Swing Analysis and Electoral Forecasting

This paper examines the procedure by which votes are converted into seats at U.K. General Elections. In particular it seeks to answer these questions.

1. i)
   What is the meaning of swing when more than two parties fight an election?
    
2. ii)
   How can the distribution of seats at an election be determined from a prediction of national swing?
    
3. iii)
   How can swing analysis be extended to help determine an electoral strategy for political parties?
    
4. iv)
   What explanation can be provided for the swing in Scotland?
    
5. v)
   What explanation can be provided for variations in the English swing?
    

     

Operational Research Projects in Health and Welfare Services in the United Kingdom and Ireland

The paper presents a comparison of Health and Welfare O.R. in the U.K. and Ireland as represented by two registers of current projects, compiled in 1972 and 1977 respectively. The survey shows that

1. i)
   There appear to be very few O.R. studies of social services;
    
2. ii)
   The major focus has apparently moved away from hospital-based tactical studies, towards more broadly-based strategic studies;
    
3. iii)
   Related to this, the number of projects labelled as concerned with "planning" has increased;
    
4. iv)
   The number of projects involving "optimisation" has decreased, whilst those involving simulation have shown some increase; and
    
5. v)
   The majority of projects in both years were described as involving "applied common sense", rather than any specific technique.
    

     

Operational Research and Management

The purpose of this paper is to discuss and if possible to dispel the industrialist's misunderstanding of operational research. The author lists some of the varied ideas held by industrialists of operational research including several which are contradictory, as for example (a) That operational research is new and revolutionary; and (b) That operational research proves nothing new and has always been practised without being so named.A number of suggestions are then made and developed, which should lead to a better understanding. These include, clearer definitions of operational research and a better adaptation of the specialized literature to the needs of industrial management, with particular emphasis on the publication of real rather than synthetic case histories. Headings of sections are:

1. 1
   The attitude of industrialists.
    
2. 2
   The exact nature of operational research.
    
3. 3
   Problems, criteria, models.
    
4. 4
   The practice of operational research in business.
    

     

Multi-invariant measures and subsets on nilmanifolds

Given a Z^r-action α on a nilmanifold X by automorphisms and an ergodic α-invariant probability measure μ, we show that μ is the uniform measure on X unless, modulo finite index modification, one of the following obstructions occurs for an algebraic factor action

1. (1)
   the factor measure has zero entropy under every element of the action
    
2. (2)
   the factor action is virtually cyclic.
    

We also deduce a rigidity property for invariant closed subsets.

     

A Production Planning and Stock Control Problem in the Textile Industry

This paper describes the techniques used in an enquiry into production planning and stock holding problems encountered by one of our member firms.The investigation of the problem falls into three parts:

1. i)
   The analysis of demand.
    
2. ii)
   The determination of optimum stock levels.
    
3. iii)
   The solution of the allocation problem which arises if the capacity available for the production of stock lines is not equal to the amount required to bring stocks back to the optimum levels. This problem has not been fully investigated, but a proposed method of solution is given in an appendix.
    

     

Semistability and Restrictions of tangent bundle to curves

We consider all complex projective manifolds X that satisfy at least one of the following three conditions:

1. (1)
   There exists a pair \({(C\,,\varphi)}\) , where C is a compact connected Riemann surface and

   $\varphi\,:\, C\,\longrightarrow\, X$

   a holomorphic map, such that the pull back \({\varphi^* {\it TX}}\) is not semistable.
    
2. (2)
   The variety X admits an étale covering by an abelian variety.
    
3. (3)
   The dimension dim X ≤ 1.
    

We prove that the following classes are among those that are of the above type.

• All X with a finite fundamental group.
• All X such that there is a nonconstant morphism from \({{\mathbb C}{\mathbb P}^1}\) to X.
• All X such that the canonical line bundle K _X is either positive or negative or \({c_1(K_X)\,\in\,H^2(X,\, {\mathbb Q})}\) vanishes.
• All X with \({{\rm dim}_{\mathbb C} X\, =\,2}\).

