Space–Time Fractional Equations and the Related Stable Processes at Random Time

In this paper, we consider the general space–time fractional equation of the form \(\sum _{j=1}^m \lambda _j \frac{\partial ^{\nu _j}}{\partial t^{\nu _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)\), for \(\nu _j \in \left( 0,1 \right] \) and \(\beta \in \left( 0,1 \right] \) with initial condition \(w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)\). We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), where \(\varvec{S}_n^{2\beta }\) is an isotropic stable process independent from \(\mathcal {L}^{\nu _1, \ldots , \nu _m}(t)\), which is the inverse of \(\mathcal {H}^{\nu _1, \ldots , \nu _m} (t) = \sum _{j=1}^m \lambda _j^{1/\nu _j} H^{\nu _j} (t)\), \(t>0\), with \(H^{\nu _j}(t)\) independent, positively skewed stable random variables of order \(\nu _j\). The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), supplies a probabilistic representation for the solutions of the fractional equations above and coincides for \(\beta = 1\) with the n-dimensional Brownian motion at the random time \(\mathcal {L}^{\nu _1, \ldots , \nu _m} (t)\), \(t>0\). The iterated process \(\mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t)\), \(t>0\), inverse to \(\mathfrak {H}^{\nu _1, \ldots , \nu _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/\nu _j} \, _1H^{\nu _j} \left( \, _2H^{\nu _j} \left( \, _3H^{\nu _j} \left( \ldots \, _{r}H^{\nu _j} (t) \ldots \right) \right) \right) \), \(t>0\), permits us to construct the process \(\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t) \right) \), \(t>0\), the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For \(r \rightarrow \infty \) and \(\beta = 1\), we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation \(\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)\). Our analysis represents a general framework of the interplay between fractional differential equations and composition of processes of which the iterated Brownian motion is a very particular case.     

Local Existence and WKB Approximation of Solutions to Schrödinger–Poisson System in the Two-Dimensional Whole Space

We consider the Schrödinger–Poisson system in the two-dimensional whole space. A new formula of solutions to the Poisson equation is used. Although the potential term solving the Poisson equation may grow at the spatial infinity, we show the unique existence of a time-local solution for data in the Sobolev spaces by an analysis of a quantum hydrodynamical system via a modified Madelung transform. This method has been used to justify the WKB approximation of solutions to several classes of nonlinear Schrödinger equation in the semiclassical limit.     

Branching Rules, Kostka–Foulkes Polynomials and q-multiplicities in Tensor Product for the Root Systems B n , C n and D n

The Kostka–Foulkes polynomials related to a root system can be defined as alternating sums running over the Weyl group associated to . By restricting these sums over the elements of the symmetric group when is of type or , we obtain again a class of Kostka–Foulkes polynomials. When is of type or there exists a duality between these polynomials and some natural -multiplicities and in tensor products [11]. In this paper we first establish identities for the which implies in particular that they can be decomposed as sums of Kostka–Foulkes polynomials with nonnegative integer coefficients. Moreover these coefficients are branching coefficients This allows us to clarify the connection between the -multiplicities and the polynomials defined by Shimozono and Zabrocki. Finally we show that and coincide up to a power of with the one dimension sum introduced by Hatayama and co-workers when all the parts of are equal to , which partially proves some conjectures of Lecouvey and Shimozono and Zabrocki.Presented by P. Littelmann.     

A Note on the Penrose–Mackay Tiling,HirzbruchSignature of M 4 and Knots in e (∞) Space

Mackays generalization of Penrose tiling is shown to be related to the Hirzbruchsignature, δ, of four manifolds in the case of a Murray–von Neumann continuousdimension in E ^(∞) space. It is further shown that σ is numerically identicalto the Jones knot invariant, V_L, for the right-handed trefoil. Finally, aconjecture regarding the theorem of local isomorphism is presented.     

Organizational capability,efficiency, and effectiveness in Major League Baseball: 1901–2002

Effective organizations need capabilities relevant to their missions and must manage those capabilities efficiently. We anticipate capability is more important in industries in which labor is highly paid, while efficiency is more important in industries in which labor is inexpensive. We explore the contributions of capability and efficiency to effectiveness for Major League Baseball teams from 1901 through 2002. Our analysis measures team capability using offensive and defensive statistics and uses Network Data Envelopment Analysis to derive efficiency scores to capture managerial performance. We define effectiveness as the team’s winning percentage. Both capability and efficiency are significant contributors to regular season effectiveness. Capability is more important. Finally, we examine the post-season performance of post-season teams between 1903 and 2002. Our analysis measures post-season performance based on the team’s winning percentage and that of its opponent. Post-season performance is unrelated to capability and managerial performance, accounting for about 1% of post-season success.     

