A challenging goal in materials chemistry and physics is spontaneously to form intended superstructures from designed building blocks. In fields such as crystal engineering and the design of porous materials, this typically involves building blocks of organic molecules, sometimes operating together with metallic ions or clusters. The translation of such ideas to nanoparticles and colloidal-sized building blocks would potentially open doors to new materials and new properties, but the pathways to achieve this goal are still undetermined. Here we show how colloidal spheres can be induced to self-assemble into a complex predetermined colloidal crystal-in this case a colloidal kagome lattice-through decoration of their surfaces with a simple pattern of hydrophobic domains. The building blocks are simple micrometre-sized spheres with interactions (electrostatic repulsion in the middle, hydrophobic attraction at the poles, which we call 'triblock Janus') that are also simple, but the self-assembly of the spheres into an open kagome structure contrasts with previously known close-packed periodic arrangements of spheres. This open network is of interest for several theoretical reasons. With a view to possible enhanced functionality, the resulting lattice structure possesses two families of pores, one that is hydrophobic on the rims of the pores and another that is hydrophilic. This strategy of 'convergent' self-assembly from easily fabricated colloidal building blocks encodes the target supracolloidal architecture, not in localized attractive spots but instead in large redundantly attractive regions, and can be extended to form other supracolloidal networks.

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Directed self-assembly of a colloidal kagome lattice

Lecture Notes in Economics and Mathematical Systems

From the Publisher:
While most algorithm designs are finalized toward worst case scenarios where they have to cope efficiently with unrealistic inputs, the average case solution is a probabilistic approach that allows for the possibility that a simple algorithm would suffice. This book provides a unique overview of the tools and techniques used in average case analysis of algorithms.

Average Case Analysis of Algorithms on Sequences

The statistical processes of evolutionary theory

This is the first comprehensive introduction to the powerful moment approach for solving global optimization problems (and some related problems) described by polynomials (and even semi-algebraic functions). In particular, the author explains how to use relatively recent results from real algebraic geometry to provide a systematic numerical scheme for computing the optimal value and global minimizers. Indeed, among other things, powerful positivity certificates from real algebraic geometry allow one to define an appropriate hierarchy of semidefinite (SOS) relaxations or LP relaxations whose optimal values converge to the global minimum. Several extensions to related optimization problems are also described. Graduate students, engineers and researchers entering the field can use this book to understand, experiment with and master this new approach through the simple worked examples provided.

An Introduction to Polynomial and Semi-Algebraic Optimization

Copositive optimization is a quickly expanding scientific research domain with wide-spread applications ranging from global nonconvex problems in engineering to NP-hard combinatorial optimization. It falls into the category of conic programming (optimizing a linear functional over a convex cone subject to linear constraints), namely the cone $${\mathcal{C}}$$ of all completely positive symmetric n × n matrices (which can be factorized into $${FF^\top}$$ , where F is a rectangular matrix with no negative entry), and its dual cone $${\mathcal{C}^*}$$ , which coincides with the cone of all copositive matrices (those which generate a quadratic form taking no negative value over the positive orthant). We provide structural algebraic properties of these cones, and numerous (counter-)examples which demonstrate that many relations familiar from semidefinite optimization may fail in the copositive context, illustrating the transition from polynomial-time to NP-hard worst-case behaviour. In course of this development we also present a systematic construction principle for non-attainability phenomena, which apparently has not been noted before in an explicit way. Last but not least, also seemingly for the first time, a somehow systematic clustering of the vast and scattered literature is attempted in this paper.

/pdf/think-co-mpletely-positive-matrix-properties-examples-and-a-3t1zqk1upr.pdf

Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

We show that the maximal cp-rank of $n\times n$ completely positive matrices is attained at a positive-definite matrix on the boundary of the cone of $n\times n$ completely positive matrices, thus answering a long-standing question. We also show that the maximal cp-rank of $5\times 5$ matrices equals six, which proves the famous Drew--Johnson--Loewy conjecture [Linear Multilinear Algebra, 37 (1994), pp. 303--310] for matrices of this order. In addition we present a simple scheme for generating completely positive matrices of high cp-rank and investigate the structure of a minimal cp factorization.

On the cp-Rank and Minimal cp Factorizations of a Completely Positive Matrix

Let $p_n$ denote the largest possible cp-rank of an $n\times n$ completely positive matrix. This matrix parameter has its significance both in theory and applications, as it sheds light on the geometry and structure of the solution set of hard optimization problems in their completely positive formulation. Known bounds for $p_n$ are $s_n=\binom{n+1}2-4$, the current best upper bound, and the Drew--Johnson--Loewy (DJL) lower bound $d_n=\lfloor\frac{n^2}4\rfloor$. The famous DJL conjecture (1994) states that $p_n=d_n$. Here we show $p_n=\frac {n^2}2 +{\mathcal O}(n^{3/2}) = 2d_n+{\mathcal O}(n^{3/2}){{p_n=\frac {n^2}2 +{\mathcal O}(n^{3/2}) = 2d_n+{\mathcal O}(n^{3/2})}}$, and construct counterexamples to the DJL conjecture for all $n\ge {12}$ (for orders seven through eleven counterexamples were already given in [I. M. Bomze, W. Schachinger, and R. Ullrich, Linear Algebra Appl., 459 (2014), pp. 208--221].

http://www.optimization-online.org/DB_FILE/2014/06/4392.pdf

New Lower Bounds and Asymptotics for the cp-Rank

http://www.optimization-online.org/DB_FILE/2014/01/4206.pdf

From seven to eleven: Completely positive matrices with high cp-rank

A claim of Leland (1985) states that in the presence of transaction costs a call option on a stock S, described by geometric Brownian motion, can be perfectly hedged using Black-Scholes delta hedging with a modified volatility. Recently Kabanov and Safarian (1997) disproved this claim, giving an explicit (up to an integral) expression of the limiting hedging error, which appears to be strictly negative and depends on the path of the stock price only via the stock price at expiry ST . We prove in this paper that the limiting hedging error, considered as a function of ST, exhibits a removable discontinuity at the exercise price. Furthermore, we provide a quantitative result describing the evolution of the discontinuity, which shows that its precursors can very well be observed also in cases of reasonable length of revision intervals. (author's abstract)

Werner Schachinger

Papers

Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

On the cp-Rank and Minimal cp Factorizations of a Completely Positive Matrix

New Lower Bounds and Asymptotics for the cp-Rank

From seven to eleven: Completely positive matrices with high cp-rank

Leland's Approach to Option Pricing: The Evolution of a Discontinuity