     

Regularly weakly based modules over right perfect rings and Dedekind domains

A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contains a weak basis. We study

1. (1)
   rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, and
    
2. (2)
   regularly weakly based modules over Dedekind domains.
    

     

Speed,Gradient and Workrate in Uphill Running

A series of treadmill experiments is described concerned with a runner's speed, heart-rate and the gradient. Together with the results of similar experiments, some of them carried out over 50 years ago, the results suggest that for a given heart-rate, log(speed) is linearly related to gradient, and that for a given gradient, heart-rate is linearly related to speed. The results suggest:

1. 1)
   that athletes who run p% faster on the level will run p% faster up a slope, if they maintain the same heart-rate;
    
2. 2)
   that athletes will use the same number of heart beats running up a hill of uniform slope no matter how fast or slowly they run;
    
3. 3)
   that athletes should run directly up any slope of less than about 20° and try to zigzag up slopes greater than this.
    

     

Robust solutions and risk measures for a supply chain planning problem under uncertainty

We consider a strategic supply chain planning problem formulated as a two-stage stochastic integer programming (SIP) model. The strategic decisions include site locations, choices of production, packing and distribution lines, and the capacity increment or decrement policies. The SIP model provides a practical representation of real-world discrete resource allocation problems in the presence of future uncertainties which arise due to changes in the business and economic environment. Such models that consider the future scenarios (along with their respective probabilities) not only identify optimal plans for each scenario, but also determine a hedged strategy for all the scenarios. We

1. 1)
   exploit the natural decomposable structure of the SIP problem through Benders’ decomposition,
    
2. 2)
   approximate the probability distribution of the random variables using the generalized lambda distribution, and
    
3. 3)
   through simulations, calculate the performance statistics and the risk measures for the two models, namely the expected-value and the here-and-now.
    

     

Linear Programming in Air Defence Modelling

The paper describes the use of an optimizing model, together with a simulation model, to determine the most effective mix and deployment of air defence weapons to defend a given set of assets against a range of air threats. The advantages of using an optimizing model in such studies are that it:

1. a
   —Gives more consistent results by reducing player variance.
    
2. b
   —Can generate solutions that are robust to variations in threat and environment.
    
3. c
   —Ensures that good solutions are not overlooked.
    
4. d
   —Reduces the total processing time required to complete the study.
    

     

The Riesz transform and quantitative rectifiability for general Radon measures

In this paper we show that if \(\mu \) is a Borel measure in \({{\mathbb {R}}}^{n+1}\) with growth of order n, such that the n-dimensional Riesz transform \({{\mathcal {R}}}_\mu \) is bounded in \(L^2(\mu )\), and \(B\subset {{\mathbb {R}}}^{n+1}\) is a ball with \(\mu (B)\approx r(B)^n\) such that:

1. (a)
   there is some n-plane L passing through the center of B such that for some \(\delta >0\) small enough, it holds

   $$\begin{aligned}\int _B \frac{\mathrm{dist}(x,L)}{r(B)}\,d\mu (x)\le \delta \,\mu (B),\end{aligned}$$
    
2. (b)
   for some constant \({\varepsilon }>0\) small enough,

   $$\begin{aligned}\int _{B} |{{\mathcal {R}}}_\mu 1(x) - m_{\mu ,B}({{\mathcal {R}}}_\mu 1)|^2\,d\mu (x) \le {\varepsilon }\,\mu (B),\end{aligned}$$

   where \(m_{\mu ,B}({{\mathcal {R}}}_\mu 1)\) stands for the mean of \({{\mathcal {R}}}_\mu 1\) on B with respect to \(\mu \),
    

then there exists a uniformly n-rectifiable set \(\Gamma \), with \(\mu (\Gamma \cap B)\gtrsim \mu (B)\), and such that \(\mu |_\Gamma \) is absolutely continuous with respect to \({{\mathcal {H}}}^n|_\Gamma \). This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.

