共找到 20 条结果
Family algebraic structures indexed by a semigroup first appeared in the algebraic aspects of renormalizations in quantum field theory. The concept of the Rota-Baxter family and its relation with (tri)dendriform family algebras have been recently discovered. In this paper, we first consider a notion of $\mathcal{O}$-operator family as a generalization of the Rota-Baxter family and define two variations of associative Yang-Baxter family that produce $\mathcal{O}$-operator families. Given a Hochschild $2$-cocycle on the underlying algebra, we also define a notion of twisted $\mathcal{O}$-operator family (in particular twisted Rota-Baxter family). We also introduce and study NS-family algebras as the underlying structure of twisted $\mathcal{O}$-operator families. Finally, we define suitable cohomology of twisted $\mathcal{O}$-operator families and NS-family algebras (in particular cohomology of Rota-Baxter families and dendriform family algebras) that govern their deformations.
Cooperative play (co-play) is often positioned as a family-beneficial practice that can strengthen parent-child bonds and support parental mediation in games. Yet co-play in user-generated virtual worlds (UGVWs) can be disrupted by real-time harms that parents cannot easily prevent. Roblox, a platform with millions of user-generated virtual worlds and a large child player base, illustrates this challenge. Prior work on harmful UGVW design highlights risks beyond content problems, including manipulative monetization prompts, unmoderated social interactions, emergent in-world behaviors, and narrative designs that may normalize harmful ideologies. Current governance and moderation approaches, largely adapted from social media, focus on static artifacts and often fail to capture interactive and emergent harms in virtual worlds. This workshop paper asks: how might UGVWs and their platforms be designed to minimize harms that specifically impair family co-play experiences?
We extend Gromov's conjecture on the sharp width estimate for Riemannian bands with positive scalar curvature to the family case and prove that it holds for fiber bundles with infinite family A-hat area. The method we employ is based on Dirac operators and the family index theory. Our proof relies on a product formula for index bundles established in this paper.
Motivated by the commitments from the Talmud in Judaism, we consider the family planning rules which require a couple to get children till certain numbers of boys and girls are reached. For example, the rabbinical school of Beit Hillel says that one boy and one girl are necessary, whereas Beit Shammai urges for two boys. Surprisingly enough, although the corresponding average family sizes differ in both cases, the gender ratios remain constant. We show more that for any family planning rule the gender ratio is equal to the birth odds. The proof of this result is given by using different mathematical techniques, such as induction principle, Doob's optional-stopping theorem, and brute-force. We conclude that, despite possible asymmetries in the religiously motivated family planning rules, they discriminate neither boys nor girls.
In this paper we study pseudorandomness of a family of sequences in terms of two measures, the family complexity ($f$-complexity) and the cross-correlation measure of order $\ell$. We consider sequences not only on binary alphabet but also on $k$-symbols ($k$-ary) alphabet. We first generalize some known methods on construction of the family of binary pseudorandom sequences. We prove a bound on the $f$-complexity of a large family of binary sequences of Legendre-symbols of certain irreducible polynomials. We show that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order. Next, we present another family of binary sequences having high $f$-complexity and low cross-correlation measure. Then we extend the results to the family of sequences on $k$-symbols alphabet.
