arXiv subjects

Achim Jung

Publications and source records attributed to Achim Jung.

9 recordsLinked to original sources

arXiv2025

On the symmetry behind duality

Dualities such as Stone duality and the duality between sober spaces and spatial frames hinge on an interaction between open sets and compact saturated sets. In several important classes of spaces-Stone spaces, spectral spaces, and stably compact spaces-this interaction forms a perfect symmetry, reflected dually as order self-duality. But the class of sober spaces, despite being central to Stone-like dualities, exhibits only a partial symmetry between openness and compactness. This raises a central question: can we enlarge the setting enough to recover a perfect symmetry, while still retaining sober spaces and preserving the conditions that make the sober-spatial-frame duality work? We answer this question affirmatively. We introduce ko-spaces, whose families of open and compact saturated sets satisfy the compatibility needed for duality, and bi-dcpos, a pointfree companion generalizing both spatial frames and continuous domains. We prove that the categories of ko-spaces and distributive bi-dcpos are equivalent (and dually equivalent, too), and that each category carries a symmetry in the form of a self-duality. On spaces, this extends de Groot duality; on domains, it extends Lawson duality. Classical results fall out as special cases: the sober-spatial-frame duality reappears inside our symmetric framework, and continuous domains acquire a presentation akin to that of d-frames. Our work suggests that an appropriate home for Stone-like duality is a fully symmetric two-sorted world in which openness and compactness play on equal footing.

math.LO↗

arXiv2024

Continuous Domains for Function Spaces Using Spectral Compactification

We introduce a continuous domain for function spaces over topological spaces which are not core-compact. Notable examples of such topological spaces include the real line with the upper limit topology, which is used in solution of initial value problems with temporal discretization, and various infinite dimensional Banach spaces which are ubiquitous in functional analysis and solution of partial differential equations. If a topological space $\mathbb{X}$ is not core-compact and $\mathbb{D}$ is a non-singleton bounded-complete domain, the function space $[\mathbb{X} \to \mathbb{D}]$ is not a continuous domain. To construct a continuous domain, we consider a spectral compactification $\mathbb{Y}$ of $\mathbb{X}$ and relate $[\mathbb{X} \to \mathbb{D}]$ with the continuous domain $[\mathbb{Y} \to \mathbb{D}]$ via a Galois connection. This allows us to perform computations in the native structure $[\mathbb{X} \to \mathbb{D}]$ while computable analysis is performed in the continuous domain $[\mathbb{Y} \to \mathbb{D}]$, with the left and right adjoints used for moving between the two function spaces.

cs.LO↗

arXiv2021

Stone Duality for Relations

We show how Stone duality can be extended from maps to relations. This is achieved by working order enriched and defining a relation from A to B as both an order-preserving function from the opposite of A times B to the 2-element chain and as a subobject of A times B. We show that dual adjunctions and equivalences between regular categories, taken in a suitably order enriched sense, extend to (framed bi)categories of relations.

cs.LO↗

arXiv2018

The Ho-Zhao Problem

Given a poset $P$, the set, $Γ(P)$, of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory $\mathbf{C}$ of $\mathbf{Pos}_d$ (the category of posets and Scott-continuous maps) is said to be $Γ$-faithful if for any posets $P$ and $Q$ in $\mathbf{C}$, $Γ(P) \cong Γ(Q)$ implies $P \cong Q$. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are $Γ$-faithful, while $\mathbf{Pos}_d$ is not. Ho & Zhao (2009) asked whether the category $\mathbf{DCPO}$ of dcpos is $Γ$-faithful. In this paper, we answer this question in the negative by exhibiting a counterexample. To achieve this, we introduce a new subcategory of dcpos which is $Γ$-faithful. This subcategory subsumes all currently known $Γ$-faithful subcategories. With this new concept in mind, we construct the desired counterexample which relies heavily on Johnstone's famous dcpo which is not sober in its Scott topology.

cs.LO↗

arXiv2017

Free constructions and coproducts of d-frames

A general theory of presentations for d-frames does not yet exist. We review the difficulties and give sufficient conditions for when they can be overcome. As an application we prove that the category of d-frames is closed under coproducts.

cs.LO↗

arXiv2017

Diagrammatic Semantics for Digital Circuits

We introduce a general diagrammatic theory of digital circuits, based on connections between monoidal categories and graph rewriting. The main achievement of the paper is conceptual, filling a foundational gap in reasoning syntactically and symbolically about a large class of digital circuits (discrete values, discrete delays, feedback). This complements the dominant approach to circuit modelling, which relies on simulation. The main advantage of our symbolic approach is the enabling of automated reasoning about abstract circuits, with a potentially interesting new application to partial evaluation of digital circuits. Relative to the recent interest and activity in categorical and diagrammatic methods, our work makes several new contributions. The most important is establishing that categories of digital circuits are Cartesian and admit, in the presence of feedback expressive iteration axioms. The second is producing a general yet simple graph-rewrite framework for reasoning about such categories in which the rewrite rules are computationally efficient, opening the way for practical applications.

cs.PL↗

arXiv2016

Join-continuity + Hypercontinuity = Prime continuity

A remarkable result due to Kou, Liu & Luo states that the condition of continuity for a dcpo can be split into quasi-continuity and meet-continuity. Their argument contained a gap, however, which is probably why the authors of the monograph Continuous Lattices and Domains used a different (and fairly sophisticated) sequence of lemmas in order to establish the result. In this note we show that by considering the Stone dual, that is, the lattice of Scott-open subsets, a straightforward proof may be given. We do this by showing that a complete lattice is prime-continuous if and only if it is join-continuous and hypercontinuous. A pleasant side effect of this approach is that the characterisation of continuity by Kou, Liu & Luo also holds for posets, not just dcpos.

cs.LO↗

arXiv2011

Stimulation of human red blood cells leads to Ca2+-mediated intercellular adhesion

Red blood cells (RBCs) are a major component of blood clots, which form physiologically as a response to injury or pathologically in thrombosis. The active participation of RBCs in thrombus solidification has been previously proposed but not yet experimentally proven. Holographic optical tweezers and single-cell force spectroscopy were used to study potential cell-cell adhesion between RBCs. Irreversible intercellular adhesion of RBCs could be induced by stimulation with lysophosphatidic acid (LPA), a compound known to be released by activated platelets. We identified Ca2+ as an essential player in the signaling cascade by directly inducing Ca2+ influx using A23187. Elevation of the internal Ca2+ concentration leads to an intercellular adhesion of RBCs similar to that induced by LPA stimulation. Using single-cell force spectroscopy, the adhesion of the RBCs was identified to be approximately 100 pN, a value large enough to be of significance inside a blood clot or in pathological situations like the vasco-occlusive crisis in sickle cell disease patients.

physics.bio-ph↗

arXiv2003

Ferrofluids as thermal ratchets

Colloidal suspensions of ferromagnetic nano-particles, so-called ferrofluids, are shown to be suitable systems to demonstrate and investigate thermal ratchet behavior: By rectifying thermal fluctuations, angular momentum is transferred to a resting ferrofluid from an oscillating magnetic field without net rotating component. Via viscous coupling the noise driven rotation of the microscopic ferromagnetic grains is transmitted to the carrier liquid to yield a macroscopic torque. For a simple setup we analyze the rotation of the ferrofluid theoretically and show that the results are compatible with the outcome of a simple demonstration experiment.

cond-mat.mtrl-sci↗