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Given an acyclic map $X\to Y$ of closed manifolds dimension $d$, we study the relationship between the embeddings of $Y$ in $S^{n}$ with those of $X$ in $S^{n}$ when $n-d \ge 3$. The approach taken here is to first solve the Poincar\'e duality space variant of the problem. We then apply the surgery machine to obtain the corresponding results for manifolds. Thereafter, we focus on the case when $X$ is a smooth homology sphere and deduce results about the homotopy type of the space of block embeddings of $X$ in a sphere.

For an $n$-qubit system, a rational function on the space of mixed states which is invariant with respect to the action of the group of local symmetries may be viewed as a detailed measure of entanglement. We show that the field of all such invariant rational functions is purely transcendental over the complex numbers and has transcendence degree $4^n - 2n-1$. An explicit transcendence basis is also exhibited., Comment: greatly simplified the treatment of quotients

We introduce the notion of an X-state on $n$-qubits. After taking the Zariski closure of the set of X-states in the space of all mixed states, we obtain a complex algebraic variety $\scr X$ that is equipped with the action of the Lie group of local symmetries $G$. We show that the field of $G$-invariant rational functions on $\scr X$ is purely transcendental over the complex numbers of degree $2^{2n-1}-n-1$.

We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality statements for the local unitary invariants of symmetric mixed states of two qubits. Our results apply to both complex-valued and real-valued invariants.

We compute the field of rational local unitary invariants for locally maximally mixed states and symmetrically mixed states of two qubits. In both cases, we prove that the field of rational invariants is purely transcendental. We also construct explicit geometric quotients and prove that they are always rational. All the results are obtained by working over the field of real numbers, employing methods from classical and geometric invariant theory over arbitrary fields of characteristic zero.

Software is a critical aspect of large-scale science, providing essential capabilities for making scientific discoveries. Large-scale scientific projects are vast in scope, with lifespans measured in decades and costs exceeding hundreds of millions of dollars. Successfully designing software that can exist for that span of time, at that scale, is challenging for even the most capable software companies. Yet scientific endeavors face challenges with funding, staffing, and operate in complex, poorly understood software settings. In this paper we discuss the practice of early-phase software architecture in the Square Kilometre Array Observatory's Science Data Processor. The Science Data Processor is a critical software component in this next-generation radio astronomy instrument. We customized an existing set of processes for software architecture analysis and design to this project's unique circumstances. We report on the series of comprehensive software architecture plans that were the result. The plans were used to obtain construction approval in a critical design review with outside stakeholders. We conclude with implications for other long-lived software architectures in the scientific domain, including potential risks and mitigations., Comment: published at Journal of Systems and Software as In Practice article. Data package at doi:10.5281/zenodo.7868987

In this short paper, we give two proofs that the Euler characteristic is multiplicative, for fiber sequences of finitely dominated spaces. This is equivalent to proving that the Becker-Gottlieb transfer is functorial on $\pi_0$., Comment: Accepted version. 13 pages plus references

The object of this paper is to show that non-homotopy finite Poincar\'e duality spaces are plentiful. Let $\pi$ be finitely presented group. Assuming that the reduced Grothendieck group $\tilde K_0(\Bbb Z[\pi])$ has a non-trivial 2-divisible element, we construct a finitely dominated Poincar\'e space $X$ with fundamental group $\pi$ such that $X$ is not homotopy finite. The dimension of $X$ can be made arbitrarily large. Our proof relies on a result which says that every finitely dominated space possesses a stable Poincar\'e duality thickening., Comment: Final version. To appear in Homology, Homotopy and Applications

For a finitely dominated Poincar\'e duality space $M$, we show how the author's total obstruction to the existence of a Poincar\'e embedding of the diagonal map $M \to M \times M$ relates to the Reidemeister trace of the identity map of $M$. We also show that if the dimension of $M$ is even and at least four, and if $\pi_1(M)$ is a finite direct product of cyclic groups of order two, then the diagonal map admits a Poincar\'e embedding., Comment: Final prepublication version. This paper is to appear in, "Manifolds and K-Theory: in honour of Andrew Ranicki of the journal Proceedings of the Royal Society of Edinburgh Section A: Mathematics."

