Search

Massive main-sequence stars have convective cores and radiative envelopes, but also sub-surface convection zones caused by partial ionisation. However, the convective properties depend on opacity and a star's metallicity. Non-rotating 1D evolution models of main-sequence stars with the metallicity of the SMC suggest tenuous sub-surface convection zones using the Rayleigh number as a criterion for convection owing to their lower metallicity. We test whether massive stars of different metallicities both inside and outside of asteroseismically calibrated stability windows for sub-surface convection exhibit different properties in stochastic low-frequency (SLF) variability. We extracted customised light curves from the TESS mission for a sample of massive stars using an effective point spread function (ePSF) method, and compared their morphologies in terms of characteristic frequency and amplitude using a Gaussian process (GP) regression methodology. We demonstrate that the properties of SLF variability are generally consistent across the metallicity range from the Milky Way down to the SMC, for stars both inside and outside of the sub-surface stability windows. We conclude that non-rotating 1D stellar structure models cannot alone be used to explain SLF variability in light curves of massive stars. The similar properties of SLF variability across a range of metallicity values, which follow the same trends in mass and age in the HR diagram at both high and low metallicity, support a transition in the dominant mechanism causing SLF variability from younger to more evolved stars. Specifically, core-excited internal gravity waves (IGWs) are favoured for younger stars lacking sub-surface convection zones, especially at low metallicity, and sub-surface convection zones are favoured for more evolved massive stars. (abstract abridged for arXiv), Comment: Accepted and in press, A&A (https://doi.org/10.1051/0004-6361/202451419)

Let $A$ be an algebra with identity and $\Delta:A\to A\otimes A$ a coproduct that admits a counit. If there exist a faithful left integral and a faithful right integral, one can construct an antipode and $(A,\Delta)$ is a Hopf algebra. This is the Larson-Sweedler theorem. There are generalizations of this result for multiplier Hopf algebras, weak Hopf algebras and weak multiplier Hopf algebras. In the case of a multiplier Hopf algebra, the existence of a counit can be weakened and can be replaced by the requirement that the coproduct is full. A similar result is true for weak multiplier Hopf algebras. What we show in this note is that in fact the result for multiplier Hopf algebras can still be obtained without the condition of fullness of the coproduct. As it turns out, this property will already follow from the other conditions. Consequently, also in the original theorem for Hopf algebras, the existence of a counit is a consequence of the other conditions. This slightly generalizes the original result. The situation for weak multiplier Hopf algebras seems to be more subtle. We discuss the problems and see what is still possible here. We consider these results in connection with the development of the theory of locally compact quantum groups. This is discussed in an appendix.

Let $A$ be a non-degenerate algebra over the complex numbers and $\Delta$ a homomorphism from $A$ to the multiplier algebra $M(A\otimes A)$. Consider the linear maps $T_1$ and $T_2$ from $A\otimes A$ to $M(A\otimes A)$ defined by \begin{equation*} T_1(a\otimes b)=\Delta(a)(1\otimes b) \qquad\text{and}\qquad T_2(c\otimes a)=(c\otimes 1)\Delta(a). \end{equation*} The pair $(A,\Delta)$ is a multiplier Hopf algebra if these two maps have range in $A\otimes A$ and are bijections from $A\otimes A$ to itself. In our recent paper on the Larson-Sweedler theorem, single sided multiplier Hopf algebras emerge in a natural way. For this case, instead of requiring the above for the maps $T_1$ and $T_2$, we now have this property for the maps $T_1$ and $T_4$ or for $T_2$ and $T_3$ where \begin{equation*} T_3(a\otimes b)=(1\otimes b)\Delta(a) \qquad\text{and}\qquad T_4(c\otimes a)=\Delta(a)(c\otimes 1). \end{equation*} As it turns out, also for these single sided multiplier Hopf algebras, the existence of a unique counit and antipode can be proven. In fact, rather surprisingly, using the properties of the antipode, one can actually show that for a single sided multiplier Hopf algebra all four canonical maps are bijections from $A\otimes A$ to itself. In other words, $(A,\Delta)$ is automatically a regular multiplier Hopf algebra. We take the advantage of this approach to reconsider some of the known results for a regular multiplier Hopf algebra.

