Combinatorial Line Research Articles

Multi-objective collaborative assembly line design problem with the optimisation of ergonomics and economics

Manufacturing systems are socio-technical systems, with explicit interactions between humans and technologies in shared workspaces. These shared workspaces could also be called hybrid collaborative manufacturing systems, which involve workers as well as technological equipment and combine the benefits of human workers and new Industry 4.0 technologies, such systems are particularly useful in a context requiring flexibility and adaptability. Furthermore, the new Industry 5.0 approach has the objective to shift toward more human-centric and resilient manufacturing systems. The key problems to solve in the design of collaborative manufacturing systems are the combinatorial assembly line balancing problem and the equipment selection problem. An efficient and sustainable line requires a cost-effective choice of equipment while improving the ergonomics and the safety of workers. Both decisions of balancing workload and the assignment of equipment impact the ergonomics of a collaborative system and present conflicting criteria. To this end, we propose a multi-objective approach, the objectives are the optimisation of the investment costs and the ergonomics with a fatigue and recovery criterion. We propose to linearise the fatigue and recovery to formulate a new Mixed Integer Linear Programming formulation. We developed an exact multi-objective solving algorithm based on the ϵ-constraint to obtain the trade-off between these objectives. We conducted numerical experiments with different instances from the literature with promising results for instances with up to 45 operations. Finally, we discuss insightful managerial conclusions and future research perspectives.

Another Note on Intervals in the Hales–Jewett Theorem

The Hales–Jewett Theorem states that any $r$–colouring of $[m]^n$ contains a monochromatic combinatorial line if $n$ is large enough. Shelah's proof of the theorem implies that for $m = 3$ there always exists a monochromatic combinatorial line whose set of active coordinates is the union of at most $r$ intervals. For odd $r$, Conlon and Kamčev constructed $r$–colourings for which it cannot be fewer than $r$ intervals. However, we show that for even $r$ and large $n$, any $r$–colouring of $[3]^n$ contains a monochromatic combinatorial line whose set of active coordinates is the union of at most $r-1$ intervals. This is optimal and extends a result of Leader and Räty for $r=2$.

Optimal arrangements of classical and quantum states with limited purity

We consider sets of trace-normalized non-negative operators in Hilbert-Schmidt balls that maximize their mutual Hilbert-Schmidt distance; these are optimal arrangements in the sets of purity-limited classical or quantum states on a finite-dimensional Hilbert space. Classical states are understood to be represented by diagonal matrices, with the diagonal entries forming a probability vector. We also introduce the concept of spectrahedron arrangements which provides a unified framework for classical and quantum arrangements and the flexibility to define new types of optimal packings. Continuing a prior work, we combine combinatorial structures and line packings associated with frames to arrive at optimal arrangements of higher-rank quantum states. One new construction that is presented involves generating an optimal arrangement we call a Gabor-Steiner equiangular tight frame as the orbit of a projective representation of the Weyl-Heisenberg group over any finite abelian group. The minimal sets of linearly dependent vectors, the so-called binder, of the Gabor-Steiner equiangular tight frames are then characterized; under certain conditions these form combinatorial block designs and in one case generate a new class of block designs. The projections onto the span of minimal linearly dependent sets in the Gabor-Steiner equiangular tight frame are then used to generate further optimal spectrahedron arrangements.

Intersectional Strategies for Targeting Amacrine and Ganglion Cell Types in the Mouse Retina.

The mammalian retina harbors over 100 different cell types. To understand how retinal circuits work, it is essential to systematically access each type. A widely used approach for achieving targeted transgene expression exploits promoter-driven Cre lines. However, Cre expression in a given transgenic line in the retina and elsewhere in the brain is rarely confined to a single cell type, contributing ambiguity to the interpretation of results from broadly applied manipulations. To obtain unambiguous information about retinal processing, it is desirable to have strategies for further restricting transgene expression to a few or even to a single cell type. We employed an intersectional strategy based on a Cre/Flp double recombinase system to target amacrine and ganglion cell types in the inner retina. We analyzed expression patterns in seven Flp drivers and then created combinational mouse lines by selective cross breeding with Cre drivers. Breeding with Flp drivers can routinely remove labeling from more than 90% of the cells in Cre drivers, leading to only a handful cell types, typically 2–3, remaining in the intersection. Cre/Flp combinatorial mouse lines enabled us to identify and anatomically characterize retinal cell types with greater ease and demonstrated the feasibility of intersectional strategies in retinal research. In addition to the retina, we examined Flp expression in the lateral geniculate nucleus and superior colliculus. Our results establish a foundation for future application of intersectional strategies in the retina and retino-recipient regions.