Background: Accurate segmentation of microscopic structures such as bio-artificial capsules in microscopy imaging is a prerequisite to the computer-aided understanding of important biomechanical phenomenons. State-of-the-art segmentation performances are achieved by deep neural networks and related data-driven approaches. Training these networks from only a few annotated examples is challenging while producing manually annotated images that provide supervision is tedious. Method: Recently, self-supervision, i.e. designing a neural pipeline providing synthetic or indirect supervision, has proved to significantly increase generalization performances of models trained on few shots. The objective of this paper is to introduce one such neural pipeline in the context of micro-capsule image segmentation. Our method leverages the rather simple content of these images so that a trainee network can be mentored by a referee network which has been previously trained on synthetically generated pairs of corrupted/correct region masks. Results: Challenging experimental setups are investigated. They involve from only 3 to 10 annotated images along with moderately large amounts of unannotated images. In a bio-artificial capsule dataset, our approach consistently and drastically improves accuracy. We also show that the learnt referee network is transferable to another Glioblastoma cell dataset and that it can be efficiently coupled with data augmentation strategies. Conclusions: Experimental results show that very significant accuracy increments are obtained by the proposed pipeline, leading to the conclusion that the self-supervision mechanism introduced in this paper has the potential to replace human annotations.

The particle filter (PF) is a powerful inference tool widely used to estimate the filtering distribution in non-linear and/or non-Gaussian problems. To overcome the curse of dimensionality of PF, the block PF (BPF) inserts a blocking step to partition the state space into several subspaces or blocks of smaller dimension so that the correction and resampling steps can be performed independently on each subspace. Using blocks of small size reduces the variance of the filtering distribution estimate, but in turn the correlation between blocks is broken and a bias is introduced. When the dependence relationships between state variables are unknown, it is not obvious to decide how to split the state space into blocks and a significant error overhead may arise from a poor choice of partitioning. In this paper, we formulate the partitioning problem in the BPF as a clustering problem and we propose a state space partitioning method based on spectral clustering (SC). We design a generalized BPF algorithm that contains two new steps: (i) estimation of the state vector correlation matrix from predicted particles, (ii) SC using this estimate as the similarity matrix to determine an appropriate partition. In addition, a constraint is imposed on the maximal cluster size to prevent SC from providing too large blocks. We show that the proposed method can bring together in the same blocks the most correlated state variables while successfully escaping the curse of dimensionality.

In practice, data augmentation is assigned a predefined budget in terms of newly created samples per epoch. When using several types of data augmentation, the budget is usually uniformly distributed over the set of augmentations but one can wonder if this budget should not be allocated to each type in a more efficient way. This paper leverages online learning to allocate on the fly this budget as part of neural network training. This meta-algorithm can be run at almost no extra cost as it exploits gradient based signals to determine which type of data augmentation should be preferred. Experiments suggest that this strategy can save computation time and thus goes in the way of greener machine learning practices.

The aim of this note is to construct a probability measure on the space of trajectories in a continuous time Markov chain having a finite state diagram, or more generally which admits a global bound on its degree and rates. Our approach is elementary. Our main intention is to fill a gap in the literature., Comment: Fixed a zoo of typos

We introduce the notion of a protocol, which consists of a space whose points are labeled by real numbers indexed by the set of cells of a fixed CW complex in prescribed degrees, where the labels are required to vary continuously. If the space is a one-dimensional manifold, then a protocol determines a continuous time Markov chain. When a homological gap condition is present, we associate to each protocol a 'characteristic' cohomology class which we call the hypercurrent. The hypercurrent comes in two flavors: one algebraic topological and the other analytical. For generic protocols we show that the analytical hypercurrent tends to the topological hypercurrent in the low temperature limit. We also exhibit examples of protocols having nontrivial hypercurrent., Comment: This version fixes some references

We generalize to higher algebraic $K$-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion., Comment: 39 pages incl. references. Comments welcome

We investigate a problem in which each member of a group of learners is trained separately to solve the same classification task. Each learner has access to a training dataset (possibly with overlap across learners) but each trained classifier can be evaluated on a validation dataset. We propose a new approach to aggregate the learner predictions in the possibility theory framework. For each classifier prediction, we build a possibility distribution assessing how likely the classifier prediction is correct using frequentist probabilities estimated on the validation set. The possibility distributions are aggregated using an adaptive t-norm that can accommodate dependency and poor accuracy of the classifier predictions. We prove that the proposed approach possesses a number of desirable classifier combination robustness properties.