Short videos on platforms such as TikTok, Instagram Reels, and YouTube Shorts (i.e. short-form videos) have become a primary source of information and entertainment. Many short-form videos are inaccessible to blind and low vision (BLV) viewers due to their rapid visual changes, on-screen text, and music or meme-audio overlays. In our formative study, 7 BLV viewers who regularly watched short-form videos reported frequently skipping such inaccessible content. We present ShortScribe, a system that provides hierarchical visual summaries of short-form videos at three levels of detail to support BLV viewers in selecting and understanding short-form videos. ShortScribe allows BLV users to navigate between video descriptions based on their level of interest. To evaluate ShortScribe, we assessed description accuracy and conducted a user study with 10 BLV participants comparing ShortScribe to a baseline interface. When using ShortScribe, participants reported higher comprehension and provided more accurate summaries of video content., Comment: To appear at CHI 2024

A coproduct on a vector space $A$ is defined as a linear map $\Delta:A\to A\otimes A$ satisfying coassociativity $(\Delta\otimes\iota)\Delta=(\iota\otimes\Delta)\Delta$. We use $\iota$ for the identity map. If $G$ is a finite group and if $A$ is the space of all complex functions on $G$, a coproduct on $A$ is defined by $\Delta(f)(p,q)=f(pq)$ where $p,q\in G$. We identify $A\otimes A$ with complex functions on the Cartesian product $G\times G$. Coassociativity follows from the associativity of the product in $G$. Unfortunately, sometimes this notion of a coproduct is not the appropriate one. In this note, we consider the case of an algebra $A$, not necessarily unital but with a non-degenerate product. Now a coproduct is a linear map from $A$ to $M(A\otimes A)$, the multiplier algebra of $A\otimes A$. Unfortunately, it is no longer possible to express coassociativity in its usual form as the maps $\Delta\otimes\iota$ and $\iota\otimes \Delta$, defined on $A\otimes A$, may no longer be defined on the multiplier algebra $M(A\otimes A)$. Similar problems occurs when we want to define a useful notion of a coaction in the case of non-unital algebras. We discuss this in another paper. Not all the results we present in this paper are new. We provide a number of references to the original papers where some of this material is treated. However, in the original papers, results are not always found in an organized form and we hope to improve that here. Further a few solutions to some open questions are included as well as some more peculiar examples. Finally, we discuss some open problems and possible further research.

Understanding the combined effects of climate change and habitat fragmentation on the adaptive potential of plant populations is essential for devising effective conservation strategies. This is particularly important where mating system variation impacts the evolutionary consequences of habitat fragmentation. Here we aimed to reveal how habitat fragmentation and climate adaptation jointly influence the evolutionary trajectories in Primula elatior, a heterostylous self-incompatible and dispersal-constrained forest herb. We quantified the genomic variation and degree of herkogamy, a floral trait reducing self-pollination, across 60 geographically paired populations of Primula elatior across Europe, each pair featuring contrasting levels of habitat fragmentation. Our findings revealed a large and unique set of adaptive outliers in more fragmented landscapes, compared to high-connectivity ones, despite the geographic proximity of the sampling pairs. This suggests elevated selective pressures in fragmented habitats, mirrored by a reduced adaptive potential to cope with climate change. Finally, a minority of genetic variants associated with herkogamy were influenced by current levels of habitat fragmentation and population size, potentially signalling early indicators of evolutionary mating system changes in response to pollinator limitation. Because evolutionary trajectories and adaptive potential are expected to be increasingly affected by habitat fragmentation, our findings underscore the importance of considering both habitat fragmentation and climate adaptation in conservation research and planning., Comment: 26 pages, 5 figures

Consider a locally compact group $G$ and a locally compact space $X$. A local right action of $G$ on $X$ is a continuous map $(x,p)\mapsto x\cdot p$ from an open subset $\Gamma$ of the Cartesian product $X\times G$ to $X$ satisfying certain obvious properties. A global right action of $G$ on $X$ gives rise to a global left action of $G$ on the space $C_c(X)$ of continuous complex functions with compact support in $X$ by the formula $p\,\cdot f:x\mapsto f(x\cdot p)$. In the case of a local action, one still can define $p\,\cdot f$ in $C_c(X)$ by this formula for $f\in C_c(X)$ and $p$ in a neighborhood $V_f$ of the identity in $G$. This yields a local left action of $G$ on $C_c(X)$. Given a local right action of $G$ on $X$, a function $f\in C_c(X)$ is called polynomial if there is a neighborhood $V$ of the identity, contained in $V_f$, and a finite-dimensional subspace $F$ of $C_c(X)$ containing all the functions $v\cdot f$ for $v\in V$. In this paper we study such polynomial functions. If $G$ acts on itself by multiplication, we are also interested in the local actions obtained by restricting it to an open subset of $G$. This is the typical situation that is encountered in our paper on bicrossproducts of groups with a compact open subgroup. In fact, the need for a better understanding of polynomial functions for that case has led us to develop the theory in general here.