A note on multiparty communication complexity and the Hales–Jewett theorem

For integers n and k, the density Hales–Jewett numbercn,k is defined as the maximal size of a subset of [k]n that contains no combinatorial line. We show that for k≥3 the density Hales–Jewett number cn,k is equal to the maximal size of a cylinder intersection in the problem Partn,k of testing whether k subsets of [n] form a partition. It follows that the communication complexity, in the Number On the Forehead (NOF) model, of Partn,k, is equal to the minimal size of a partition of [k]n into subsets that do not contain a combinatorial line. Thus, the bound in [7] on Partn,k using the Hales–Jewett theorem is in fact tight, and the density Hales–Jewett number can be thought of as a quantity in communication complexity. This gives a new angle to this well studied quantity.As a simple application we prove a lower bound on cn,k, similar to the lower bound in [19] which is roughly cn,k/kn≥exp⁡(−O(log⁡n)1/⌈log2⁡k⌉). This lower bound follows from a protocol for Partn,k. It is interesting to better understand the communication complexity of Partn,k as this will also lead to the better understanding of the Hales–Jewett number. The main purpose of this note is to motivate this study.

Theta function solutions of the quantum Knizhnik–Zamolodchikov–Bernard equation for a face model

We consider the quantum Knizhnik–Zamolodchikov–Bernard equation for a face model with elliptic weights, the SOS model. We provide explicit solutions as theta functions. On the so-called combinatorial line, in which the model is equivalent to the three-colour model, these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions.

An Upper Bound for the Hales--Jewett Number $\mathrm{HJ}(4,2)$

We show that for $n$ at least $10^{11}$, any 2-coloring of the $n$-dimensional grid $[4]^n$ contains a monochromatic combinatorial line. This is a special case of the Hales--Jewett theorem [Hales and Jewett, Trans. Amer. Math., 106 (1963), pp. 222--229], to which the best known general upper bound is due to Shelah [J. Amer. Math. Soc., 1 (1988), pp. 683--697]; Shelah's recursion gives an upper bound between $2 \uparrow \uparrow 7$ and $2 \uparrow \uparrow 8$ for the case we consider, and no better value was previously known.

On topological and geometric [formula omitted] configurations

An (nk)configuration is a set of n points and n lines such that each point lies on k lines while each line contains k points. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of (nk)configurations for a given k has been subject to active research. A current front of research concerns geometric (n4)configurations: it is now known that geometric (n4)configurations exist for all n≥18, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of (194)configurations: we obtain all topological (194)configurations among which none are geometrically realizable.

Optogenetic sensors and effectors: CHROMus-the Cornell Heart Lung Blood Institute Resource for Optogenetic Mouse Signaling.

Significant progress has been made in the last decade in the development of optogenetic effectors and sensors that can be deployed to understand complex biological signaling in mammals at a molecular level, without disrupting the distributed, lineage specific signaling circuits that comprise nuanced physiological responses. A major barrier to the widespread exploitation of these imaging tools, however, is the lack of readily available genetic reagents that can be easily combined to probe complex biological processes. Ideally, one could envision purpose–produced mouse lines expressing optically compatible sensors and effectors, sensor pairs in distinct lineages, or sensor pairs in discrete subcellular compartments, such that they could be crossed to enable in vivo imaging studies of unprecedented scientific power. Such lines could also be combined with mice to determine the alteration in signaling accompanying targeted gene deletion or addition. In order to address this lack, the National Heart Lung and Blood Institute has recently funded an optogenetic resource designed to create optically compatible, combinatorial mouse lines that will advance NHLBI research. Here we review recent advances in optogenetic sensor and effectors and describe the rationale and goals for the establishment of the Cornell/National Heart Lung Blood Resource for Optogenetic Mouse Signaling (CHROMus).

A new proof of the density Hales-Jewett theorem

The Hales-Jewett theorem asserts that for every r and every k there exists n such that every r-colouring of the n-dimensional grid {1, . . . , k} contains a monochromatic combinatorial line. This result is a generalization of van der Waerden’s theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van der Waerden has a famous density version, conjectured by Erdős and Turan in 1936, proved by Szemeredi in 1975, and given a different proof by Furstenberg in 1977. The Hales-Jewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemeredi’s theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson and the first to provide a quantitative bound on how large n needs to be. In particular, we show that a subset of {1, 2, 3} of density δ contains a combinatorial line if n is at least as big as a tower of 2s of height O(1/δ). Our proof is surprisingly simple: indeed, it gives arguably the simplest known proof of Szemeredi’s theorem.

