Feynman’s operational calculi for noncommuting operators: The monogenic calculus

In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.     

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.     

The Buchstab’s function and the operational Tau Method

In this article we discuss some aspects of operational Tau Method on delay differential equations and then we apply this method on the differential delay equation defined byw(u) = 1/u for 1 ≤u ≤ 2 and(uw(u))′ = w(u-1) foru ≥ 2, which was introduced by Buchstab. As Khajah et al.[l] applied the Recursive Tau Method on this problem, they had to apply that Method under theMathematica software to get reasonable accuracy. We present very good results obtained just by applying the Operational Tau Method using a Fortran code. The results show that we can obtain as much accuracy as is allowed by the Fortran compiler and the machine-limitations. The easy applications and reported results concerning the Operational Tau are again confirming the numerical capabilities of this Method to handle problems in different applications.     

‘The unplanned impact of mathematics’ and its implications for research funding: a discussion-led educational activity

‘The unplanned impact of mathematics’ refers to mathematics which has an impact that was not planned by its originator, either as pure maths that finds an application or applied maths that finds an unexpected one. This aspect of mathematics has serious implications when increasingly researchers are asked to predict the impact of their research before it is funded and research quality is measured partly by its short term impact.

A session on this topic has been used in a UK undergraduate mathematics module that aims to consider topics in the history of mathematics and examine how maths interacts with wider society. First, this introduced the ‘unplanned impact’ concept through historical examples. Second, it provoked discussion of the concept through a fictionalized blog comments discussion thread giving different views on the development and utility of mathematics. Finally, a mock research funding activity encouraged a pragmatic view of how research funding is planned and funded.

The unplanned impact concept and the structure and content of the taught session are described.     

The ‘Three-Line’ Theorem for the Vinogradov C-Spectral Sequence of the Yang–Mills Equations

The Vinogradov C-spectral sequence for the Yang–Mills equations is considered and the three-line theorem for the term E₁ of the C-spectral sequence is proved: E₁ ^p,q = 0 if p > 0 and q < n – 2, where n is the dimension of spacetime.     

The ‘corrected Durfee’s inequality’ for homogeneous complete intersections

We address the conjecture of Durfee (Math Ann 232:85–98, 1978), bounding the singularity genus $p_g$ by a multiple of the Milnor number $\mu $ for an $n$ -dimensional isolated complete intersection singularity. We show that the original conjecture of Durfee, namely $(n +1)!\cdot p_g \le \mu $ , fails whenever the codimension $r$ is greater than one. Moreover, we propose a new inequality $C_{n,r}\cdot p_g \le \mu $ , and we verify it for homogeneous complete intersections. In the homogeneous case the inequality is guided by a ‘combinatorial inequality’, that might have an independent interest.     

The ‘window problem’ for series of complex exponentials

Under a suitable sparsity condition on the exponents Λ=(λ_k=τ_k+iσ_k), it is shown that the individual terms can be obtained from observation of the L² function through the ‘window’ t∈[0, δ]—with an l² estimate (uniform for such Λ) asymptotically as t, δ→0. Some applications are given to control theory for partial differential equations.     

The ‘electromagnetic’ Wigner equation for an electron with spin

We derive and analyse the three-dimensional quantum Liouville equation for an electron with spin in an external electromagnetic field. By using methods of semigroup theory, we prove existence and uniqueness of the initial value problem. Expanding the solution into a series of pure states, we derive the existence of a generalized particle density and anL -estimate on the solution. The last section is devoted to an analysis of the classical and electrostatic limits.

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Zusammenfassung Die drei-dimensionale Quanten-Liouville Gleichung für ein Elektron mit Spin in einem äueren elektromagnetischen Feld wird hergeleitet und analysiert. Mit Hilfe der Halbgruppen-Theorie werden Existenz und Eindeutigkeit des Anfangswertproblems bewiesen. Aus der Entwicklung der Lösung in eine Reihe von reinen Zuständen schließen wir die Existenz einer verallgemeinerten Teilchendichte und erhalten eineL -Abschätzung für die Lösung. Im letzten Kapitel werden der klassische und elektrostatische Limes untersucht.

     

A stylized model of ‘Momentum’ processes: a research note

Several phenomena of interest like stock price movements, online auction bidprices or inventory levels, can be stylized as ‘momentum processes’,whereby the very start of activity (‘up’ or ‘down’) cantrigger subsequent activity in that direction. We calculate the transientprobabilities of being in a particular state (eg, stock price or current auctionbid level) in such activities. The calculations can be used to estimate theexpected time to reach a certain price, inventory level or bid. The processescan be modified to show the phenomenon where the lack of activity can have an‘inertia of rest’.     

A good foundation for number learning for five-year-olds? An evaluation of the English Early Learning ‘Numbers’ Goal in the light of research

This article sets out to evaluate the English Early Years Foundation Stage Goal for Numbers, in relation to research evidence. The Goal, which sets out to provide ‘a good foundation in mathematics’, has greater breadth of content and higher levels of difficulty than previous versions. Research suggests that the additional expectations are unachievable by the majority of rising fives, but that a more effective foundation to ensure progress and prevent later difficulties would prioritise number sense with numbers to 10, including subitising, numeral comprehension, relative number size, one more/less than and part–whole relationships.     

