitsuma , H. ( 2002 ). Regularity criterion of Noetherian local rings of prime characteristic . J. Algebra 247 : 219 – 230 .[Crossref], [Web of Science ®] , [Google Scholar]], and the concept of an pⁿ-admissible field for an algebraic function field, we give regularity criteria and absolute regularity criteria for an analytic algebra A/k in terms of the higher differential algebra and analytic pⁿ-basis. The results are partial extension of our previous results for affine algebras to the case of analytic algebras (cf. [1 Furuya , M. , Niitsuma , H. ( 2002 ). Regularity criterion of Noetherian local rings of prime characteristic . J. Algebra 247 : 219 – 230 .[Crossref], [Web of Science ®] , [Google Scholar], 3 Furuya , M. , Niitsuma , H. (2006). Regular local rings essentially of finite type over fields of prime characteristic. J. Algebra 306:703–711.[Crossref], [Web of Science ®] , [Google Scholar]]), and these are partial generalization of results of Orbanz in the analytic case (cf. [9 Orbanz , U. ( 1973 ). Höhere Derivationen und Regularität . J. Reine Angew. Mathematik 262/263 : 194 – 204 .[Crossref] , [Google Scholar]]).     

The Complement Graph for Gaussian Integers Modulo n

This article is a continuation for the work done in [1 Abu Osba , E. , Al-Addasi , S. , Abu Jaradeh , N. ( 2008 ). Zero divisor graph for the ring of Gaussian integers modulo . n. Comm. Algebra 36 ( 10 ): 3865 – 3877 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2 Abu Osba , E. , Al-Addasi , S. , Al-Khamaiseh , B. ( 2011 ). Some properties of the zero divisor graph for the ring of gaussian integers modulo . n. Glasgow Journal of Mathematics 53 : 391 – 399 .[Crossref], [Web of Science ®] , [Google Scholar]] on the zero divisor graph for the ring of Gaussian integers modulo n. It investigates when the complement graph of the zero divisor graph for the Gaussian integers modulo n connected, planar, regular, or Eulerian. The girth and diameter were also studied.     

Virtually Biautomatic Groups

We introduce virtually biautomatic groups (groups with finite index biautomatic subgroups) and generalize results of Gersten and Short [3 Gersten , S. , Short , H. ( 1991 ). Rational subgroups of biautomatic groups . Annals of Mathematics 134 : 125 – 128 .[Crossref], [Web of Science ®] , [Google Scholar]] and Mosher [5 Mosher , L. ( 1997 ). Central quotients of biautomatic groups . Comment. Math. Helv. 72 ( 1 ): 16 – 29 .[Crossref], [Web of Science ®] , [Google Scholar]] on centralizers, normalizers, and quotients to virtually biautomatic groups.