This is the last part of a series of three papers on the subject. In the first part we have considered the duality of algebraic quantum groups. In that paper, we use the term algebraic quantum group for a regular multiplier Hopf algebra with integrals. We treat the duality as studied in that theory. In the second part we have considered the duality for multiplier Hopf $^*$-algebras with positive integrals. The main purpose of that paper is to explain how it gives rise to a locally compact quantum group in the sense of Kustermans and Vaes. In a preliminary section of the second part we have mentioned the analytic structure of such a $^*$-algebraic quantum group. We have not gone into the details, except for that part needed to obtain that the scaling constant is trivial and for proving that the composition of a positive left integral with the antipode gives a right integral that is again positive. Also the construction of the square root. of the modular element is discussed and used to obtain the two Haar weights on the associated locally compact quantum group. In this paper we treat the analytic structure of the $^*$-algebraic quantum group in detail. The analytic structure is intimately related with the modular structure. We obtain a collection of interesting formulas, relating the two structures. Further we compare these results with formulas obtained in the framework of general locally compact quantum groups. As for the first two papers in this series, also here no new results are obtained. On the other hand the approach is different, more direct and instructive. It might help the reader for understanding the more general theory of locally compact quantum groups.

In the paper Algebraic quantum groups and duality I, we consider a pairing $(a,b)\mapsto\langle a,b\rangle$ of regular multiplier Hopf algebras $A$ and $B$. When $A$ has integrals and when $B$ is the dual of $A$, we can describe the duality with an element $V$ in the multiplier algebra $M(B\otimes A)$ satisfying and defined by $\langle V, a\otimes b\rangle=\langle a,b\rangle$ for all $a,b$. Properties of the dual pair are formulated in terms of this multiplier $V$. It acts, in a natural way, on $A\otimes A$ as the canonical map $T$, given by $T(a\otimes a')=\Delta(a)(1\otimes a')$. In this second paper on the subject, we assume that the pairing is coming from a multiplier Hopf $^*$-algebra with positive integrals. In this case, the positive right integral on $A$ can be used to construct a Hilbert space $\mathcal H$. The duality $V$ now acts as a unitary operator on the Hilbert space tensor product $\mathcal H\otimes\mathcal H$. This eventually makes it possible to complete the algebraic quantum group to a locally compact quantum group. The procedure to pass from the algebraic quantum group to the operator algebraic completion has been treated in the literature but the construction is rather involved because of the necessary use of left Hilbert algebras. In this paper, we give a comprehensive, yet concise and somewhat simpler approach. It should be considered as a springboard to the more complicated theory of locally compact quantum groups.

Let $(A,\Delta)$ be a finite-dimensional Hopf algebra. The linear dual $B$ of $A$ is again a finite-dimensional Hopf algebra. The duality is given by an element $V\in B\otimes A$, defined by $\langle V,a\otimes b\rangle=\langle a,b\rangle$ where $a\in A$ and $b\in B$. We use $\langle\,\cdot\, , \,\cdot\,\rangle$ for the pairings. In the introduction of this paper, we recall the various properties of this element $V$ as sitting in the algebra $B\otimes A$. More generally, we can consider an algebraic quantum group $(A,\Delta)$. We use the term here for a regular multiplier Hopf algebra with integrals. For $B$ we now take the dual $\widehat A$ of $A$. It is again an algebraic quantum group. In this case, the duality gives rise to an element $V$ in the multiplier algebra $M(B\otimes A)$. Still, most of the properties of $V$ in the finite-dimensional case are true in this more general setting. The focus in this paper lies on various aspects of the duality between $A$ and its dual $\widehat A$. Among other things we include a number of formulas relating the objects associated with an algebraic quantum group and its dual. This note is meant to give a comprehensive, yet concise (and sometimes simpler) account of these known results. This is part I of a series of three papers on this subject. The case of a multiplier Hopf $^*$-algebra with positive integrals is treated in detail in part II and part III.