On the ‘Standard’ Condition for Noncompact Homogeneous Einstein Spaces

A nonflat Einstein solvmanifold ( , g) is said to be of standard type if in the associated metric Lie algebra , the orthogonal complement of the derived algebra is Abelian. It is an open question whether the standard condition is automatically satisfied for all nonflat Einstein solvmanifolds. We derive certain properties of the metric Lie algebra of a nonflat Einstein solvmanifold ( , g) under the assumption . In particular, we obtain some new sufficient conditions which imply standard type.     

Conway’s Question: The Chase for Completeness

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–?ech compactification β X (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory Conw _C of C-Conway spaces, that turns out to be again a bireflective subcategory of Tych. We show that every bireflective subcategory of Tych can be presented in this way by building a Galois connection between bireflective subcategories of Tych and closure operators of Top finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Ra?kov completion ${ \ifmmode\expandafter\tilde\else\expandafter\~\fi{G}}$ . The sequential completeness for topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc.     

Utility functions and the ‘lin’ operation for convex sets

For an ℵ₀-dimensional spaceE of alternatives, there is described a preference relation ≿ such that (in a very strong sense)no information about ≿ can be expressed in terms of finite-dimensional linear transformations ofE. The same construction shows that for each countable ordinalβ, E contains a convex coneK such that lin ^β K =E but lin^β K ≡E forα<β.     

Holistic oil field value management: using system dynamics for ‘intermediate level’ and ‘value-based’ modelling in the oil industry

System dynamics has been seen primarily as a strategic tool, most effectively used at the highest level of strategy to identify robust policy interventions under a wide range of scenarios. However, an alternative, complementary and powerful role is emerging. This is at an ‘intermediate level’ in organisations to coordinate and integrate policies across the value chain. It is at this level where business value, as defined by the discounted value of future free cash flow, is both created and destroyed. This paper introduces the need for ‘intermediate-level’ and ‘value-based’ modelling and emphasises the natural role of system dynamics in supporting a methodology to fulfil the need. It describes the development of an approach and its application in the oil industry to coordinate the response of people and tools within operational, financial and commercial functions across the value chain to address a variety of problems and issues.     

The ‘similar’ solution of the hydromagnetic free-jet boundary layer equations

In the present note the similar solution of the boundary layer equations for flow of a plane free-jet of a viscous, incompressible, electrically conducting fluid in the presence of a transverse uniform magnetic field is studied. The corresponding hydromagnetic stream function is obtained in a compact form, which is remarkably simple and enables us to locate the distance from the slit, exactly, where the width of the jet becomes infinite.

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Zusammenfassung Es wird eine ähnliche Lösung der Grenzschichtgleichungen angegeben für die ebene Strömung eines Strahles in einer zähen, inkompressiblen, elektrisch leitenden Flüssigkeit in der Gegenwart eines quergerichteten gleichförmigen Magnetfeldes. Die ensprechende hydromagnetische Stromfunktion wird in einer geschlossenen Form angegeben, die besonders einfach ist und eine exakte Bestimmung der Distanz vom Ursprung ermöglicht, bei der die Strahlweite unendlich wird.

     

Stabilization for the vibrations modeled by the ‘standard linear model’ of viscoelasticity

We study the stabilization of vibrations of a flexible structure modeled by the ‘standard linear model’ of viscoelasticity in a bounded domain in ℝ ⁿ with a smooth boundary. We prove that amplitude of the vibrations remains bounded in the sense of a suitable norm in a space $ mathbb{X} $ mathbb{X} , defined explicitly in (22) subject to a restriction on the uncertain disturbing forces on $ mathbb{X} $ mathbb{X} . We also estimate the total energy of the system over time interval [0, T] for any T > 0, with a tolerance level of the disturbances. Finally, when the input disturbances are insignificant, uniform exponential stabilization is obtained and an explicit form for the energy decay rate is derived. These results are achieved by a direct method under undamped mixed boundary conditions.     

Defining a mathematical research school: the case of algebra at the University of Chicago, 1892–1945

Historians of science have long considered the concept of the “research school” as a potent analytical construct for understanding the development of the laboratory sciences. Unfortunately, their definitions fall short in the case of mathematics. Here, a definition of “mathematical research school” is proposed in the context of a case study of algebraic work associated with the University of Chicago's Department of Mathematics from the University's founding in 1892 through 1945.     

An ‘odd’ formula for the volume of three-dimensional lattice polyhedra

Let be the tiling of R ³ with unit cubes whose vertices belong to the fundamental lattice L ₁ of points with integer coordinates. Denote by L _n the lattice consisting of all points x in R ³ such that nx belongs to L ₁. When the vertices of a polyhedron P in R ³ are restricted to lie in L ₁ then there is a formula which relates the volume of P to the numbers of all points of two lattices L ₁ and L _n lying in the interior and on the boundary of P. In the simplest case of the lattices L ₁ and L ₂ there are 27 points in each cube from whose relationships to the polyhedron P must be examined. In this note we present a new formula for the volume of lattice polyhedra in R ³ which involves only nine points in each cube of : one from L ₂ and eight belonging to L ₄. Another virtue of our formula is that it does not employ any additional parameters, such as the Euler characteristic.