The Rafita asteroid family is an S-type group located in the middle main belt, on the right side of the 3J:-1A mean-motion resonance. The proximity of this resonance to the family left side in semi-major axis caused many former family members to be lost. As a consequence, the family shape in the $(a,1/D)$ domain is quite asymmetrical, with a preponderance of objects on the right side of the distribution. The Rafita family is also characterized by a leptokurtic distribution in inclination, which allows the use of methods of family age estimation recently introduced for other leptokurtic families such as Astrid, Hansa, Gallia, and Barcelona. In this work we propose a new method based on the behavior of an asymmetry coefficient function of the distribution in the $(a,1/D)$ plane to date incomplete asteroid families such as Rafita. By monitoring the time behavior of this coefficient for asteroids simulating the initial conditions at the time of the family formation, we were able to estimate that the Rafita family should have an age of $490\pm200$ Myr, in good agreement with results from independent methods such as Monte Carlo simulations of Yarkovsky and Yorp dynamical induced evolutio
We study the relationship between two measures of pseudorandomness for families of binary sequences: family complexity and cross-correlation measure introduced by Ahlswede et al.\ in 2003 and recently by Gyarmati et al., respectively. More precisely, we estimate the family complexity of a family $(e_{i,1},\ldots,e_{i,N})\in \{-1,+1\}^N$, $i=1,\ldots,F$, of binary sequences of length $N$ in terms of the cross-correlation measure of its dual family $(e_{1,n},\ldots,e_{F,n})\in \{-1,+1\}^F$, $n=1,\ldots,N$. We apply this result to the family of sequences of Legendre symbols with irreducible quadratic polynomials modulo $p$ with middle coefficient $0$, that is, $e_{i,n}=\left(\frac{n^2-bi^2}{p}\right)_{n=1}^{(p-1)/2}$ for $i=1,\ldots,(p-1)/2$, where $b$ is a quadratic nonresidue modulo $p$, showing that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order.
Influence maximization (IM) is a classic problem that aims to identify a small group of critical individuals, known as seeds, who can influence the largest number of users in a social network through word-of-mouth. This problem finds important applications including viral marketing, infection detection, and misinformation containment. The conventional IM problem is typically studied with the oversimplified goal of selecting a single seed set. Many real-world scenarios call for multiple sets of seeds, particularly on social media platforms where various viral marketing campaigns need different sets of seeds to propagate effectively. To this end, previous works have formulated various IM variants, central to which is the requirement of multiple seed sets, naturally modeled as a matroid constraint. However, the current best-known solutions for these variants either offer a weak $(1/2-ε)$-approximation, or offer a $(1-1/e-ε)$-approximation algorithm that is very expensive. We propose an efficient seed selection method called AMP, an algorithm with a $(1-1/e-ε)$-approximation guarantee for this family of IM variants. To further improve efficiency, we also devise a fast implementation, c
Statistical hypotheses are translations of scientific hypotheses into statements about one or more distributions, often concerning their centre. Tests that assess statistical hypotheses of centre implicitly assume a specific centre, e.g., the mean or median. Yet, scientific hypotheses do not always specify a particular centre. This ambiguity leaves the possibility for a gap between scientific theory and statistical practice that can lead to rejection of a true null. In the face of replicability crises in many scientific disciplines, significant results of this kind are concerning. Rather than testing a single centre, this paper proposes testing a family of plausible centres, such as that induced by the Huber loss function (the Huber family). Each centre in the family generates a testing problem, and the resulting family of hypotheses constitutes a familial hypothesis. A Bayesian nonparametric procedure is devised to test familial hypotheses, enabled by a novel pathwise optimization routine to fit the Huber family. The favourable properties of the new test are demonstrated theoretically and experimentally. Two examples from psychology serve as real-world case studies.
We consider the classical problem of learning, with arbitrary accuracy, the natural parameters of a $k$-parameter truncated \textit{minimal} exponential family from i.i.d. samples in a computationally and statistically efficient manner. We focus on the setting where the support as well as the natural parameters are appropriately bounded. While the traditional maximum likelihood estimator for this class of exponential family is consistent, asymptotically normal, and asymptotically efficient, evaluating it is computationally hard. In this work, we propose a novel loss function and a computationally efficient estimator that is consistent as well as asymptotically normal under mild conditions. We show that, at the population level, our method can be viewed as the maximum likelihood estimation of a re-parameterized distribution belonging to the same class of exponential family. Further, we show that our estimator can be interpreted as a solution to minimizing a particular Bregman score as well as an instance of minimizing the \textit{surrogate} likelihood. We also provide finite sample guarantees to achieve an error (in $\ell_2$-norm) of $α$ in the parameter estimation with sample compl
The Hoffmeister family is a C-type group located in the central main belt. Dynamically, it is important because of its interaction with the $ν_{1C}$ nodal secular resonance with Ceres, that significantly increases the dispersion in inclination of family members at lower semi-major axis. As an effect, the distribution of inclination values of the Hoffmeister family at semi-major axis lower than its center is significantly leptokurtic, and this can be used to set constraints on the terminal ejection velocity field of the family at the time it was produced. By performing an analysis of the time behaviour of the kurtosis of the $v_W$ component of the ejection velocity field ($γ_2(v_W)$), as obtained from Gauss' equations, for different fictitious Hoffmeister families with different values of the ejection velocity field, we were able to exclude that the Hoffmeister family should be older than 335 Myr. Constraints from the currently observed inclination distribution of the Hoffmeister family suggest that its terminal ejection velocity parameter $V_{EJ}$ should be lower than 25~m/s. Results of a Yarko-YORP Monte Carlo method to family dating, combined with other constraints from inclinati
In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebraic (family) Seiberg-Witten invariants for algebraic surfaces and then give an algebraic derivation of the family blowup formula for the algebraic family Seiberg-Witten invariants.