We establish a number of foundational results on Poincar\'e spaces which result in several applications. One application settles an old conjecture of C.T.C. Wall in the affirmative. Another result shows that for any natural number n, there exists a finite CW pair $(X,Y)$ satisfying relative Poincar\'e duality in dimension n with the property that $Y$ fails to satisfy Poincar\'e duality. We also prove a relative version of a result of Gottlieb about Poincar\'e duality and fibrations., Comment: Final version. To appear in Trans. Amer. Math. Soc

We prove that intersections and unions of independent random sets in finite spaces achieve a form of Lipschitz continuity. More precisely, given the distribution of a random set $\Xi$, the function mapping any random set distribution to the distribution of its intersection (under independence assumption) with $\Xi$ is Lipschitz continuous with unit Lipschitz constant if the space of random set distributions is endowed with a metric defined as the $L_k$ norm distance between inclusion functionals also known as commonalities. Moreover, the function mapping any random set distribution to the distribution of its union (under independence assumption) with $\Xi$ is Lipschitz continuous with unit Lipschitz constant if the space of random set distributions is endowed with a metric defined as the $L_k$ norm distance between hitting functionals also known as plausibilities. Using the epistemic random set interpretation of belief functions, we also discuss the ability of these distances to yield conflict measures. All the proofs in this paper are derived in the framework of Dempster-Shafer belief functions. Let alone the discussion on conflict measures, it is straightforward to transcribe the proofs into the general (non necessarily epistemic) random set terminology.

We examine a network of learners which address the same classification task but must learn from different data sets. The learners cannot share data but instead share their models. Models are shared only one time so as to preserve the network load. We introduce DELCO (standing for Decentralized Ensemble Learning with COpulas), a new approach allowing to aggregate the predictions of the classifiers trained by each learner. The proposed method aggregates the base classifiers using a probabilistic model relying on Gaussian copulas. Experiments on logistic regressor ensembles demonstrate competing accuracy and increased robustness in case of dependent classifiers. A companion python implementation can be downloaded at https://github.com/john-klein/DELCO

Context: The systems comprising a system of systems (SoS) are inde- pendently acquired, operated, and managed. Frequently, the architecture documentation of these existing systems addresses only a stand-alone perspective, and must be augmented to address concerns that arise in the integrated SoS. Objective: We evaluated an architecture documentation viewpoint to address the concerns of a SoS architect about a constituent system, to support SoS design and analysis involving that constituent system. Method: We performed an expert review of documentation produced by applying the viewpoint to a system, using the active review method. Results: The expert panel was able to used a view constructed using the baseline version of the viewpoint to answer questions related to all SoS architect concerns about a constituent system, except for questions concerning the interaction of the constituent system with the platform and network infrastructure. Conclusions: We found that the expert panel was unable to answer certain questions because the baseline version of the viewpoint had a gap in coverage related to relationship of software units of execution (e.g., processes or services) to computers and networks. The viewpoint was revised to add a Deployment Model to address these concerns, and is included in an appendix., Comment: 49 pages

We use algebraic topology to study the stochastic mo- tion of cellular cycles in a finite CW complex. Inspired by statis- tical mechanics, we introduce a homological observable called the average current. The latter measures the average flux of the prob- ability in the process. In the low temperature, adiabatic limit, we prove that the average current fractionally quantizes, in which the denominators are combinatorial invariants of the CW complex., Comment: Added an example showing that the average current in the low temperature adiabatic limit can be non-trivial. Corrected typos

