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## On the foundations of linear and integer linear programming I

### Mathematical Programming (1975-12-01) 9: 207-226 , December 01, 1975

In this paper we consider the question: how does the flow algorithm and the simplex algorithm work? The usual answer has two parts: first a description of the “improvement process”, and second a proof that if no further improvement can be made by this process, an optimal vector has been found. This second part is usually based on duality -a technique not available in the case of an arbitrary integer programming problem. We wish to give a general description of “improvement processes” which will include both the simplex and flow algorithms, which will be applicable to arbitrary integer programming problems, and which will in*themselves assure convergence* to a solution.

Geometrically both the simplex algorithm and the flow algorithm may be described as follows. At the ith stage, we have a vertex (or feasible flow) to which is associated a finite set of vectors, namely the set of edges leaving that vertex (or the set of unsaturated paths). The algorithm proceeds by searching among this special set for a vector along which the gain function is increasing. If such a vector is found, the algorithm continues by moving along this vector as far as is possible while still remaining feasible. The search is then repeated at this new feasible point.

We give a precise definition for sets of vectors, called test sets, which will include those sets described above arising in the simplex and flow algorithms. We will then prove that any “improvement process” which searches through a test set at each stage converges to an optimal point in a finite number of steps. We also construct specific test sets which are the natural extensions of the test sets employed by the flow algorithm to arbitrary linear and integer linear programming problems.

## The Width-Volume Inequality

### Geometric and Functional Analysis (2007-11-01) 17: 1139-1179 , November 01, 2007

### Abstract.

We prove that a bounded open set *U* in
$${\mathbb{R}}^n$$
has *k*-width less than *C*(*n*) Volume(*U*)^{k/n}. Using this estimate, we give lower bounds for the *k*-dilation of degree 1 maps between certain domains in
$${\mathbb{R}}^n$$
. In particular, we estimate the smallest (*n* – 1)-dilation of any degree 1 map between two *n*-dimensional rectangles. For any pair of rectangles, our estimate is accurate up to a dimensional constant *C*(*n*). We give examples in which the (*n* – 1)-dilation of the linear map is bigger than the optimal value by an arbitrarily large factor.

## Biquadratic Eisenstein Series

### Journal of Mathematical Sciences (2003-07-01) 116: 2993-3009 , July 01, 2003

Biquadratic metaplectic Eisenstein series on the group *SL*_{3}ℂ are studied, and the “*sl*(2)-triples” technique for the computation of the Fourier coefficients of these series is used. The biquadratic theta function on the group *SL*_{3}ℂ is also constructed. Bibliography: 10 titles.

## Regular schemes over a finite Abelian group

### Geometriae Dedicata (1979-12-01) 8: 477-490 , December 01, 1979

## Schrödinger Operators and Canonical Systems

### Operator Theory (2015-01-01): 623-630 , January 01, 2015

This paper discusses the inverse spectral theory of Schrödinger equations from the point of view of canonical systems and de Branges’s theory of Hilbert spaces of entire functions. The basic idea is to view Schrödinger equations as special canonical systems. For canonical systems, a complete inverse spectral theory is available: there is a one-to-one correspondence between the coefficient functions, on the one hand, and suitable spectral data, on the other hand. The task then is to identify those subclasses that correspond to Schrödinger equations.

## Quadrature and Schatz’s Pointwise Estimates for Finite Element Methods

### BIT Numerical Mathematics (2005-12-01) 45: 695-707 , December 01, 2005

We investigate numerical integration effects on weighted pointwise estimates. We prove that local weighted pointwise estimates will hold, modulo a higher order term and a negative-order norm, as long as we use an appropriate quadrature rule. To complete the analysis in an application, we also prove optimal negative-order norm estimates for a corner problem taking into account quadrature. Finally, we present an example to show that our result is sharp.

## Square of White Noise Unitary Evolutions on Boson Fock Space

### Proceedings of the International Conference on Stochastic Analysis and Applications (2004-01-01): 267-301 , January 01, 2004

With the help of *Mathematica* we deduce an explicit formula for bringing to normal order the product of two normally ordered monomials in the generators of the Lie algebra of SL(2, ℝ). We use this formula to prove the Itô multiplication table for the stochastic differentials of the universal enveloping algebra of the renormalized square of white noise defined on Boson Fock space. Using this Itô table we derive unitarity conditions for processes satisfying quantum stochastic differential equations driven by this noise. From these conditions we deduce in particular that a quantum stochastic differential involving only the three basic integrators (see (2.7)–(2.9) below) and *dt* cannot have a unitary solution. Computer algorithms for checking these conditions, for computing the product of stochastic differentials, and for iterating the differential of the square of white noise analogue of the Poisson-Weyl operator are also provided.

## Howe’s Correspondence and Characters

### Geometric Methods in Physics (2016-01-01): 183-190 , January 01, 2016

The purpose of this note is to explain how is Howe’s correspondence used to construct irreducible unitary representations of low Gel’fand–Kirillov dimension and to recall and motivate a conjecture concerning the distribution characters of the representations involved.

## Continuous transformations in R2

### Continuous Transformations in Analysis (1955-01-01) 75: 377-438 , January 01, 1955

Throughout Part VI, we shall be concerned with bounded continuous transformations
(1)
$$ T:D \to {R^2} $$
where *D* is a bounded domain in *R*^{2}. It will be convenient to use certain alternative representations for *T*. Note that the points of *R*^{
2} are ordered pairs (*x, y*) of real numbers *x, y*. Hence we can associate with the point (*x, y*) of *R*^{2} the complex number *z = x + iy*. In the sequel, the terms “complex number” and “point of *R*^{2}” will be used interchangeably. Using complex numbers, the transformation (1) can be represented in the form
(2)
$$ T:z = T\left( w \right),w \in D $$
where *z* is the image point of *w* under *T*. Thus *T* is thought of now as a bounded, continuous, real or complex-valued function of the complex number *w*∈*D*. If we set *w = u + iv, z = x + iy*, where *u, v, x, y* are real numbers, then we obtain for *T* the representation
(3)
$$ T:x = x\left( {u,v} \right),y = y\left( {u,v} \right),\left( {u,v} \right) \in D $$
where the coordinate functions *x(u, v), y(u, v)* are bounded, continuous, real-valued functions in *D*.

## Linear Multivariate Prediction

### Reduced Rank Regression (1995-01-01): 16-37 , January 01, 1995

In the area of QSAR, described in the preceding chapter 1, a response is measured for a given predictor. There, the main scientific question is how to find a model which allows the prediction of the multivariate response for a given multivariate predictor. The statistician faced with such a prediction problem has to consider many different aspects, for example:

*Should he/she use flexible nonparametric models or rely on more stable linear procedures?* Many very flexible nonparametric nonlinear methods have become available, such as projection pursuit regression (Friedman, 1985), ACE (Breiman and Friedman, 1985), or the methods described in Gifi (1991). All these methods have problems with overfitting, that is they show spurious precision. On the other hand, linear models are generally more stable but introduce severe bias in the case of strong nonlinear relationships. In the QSAR area a relatively small number of observations and a relatively large measurement error of the responses in comparison with their variability is typical. In these circumstances, nonlinear methods can only be used with great care, and linear methods are generally the methods of choice. Hence, nonlinear methods will not be used in the following. Some further discussions about the balance of flexibility and stability, bias and variability can be found in section 4.4.4 with regard to reduced rank regression models.