Zarankiewicz’s problem for semilinear hypergraphs

A bipartite graph $H = \left (V_1, V_2; E \right )$ with $\lvert V_1\rvert + \lvert V_2\rvert = n$ is semilinear if $V_i \subseteq \mathbb {R}^{d_i}$ for some $d_i$ and the edge relation E consists of the pairs of points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination of s linear equalities and inequalities in $d_1 + d_2$ variables for some s. We show that for a fixed k, the number of edges in a $K_{k,k}$ -free semilinear H is almost linear in n, namely $\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$ for any $\varepsilon> 0$ ; and more generally, $\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$ for a $K_{k, \dotsc ,k}$ -free semilinear r-partite r-uniform hypergraph. As an application, we obtain the following incidence bound: given $n_1$ points and $n_2$ open boxes with axis-parallel sides in $\mathbb {R}^d$ such that their incidence graph is $K_{k,k}$ -free, there can be at most $O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$ incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in o-minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner).

Formulas and Properties for Families of Theories of Abelian Groups

First-order formulas reflect an information for semantic and syntactic properties. Links between formulas and properties define their existential and universal interrelations which produce both structural and topological possibilities for characteristics classifying families of semantic and syntactic objects. We adapt general approaches describing links between formulas and properties for families of Abelian groups and their theories defining possibilities for characteristics of formulas and properties including rank values. This adaptation is based on formulas reducing each formula to an appropriate Boolean combination of given ones defining Szmielew invariants for theories of Abelian groups. Using this basedness we describe a trichotomy of possibilities for the rank values of sentences defining neighbourhoods for the set of theories of Abelian groups: the rank can be equal −1, 0, or ∞. Thus the neighbourhoods are either finite or contain continuum many theories. Using the trichotomy we show that each sentence defining a neighbourhood either belongs to finitely many theories or it is generic. We introduce the notion of rich property and generalize the main results for these properties.

Revisiting separation: algorithms and complexity

Linear temporal logic (LTL) with Since and Until modalities is expressively equivalent, over the class of complete linear orders, to a fragment of first-order logic known as FOMLO (first-order monadic logic of order). It turns out that LTL, under some basic assumptions, is expressively complete if and only if it has the property, called separation, that every formula is equivalent to a Boolean combination of formulas that each refer only to the past, present or future. Herein we present simple algorithms and their implementations to perform separation of the LTL with Since and Until, over discrete and complete linear orders, and translation from FOMLO formulas into equivalent temporal logic formulas. We additionally show that the separation of a certain fragment of LTL results in at most a double exponential size growth.

Effectively Monadic Predicates

Monadic predicates play a prominent role in many decidable cases, including decision procedures for symbolic automata. We are here interested in discovering whether a formula can be rewritten into a Boolean combination of monadic predicates. Our setting is quantifier-free formulas over a decidable background theory, such as arithmetic and we here develop a semi-decision procedure for extracting a monadic decomposition of a formula when it exists.

Passive prosthetic hands and tools: A literature review

Background: The group of passive prostheses consists of prosthetic hands and prosthetic tools. These can either be static or adjustable. Limited research and development on passive prostheses has been performed although many people use these prosthesis types. Although some publications describe passive prostheses, no recent review of the peer-reviewed literature on passive prostheses is available. Objective: Review the peer-reviewed literature on passive prostheses for replacement of the hand. Study design: Literature review. Methods: Four electronic databases were searched using a Boolean combination of relevant keywords. English-language articles relevant to the objective were selected. Results: In all, 38 papers were included in the review. Publications on passive prosthetic hands describe their users, usage, functionality, and problems in activities of daily living. Publications on prosthetic tools mostly focus on sport, recreation, and vehicle driving. Conclusion: Passive hand prostheses receive little attention in prosthetic research and literature. Yet one out of three people with a limb deficiency uses this type of prosthesis. Literature indicates that passive prostheses can be improved on pulling and grasping functions. In the literature, ambiguous names are used for different types of passive prostheses. This causes confusion. We present a new and clear classification of passive prostheses. Clinical relevance This review provides information on the users of passive prosthetic hands and tools, their usage and the functionality. Passive prostheses receive very little attention and low appreciation in literature. Passive prosthetic hands and tools show to be useful to many unilateral amputees and should receive more attention and higher acceptance.