When Requirements Engineering(RE) models are unreasonably complex, they cannot support efficient decision making. SHORT is a tool to simplify that reasoning by exploiting the "key" decisions within RE models. These "keys" have the property that once values are assigned to them, it is very fast to reason over the remaining decisions. Using these "keys", reasoning about RE models can be greatly SHORTened by focusing stakeholder discussion on just these key decisions. This paper evaluates the SHORT tool on eight complex RE models. We find that the number of keys are typically only 12% of all decisions. Since they are so few in number, keys can be used to reason faster about models. For example, using keys, we can optimize over those models (to achieve the most goals at least cost) two to three orders of magnitude faster than standard methods. Better yet, finding those keys is not difficult: SHORT runs in low order polynomial time and terminates in a few minutes for the largest models., Comment: 10 pages, 5 figures, IEEE Requirements Engineering 2017

The probability distributions, as well as the mean values of stochastic currents and fluxes, associated with a driven Langevin process, provide a good and topologically protected measure of how far a stochastic system is driven out of equilibrium. By viewing a Langevin process on a compact oriented manifold of arbitrary dimension m as a theory of a random vector field associated with the environment, we are able to consider stochastic motion of higher-dimensional objects, which allow new observables, called higher-dimensional currents, to be introduced. These higher dimensional currents arise by counting intersections of a k-dimensional trajectory, produced by a evolving (k-1)-dimensional cycle, with a reference cross section, represented by a cycle of complimentary dimension (m - k). We further express the mean fluxes in terms of the solutions of the Supersymmetric Fokker-Planck (SFP), thus generalizing the corresponding well-known expressions for the conventional currents.

Driven Langevin processes have appeared in a variety of fields due to the relevance of natural phenomena having both deterministic and stochastic effects. The stochastic currents and fluxes in these systems provide a convenient set of observables to describe their non-equilibrium steady states. Here we consider stochastic motion of a (k - 1)-dimensional object, which sweeps out a k-dimensional trajectory, and gives rise to a higher k-dimensional current. By employing the low-temperature (low-noise) limit, we reduce the problem to a discrete Markov chain model on a CW complex, a topological construction which generalizes the notion of a graph. This reduction allows the mean fluxes and currents of the process to be expressed in terms of solutions to the discrete Supersymmetric Fokker-Planck (SFP) equation. Taking the adiabatic limit, we show that generic driving leads to rational quantization of the generated higher dimensional current. The latter is achieved by implementing the recently developed tools, coined the higher-dimensional Kirchhoff tree and co-tree theorems. This extends the study of motion of extended objects in the continuous setting performed in the prequel to this manuscript., Comment: To appear in Chemical Physics

We prove that the Becker-Gottlieb transfer is functorial up to homotopy, for all fibrations with finitely dominated fibers. This resolves a lingering foundational question about the transfer, which was originally defined in the late 1970s in order to simplify the proof of the Adams conjecture. Our approach differs from previous attempts in that we closely emulate the geometric argument in the case of a smooth fiber bundle. This leads to a "multiplicative'" description of the transfer, different from the standard presentation as the trace of a diagonal map., Comment: This paper contains an error: the diagram (13) in the proof of Proposition 6.5 does not strictly commute as claimed. See https://doi.org/10.1016/j.aim.2022.108752 for more details. The authors would like to thank Shachar Carmeli and Bastiaan Cnossen for finding this error and bringing it to their attention

We characterize the classical Boltzmann distribution as the unique solution to a certain combinatorial Hodge theory problem in homological degree zero on a finite graph. By substituting for the graph a CW complex and a degree d, we are able to define, by direct analogy, a higher dimensional Boltzmann distribution as a certain (d-1)-cycle on the real cellular chain complex which is characterized by appropriate constraints. We then give an explicit formula for this cycle. We explain how this circle of ideas relates to the authors' Higher Kirchhoff Network Theorem. We also give an improved version of the authors' Higher Matrix-Tree Theorems., Comment: In this revision, we have fixed the definition of the constants a_L appearing in Theorems A and C

This paper introduces an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification. The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data. We provide a lower bound for this number. The bound is shown to be sharp in a limiting case., Comment: Final version; to appear in the Journal of Topology and Analysis

We study the moduli space of A-infinity structures on a topological space as well as the moduli space of A-infinity-ring structures on a fixed module spectrum. In each case we show that the moduli space sits in a homotopy fiber sequence in which the other terms are representing spaces for Hochschild cohomology., Comment: typos corrected