A variant of the density Hales–Jewett theorem

In a recent paper “A variant of the Hales-Jewett theorem”, M. Beiglbock provides a version of the classic coloring result in which an instance of the variable in a word giving rise to a monochromatic combinatorial line can be moved around in a finite structure of specified type (for example, an arithmetic progression). We prove a density version of this result in which all instances of the variable may move. As an application, we obtain an elementary proof of a result of V. Bergelson stating that multiplicatively large subsets of N contain arbitrarily long geoarithmetic progressions.

Reading “A variant of the Hales–Jewett theorem” on its anniversary

In a recent paper “A variant of the Hales–Jewett theorem”, M. Beiglböck provides a version of the classic coloring result in which an instance of the variable in a word giving rise to a monochromatic combinatorial line can be moved around in a finite structure of specified type (for example, an arithmetic progression). We give an elementary proof and infinitary extensions.

Analysis of Combinatorial Loss-of-Function Mutants in the Arabidopsis Ethylene Receptors Reveals That the ers1 etr1 Double Mutant Has Severe Developmental Defects That Are EIN2 Dependent

Ethylene responses in Arabidopsis are controlled by the ETR receptor family. The receptors function as negative regulators of downstream signal transduction components and fall into two distinct subfamilies based on sequence similarity. To clarify the levels of functional redundancy between receptor isoforms, combinatorial mutant lines were generated that included the newly isolated ers1-2 allele. Based on the etiolated seedling growth response, all mutant combinations tested exhibited some constitutive ethylene responsiveness but also remained responsive to exogenous ethylene, indicating that all five receptor isoforms can contribute to signaling and no one receptor subtype is essential. On the other hand, light-grown seedlings and adult ers1 etr1 double mutants exhibited severe phenotypes such as miniature rosette size, delayed flowering, and sterility, revealing a distinct role for subfamily I receptors in light-grown plants. Introduction of an ein2 loss-of-function mutation into the ers1 etr1 double mutant line resulted in plants that phenocopy ein2 single mutants, indicating that all phenotypes observed in the ers1 etr1 double mutant are EIN2 dependent.

Sensitive X-ray fluorescence analysis of geological samples using synchrotron radiation

Application of synchrotron radiation for multielement X-ray fluorescence analysis of geological samples resulted in a reduction of the detection limits for most chemical elements of up to 1–2×10 −5%. Sample scanning relative to the SR beam during measuring enables analysis of rather representative sample weights and improves precision of the determinations of platinoids by several times. An absorbing filter between the sample and detector allows the influence of combinatorial lines on the results of analysis to be eliminated and the detection limits of Ru, Rh, and Pd to be reduced by a factor of 2–3. Taking the analysis of standard samples for PD and Ag as exsamples, the authors pointed to the possibility of quantitative determination of (2×10 −5)% sensitivity and relative precision of 5–10%. Preliminary chemical concentrating resulted in a detection limit of Pd and Rd equal to (1×10 −6)%.

Peculiarities of direct x-ray determination of Ru, Rh and using VEPP-4 synchrotron radiation

The first results are presented of an analysis of light platinoids (Ru, Rh, Pd) in silicate rocks, ores and references by the SRXFA technique. To improve the reproducibility of the analysis results, the sample has been scanned with a SR beam and the data obtained averaged over a large area. The influence of combinatorial lines in the SRXFA spectra, caused by the pulse nature of the radiation, has been observed. The detection limits for Ru, Rh and Pd were 0.017–0.07 ppm at a 10 3 s in the absence of conflicting spectral lines.

Behrend's Theorem for Dense Subsets of Finite Vector Spaces

The “combinatorial line conjecture” states that for all q ≧ 2 and there exists such that if , X is a q-element set, and A is any subset of X n (= cartesian product of n copies of X) with more than elements (that is, A has density greater than ), then A contains a combinatorial line. (For a definition of combinatorial line, together with statements and proofs of many results related to the combinatorial line conjecture, including all those results mentioned below, see [5]. Since we are not directly concerned with combinatorial lines in this paper, we do not reproduce the definition here.)This conjecture (which is a strengthened version of a conjecture of Moser [7]), if true, would bear the same relation to the Hales-Jewett theorem that Szemerédi's theorem bears to van der Waerden's theorem.