     

Waist and trunk of knots

Let K be a knot in the 3-sphere S ³. We define the waist of K as

$waist (K) = \mathop{\rm max}\limits_{F\in\mathcal{F}} \mathop{\rm min}\limits_{D\in\mathcal{D}_{F}} |D \cap K|,$

where \({\mathcal{F}}\) is the set of all closed incompressible surfaces in S ³?K and \({\mathcal{D}_F}\) is the set of all compressing disks for F in S ³. We define the trunk of K as

$trunk(K) = \mathop{\rm min}\limits_{h\in\mathcal{H}} \mathop{\rm max}\limits_{t\in\mathbb{R}} |h^{-1}(t) \cap K|,$

where \({\mathcal{H}}\) is the set of all Morse function \({h : S^3 \to \mathbb{R}}\) with two critical points. We show that

$waist (K) \le \frac{trunk(K)}{3}$

.

     

Lie Algebras of Slow Growth and Klein-Gordon PDE

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms:

1. 1)
   the first one is between the characteristic Lie algebra \(\chi (\sinh {u})\) of the sinh-Gordon equation \(u_{xy}=\sinh {u}\) and the non-negative part \({\mathcal {L}}({\mathfrak {sl}}(2,{\mathbb {C}}))^{\ge 0}\) of the loop algebra of \({\mathfrak {sl}}(2,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{1}^{(1)}\)

   $$\chi(\sinh{u})\cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(2,{\mathbb{C}}))^{\ge 0}={\mathfrak{s}\mathfrak{l}}(2, {\mathbb{C}}) \otimes {\mathbb{C}}[t]. $$
    
2. 2)
   the second isomorphism is for the Tzitzeica equation u_xy = e^u + e^??2u

   $$\chi(e^{u}{+}e^{-2u}) \cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(3,{\mathbb{C}}), \mu)^{\ge0}=\bigoplus_{j = 0}^{+\infty}{\mathfrak{g}}_{j (\text{mod} \; 2)} \otimes t^{j}, $$

   where \({\mathcal {L}}({\mathfrak {sl}}(3,{\mathbb {C}}), \mu )=\bigoplus _{j \in {\mathbb {Z}}}{\mathfrak {g}}_{j (\text {mod} \; 2)} \otimes t^{j}\) is the twisted loop algebra of the simple Lie algebra \({\mathfrak {sl}}(3,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{2}^{(2)}\).
    

Hence the Lie algebras \(\chi (\sinh {u})\) and χ(e^u + e^??2u) are slowly linearly growing Lie algebras with average growth rates \(\frac {3}{2}\) and \(\frac {4}{3}\) respectively.     

More on stability of almost surjective <Emphasis Type="Italic">ε</Emphasis>-isometries of Banach spaces

Let X and Y be two Banach spaces, and f: X → Y be a standard ε-isometry for some ε ≥ 0. In this paper, by using a recent theorem established by Cheng et al. (2013–2015), we show a sufficient condition guaranteeing the following sharp stability inequality of f: There is a surjective linear operator T: Y → X of norm one so that

$$\left\| {Tf(x) - x} \right\| \leqslant 2\varepsilon , for all x \in X.$$

As its application, we prove the following statements are equivalent for a standard ε-isometry f: X → Y:

1. (i)
   lim inf _t→∞ dist(ty, f(X))/|t| < 1/2, for all y ∈ S _Y ;
    
2. (ii)
   \(\tau(f)\equiv sup_{y\epsilon S_{Y}}\) lim inf _t→∞dist(ty, f(X))/|t| = 0;
    
3. (iii)
   there is a surjective linear isometry U: X → Y so that

   $$\left\| {f(x) - Ux} \right\| \leqslant 2\varepsilon , for all x \in X.$$
    

This gives an affirmative answer to a question proposed by Vestfrid (2004, 2015).