Abstract:
In order to rank investments under uncertainty, the most widely used method is mean variance analysis. Stochastic dominance is an alternative concept which ranks investments by using the whole distribution function. There exist three models: first-order stochastic dominance is used when the distribution functions do not intersect, second-order stochastic dominance is applied to situations where the distribution functions intersect only once, while third-order stochastic dominance solves the ranking problem in the case of double intersection. Almost stochastic dominance is a special model. Finally we show that the existence of arbitrage opportunities implies the existence of stochastic dominance, while the reverse does not hold.

Abstract:
Using a special metric in the space of sequences, we give a geometric description of almost periodic sets in the $k$-dimensional Euclidean space. We prove the completeness of the space of almost periodic sets and some analogue of the Bochner criterion of almost periodicity. Also, we show the connection between these sets and almost periodic measures.

Abstract:
The economic environment for financial institutions has become increasingly risky. Hence these institutions must find ways to manage risk of which one of the most important forms is credit risk. In this paper we use the mean-variance (mean-standard deviation) approach to examine a banking firm investing in risky assets and hedging opportunities. The mean-standard deviation framework can be used because our hedging model satisfies a scale and location condition. The focus of this study is on how credit risk affects optimal bank investment in the loan and deposit market when derivatives are available. Furthermore we explore the relationship among the first- and second-degree stochastic dominance efficient sets and the mean-variance efficient set.

Abstract:
Stochastic dominance is a preference relation of uncertain prospect defined over a class of utility functions. While this utility class represents basic properties of risk aversion, it includes some extreme utility functions rarely characterizing a rational decision maker's preference. In this paper we introduce reference-based almost stochastic dominance (RSD) rules which well balance the general representation of risk aversion and the individualization of the decision maker's risk preference. The key idea is that, in the general utility class, we construct a neighborhood of the decision maker's individual utility function, and represent a preference relation over this neighborhood. The RSD rules reveal the maximum dominance level quantifying the decision maker's robust preference between alternative choices. We also propose RSD constrained stochastic optimization model and develop an approximation algorithm based on Bernstein polynomials. This model is illustrated on a portfolio optimization problem.

Abstract:
We show that a set is almost periodic if and only if the associated exponential sum is concentrated in the minor arcs. Hence binary additive problems involving almost periodic sets can be solved using the circle method.

Abstract:
We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak$^{*}$ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and $L$-weakly compact sets coincide. In particular, in terms of almost Dunford-Pettis operators into $c_{0}$, we give an operator characterization of those $\sigma$-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a $\sigma$-Dedekind Banach lattice $E$, every relatively weakly compact set in $E$ is almost limited if and only if every continuous linear operator $T:E\rightarrow c_{0}$ is an almost Dunford-Pettis operator.

Abstract:
A short proof of a conjecture of Kropholler is given. This gives a relative version of Stallings' Theorem on the structure of groups with more than one end. A generalisation of the Almost Stability Theorem is also obtained, that gives information about the structure of the Sageev cubing.

Abstract:
Let $G$ be an additive group of order $v$. A $k$-element subset $D$ of $G$ is called a $(v, k, \lambda, t)$-almost difference set if the expressions $gh^{-1}$, for $g$ and $h$ in $D$, represent $t$ of the non-identity elements in $G$ exactly $\lambda$ times and every other non-identity element $\lambda+1$ times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. This paper reviews the recent work that has been done on almost difference sets and related topics. In this survey, we try to communicate the known existence and nonexistence results concerning almost difference sets. Further, we establish the link between certain almost difference sets and binary sequences with three-level autocorrelation. Lastly, we provide a thorough treatment of the tools currently being used to solve this problem. In particular, we review many of the construction methods being used to date, providing illustrative proofs and many examples.

Abstract:
In the first part, we construct a cut and project scheme from a family $\{P_\varepsilon\}$ of sets verifying four conditions. We use this construction to characterize weighted Dirac combs defined by cut and project schemes and by continuous functions on the internal groups in terms of almost periodicity. We are also able to characterise those weighted Dirac combs for which the internal function is compactly supported. Lastly, using the same cut and project construction for $\varepsilon$-dual sets, we are able to characterise Meyer sets in $\sigma$-compact locally compact Abelian groups.

Abstract:
We introduce a notion of almost antiproximinality of sets in the space L_1 which is a weakening of the notion of antiproximinality. Also we investigate properties of almost antiproximinal sets and establish a method of construction of almost antiproximinal sets.