Climate change and the resulting increased drought frequencies pose considerable threats to forest herb populations, especially when compounded by additional environmental challenges. Specifically, habitat fragmentation may disrupt climate adaptation and cause shifts in mating systems. To examine this, we conducted a garden experiment with Primula elatior offspring from 24 populations across a climate and landscape fragmentation gradient. We evaluated vegetative, regulatory, and reproductive traits under different soil moisture regimes, assessing local adaptation and phenotypic plasticity. We also conducted a field study in 60 populations along the same gradient to examine potential breakdown of reciprocal herkogamy. Our results showed an evolutionary shift from drought avoidance in southern populations to drought tolerance in northern populations for large, connected populations. However, fragmentation disrupted climate clines and adaptive responses to drought in key traits related to growth, biomass allocation and water regulation. Our findings also indicate the beginning of an evolutionary breakdown in reciprocal herkogamy. These disruptions resulted in significantly reduced flowering investment, especially in southern fragmented populations. These findings provide new evidence of how habitat fragmentation disrupts climate adaptation and drought tolerance in Primula elatior, emphasizing the need to account for habitat fragmentation in conservation strategies to preserve resilient forest herb populations amidst global changes.

A finite quantum hypergroup is a finite-dimensional unital algebra $A$ over the field of complex numbers. There is a coproduct on $A$, a coassociative map from $A$ to $A\otimes A$ assumed to be unital, but it is not required to be an algebra homomorphism. There is a counit that is supposed to be a homomorphism. Finally, the main extra requirement is the existence of a faithful left integral with the right properties. For such a finite quantum hypergroup, the dual can be constructed. It is again a finite quantum hypergroup. The more general concept of an algebraic quantum hypergroup is studied in \cite{De-VD1, De-VD2}. If the underlying algebra of an algebraic quantum hypergroup is finite-dimensional, it is a finite quantum hypergroup in the sense of this paper. Here we treat finite quantum hypergroups independently with an emphasis on the development of the notion. It is meant to clarify the various steps taken in \cite{La-VD3a} and \cite{La-VD3b}. In \cite{La-VD3b} we introduce and study a still more general concept of quantum hypergroups. It not only contains the algebraic quantum hypergroups from \cite{De-VD1}, but also some topological cases. We naturally encounter these quantum hypergroups in our work on bicrossproducts, see \cite{La-VD4, La-VD5, La-VD7}. We include some examples to illustrate the theory. One kind is coming from a finite group and a subgroup. The other examples are taken from the bicrossproduct theory.

Algebraic quantum groupoids have been developed by two of the authors (AVD and SHW) of this note in a series of papers. Regular multiplier Hopf algebroids are obtained also by two authors (TT and AVD). Integral theory and duality for those have been studied by one author here (TT). Finally, again two authors of us (TT and AVD) have investigated the relation between weak multiplier Hopf algebras and multiplier Hopf algebroids. In the paper 'Weak multiplier Hopf algebras III. Integrals and duality' (by AVD and SHW), one of the main results is that the dual of an algebraic quantum groupoid, admits a dual of the same type. In the paper 'On duality of algebraic quantum groupoids' (by TT), a result of the same nature is obtained for regular multiplier Hopf algebroids with a single faithful integral. The duality of regular weak multiplier Hopf algebras with a single integral can be obtained from the duality of regular multiplier Hopf algebroids. That is however not the obvious way to obtain this result. It is more difficult and less natural than the direct way. We will discuss this statement further in the paper. Nevertheless, it is interesting to investigate the relation between the two approaches to duality in greater detail. This is what we do in this paper. We build further on the intimate relation between weak multiplier Hopf algebras and multiplier Hopf algebroids. We now add the presence of integrals. That seems to be done best in a framework of dual pairs. It is in fact more general than the duality of these objects coming with integrals.

From a set of technical drawings and expert knowledge, we automatically learn a parser to interpret such a drawing. This enables automatic reasoning and learning on top of a large database of technical drawings. In this work, we develop a similarity based search algorithm to help engineers and designers find or complete designs more easily and flexibly. This is part of an ongoing effort to build an automated engineering assistant. The proposed methods make use of both neural methods to learn to interpret images, and symbolic methods to learn to interpret the structure in the technical drawing and incorporate expert knowledge.