Problem solving in the United States, 1970–2008: research and theory, practice and politics

Problem solving was a major focus of mathematics education research in the US from the mid-1970s though the late 1980s. By the mid-1990s research under the banner of “problem solving” was seen less frequently as the field’s attention turned to other areas. However, research in those areas did incorporate some ideas from the problem solving research, and that work continues to evolve in important ways. In curricular terms, the problem solving research of the 1970s and 1980s (see, e.g., Lester in J Res Math Educ, 25(6), 660–675, 1994, and Schoenfeld in Handbook for research on mathematics teaching and learning, MacMillan, New York, pp 334–370, 1992, for reviews) gave birth to the “reform” or “standards-based” curriculum movement. New curricula embodying ideas from the research were created in the 1990s and began to enter the marketplace. These curricula were controversial. Despite evidence that they tend to produce positive results, they may well fall victim to the “math wars” as the “back to basics” movement in the US is revitalized.     

Centrality,dominance, and interorganizational power in a network structure: Interlocking directorates among American railroads, 1886–1905

Contrary to the prevailing theory that assumes a positive relationship between an organization's structural centrality and its relational power, our analysis of the interlocking structure of major American railroad companies from 1886 to 1905 reveals that an organization's dominance within a latent class of directors is a structural variable that is more closely related to interorganizational power than overall centrality. Moreover, in one of the first longitudinal analysis of an interlocking directorate structure, we find that changes in dominance within a latent class and changes in interorganizational power are associated not only cross‐sectionally, but over time. We conclude that, while network centrality may be related to interorganizational power within a singular, monotonically hierarchical structure, in the highly clustered, or Balkanized, system like that of the American railroad system around the turn of the century, the relationship between centrality and interorganizational power dissolves. The primary theoretical implication is that the relationship between a structural variable like centrality and a power relation variable like interorganizational power is therefore a contingent relationship rather than a determinant one.     

Spreading the gospel of management science: operational research in Iron and Steel, 1950–1970

This paper builds on the author's earlier work on the history of operational research by presenting an analysis of the development of the discipline in Iron and Steel, an industry long regarded as one of the outstanding pioneers in the application of management science to decision-making processes. The contribution of Sir Charles Goodeve and BISRA to the diffusion and practice of operational research is well-documented. Less well known is the reaction to operational research within the managerial hierarchies of the private sector iron and steel companies. In the light of the development of dedicated operational research groups by the leading companies after 1950, it might be assumed that the industry was highly receptive to the discipline in terms of its perceived benefits. The present paper questions this assumption by highlighting the problems encountered by operational researchers in two of the largest Iron and Steel Companies which gave every appearance of being at the forefront of the practice of management science in British industry. Where appropriate, the paper draws contrasts and comparisons with the development of operational research within the nationalised Coal Industry.     

Nash bargaining and leader–follower models in water allocation: Application to the Zarrinehrud River basin,Iran

Mathematical models for conflict resolution are very important in integrated water resources and environmental management. This study proposes a new methodology to resolve conflicts among different water users and water suppliers while considering environmental requirements and the system’s constraints. A two-level leader–follower model is applied to maximize the net benefit with the Iran Water Resources Management Company as the leader and agricultural, domestic, and industrial users as followers subject to the system’s constraints. As a comparison, the Nash bargaining solution is also used to find a solution when simultaneous moves are assumed by the participants. The suggested method is then applied to the real case of the Zarrinehrud River basin that is one of the areas facing water shortages in Iran. For the actual optimization, Genetic Algorithm is used in order to avoid local optimum. As the contribution of this study, the results show that benefits for the leader in the leader–follower model increased in comparison with the Nash bargaining solutions.     

General Relativity in the Making, 1916–1918: Rudolf J. Humm in Göttingen and Berlin

Mathematische Semesterberichte - In 1916, the Swiss student Rudolf J. Humm arrived in Göttingen to study relativity theory under Hilbert. He enrolled in his courses, attended Klein’s...     

Corrigendum to “Semiring and semimodule issues in MV-algebras,Communications in Algebra 41:3 (2013), 1017–1048”

With the present note the authors intend to correct a mistake in the paper Semiring and semimodule issues in MV-algebras, Communications in Algebra 41:3 (2013), 1017–1048, by the same authors.