Quantum groups have been studied within several areas of mathematics and mathematical physics. This has led to different approaches, each of them with their own techniques and conventions. Starting with Hopf algebras, where there is a general consensus, moving in the direction of topological quantum groups, where there is no such consensus, it is easy to get lost. Not only many difficulties have to be overcome, but also several choices must be made. The way this is done is often confusing. Some choices even turn out to be rather annoying. As an introductory lecture at the conference on Topological quantum groups and Hopf algebras in 2016, we have explained these choices, difficulties and annoyances, encountered on the road from Hopf algebras to topological quantum groups. In these notes, we discuss more aspects of the development of locally compact quantum groups. We not only explain some of these difficulties in greater detail, but we also give background information about the different steps, combined with some historical comments. We start with finite quantum groups and continue with discrete quantum groups, compact quantum groups and algebraic quantum groups. Multiplicative unitaries are an important side track before we finally arrive at locally compact quantum groups. Along the way, we also formulate some interesting remaining problems in the theory.

This is Part II in our multi-part series of papers developing the theory of a subclass of locally compact quantum groupoids ("quantum groupoids of separable type"), based on the purely algebraic notion of weak multiplier Hopf algebras. The definition was given in Part I. The existence of a certain canonical idempotent element $E$ plays a central role. In this Part II, we develop the main theory, discussing the structure of our quantum groupoids. We will construct from the defining axioms the right/left regular representations and the antipode map., Comment: Some minor revisions made; Bibliography updated

In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we provide motivation and formulate the definition in the C*-algebra framework. Existence of a certain canonical idempotent element is required and it plays a fundamental role, including the establishment of the coassociativity of the comultiplication. This class contains locally compact quantum groups as a subclass.

Let $B$ and $C$ be non-degenerate idempotent algebras and assume that $E$ is a regular separability idempotent in $M(B\otimes C)$. Define $A=C\otimes B$ and $\Delta:A\to M(A\otimes A)$ by $\Delta(c\otimes b)=c\otimes E\otimes b$. The pair $(A,\Delta)$ is a weak multiplier Hopf algebra. Because we assume that $E$ is regular, it is a regular weak multiplier Hopf algebra. There is a faithful left integral on $(A,\Delta)$ that is also right invariant. Therefore, we call $(A,\Delta)$ a unimodular algebraic quantum groupoid. By the general theory, the dual $(\widehat A,\widehat \Delta)$ can be constructed and it is again an algebraic quantum groupoid. In this paper, we treat this algebraic quantum groupoid and its dual in great detail. The main purpose is to illustrate various aspects of the general theory. For this reason, we will also recall the basic notions and results of separability idempotents and weak multiplier Hopf algebras with integrals. The paper is to be considered as an expository note., Comment: arXiv admin note: text overlap with arXiv:1701.04951

Let $(A,\Delta)$ be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra $A$, with or without identity, and a coproduct $\Delta$ on $A$, satisfying certain properties. The main difference with multiplier Hopf algebras is that now, the canonical maps $T_1$ and $T_2$ on $A\otimes A$, defined by $$T_1(a\otimes b)=\Delta(a)(1\otimes b) \qquad\quad\text{and}\qquad\quad T_2(c\otimes a)=(c\otimes 1)\Delta(a),$$ are no longer assumed to be bijective. Also recall that a weak multiplier Hopf algebra is called regular if its antipode is a bijective map from $A$ to itself. In this paper, we introduce and study the notion of integrals on such regular weak multiplier Hopf algebras. A left integral is a non-zero linear functional on $A$ that is left invariant (in an appropriate sense). Similarly for a right integral. For a regular weak multiplier Hopf algebra $(A,\Delta)$ with (sufficiently many) integrals, we construct the dual $(\widehat A,\widehat\Delta)$. It is again a regular weak multiplier Hopf algebra with (sufficiently many) integrals. This duality extends the known duality of finite-dimensional weak Hopf algebras to this more general case. It also extends the duality of multiplier Hopf algebras with integrals, the so-called algebraic quantum groups. For this reason, we will sometimes call a regular weak multiplier Hopf algebra with enough integrals an algebraic quantum groupoid. We discuss the relation of our work with the work on duality for algebraic quantum groupoids by Timmermann. We also illustrate this duality with a particular example in a separate paper. In this paper, we only mention the main definitions and results for this example. However, we do consider the two natural weak multiplier Hopf algebras associated with a groupoid in detail and show that they are dual to each other in the sense of the above duality.

In this paper, we study the notion of a separability idempotent in the C*-algebra framework. This is analogous to the notion in the purely algebraic setting, typically considered in the case of (finite-dimensional) algebras with identity, then later also considered in the multiplier algebra framework by the second-named author. The current work was motivated by the appearance of such objects in the authors' ongoing work on locally compact quantum groupoids., Comment: Some updates made in sections 4 and 5; Corrected typos; Accepted to appear in Journal of Noncommutative Geometry