All asteroids are currently classified as either family, originating from the disruption of known bodies, or non-family. An outstanding question is the origin of these non-family asteroids. Were they formed individually, or as members of known families but with chaotically evolving orbits, or are they members of old ghost families, that is, asteroids with a common parent body but with orbits that no longer cluster in orbital element space? Here, we show that the sizes of the non-family asteroids in the inner belt are correlated with their orbital eccentricities and anticorrelated with their inclinations, suggesting that both non-family and family asteroids originate from a small number of large primordial planetesimals. We estimate that ~85% of the asteroids in the inner main belt originate from the Flora, Vesta, Nysa, Polana and Eulalia families, with the remaining ~15% originating from either the same families or, more likely, a few ghost families. These new results imply that we must seek explanations for the differing characteristics of the various meteorite groups in the evolutionary histories of a few, large, precursor bodies. Our findings also support the model that asteroid
In this paper, we define and investigate the properties of continuous family groupoids. This class of groupoids is necessary for investigating the groupoid index theory arising from the equivariant Atiyah-Singer index theorem for families, and is also required in noncommutative geometry. The class includes that of Lie groupoids, and the paper shows that, like Lie groupoids, continuous family groupoids always admit (an essentially unique) continuous left Haar system of smooth measures. We also show that the action of a continuous family groupoid $G$ on a continuous family $G$-space fibered over another continuous family $G$-space $Y$ can always be regarded as an action of the continuous family groupoid $G*Y$ on an ordinary $G*Y$-space.
In this paper, we first construct the free Rota-Baxter family algebra generated by some set $X$ in terms of typed angularly $X$-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra.
Granting that the $SU_c(3) \times SU_L(2) \times U(1) \times SU_f(3)$ Standard Model is valid (or, partially valid), for the real world, we propose the $μ^+ e^-$ collider in the $10^2\, GeV$ range as the family collider. This family collider may work efficiently in producing the family Higgs particles and detecting the effects of family gauge bosons, with the range of sub-sub-fermi's (a few $10^{-2} \, fermi$'s).
The dynamical evolution of Classical Kuiper Belt Objects (CKBOs) divides into two parts, according to the secular theory of test particle orbits. The first part is a forced oscillation driven by the planets, while the second part is a free oscillation whose amplitude is determined by the initial orbit of the test particle. We extract the free orbital inclinations and free orbital eccentricities from the osculating elements of 125 known CKBOs. The free inclinations of 32 CKBOs strongly cluster about 2 degrees at orbital semi-major axes between 44 and 45 AU. We propose that these objects comprise a collisional family, the first so identified in the Kuiper Belt. Members of this family are plausibly the fragments of an ancient parent body having a minimum diameter of \~800 km. This body was disrupted upon colliding with a comparably sized object, and generated ejecta having similar free inclinations. Our candidate family is dynamically akin to a sub-family of Koronis asteroids located at semi-major axes less than 2.91 AU; both families exhibit a wider range in free eccentricity than in free inclination, implying that the relative velocity between parent and projectile prior to impact l
Family mementos document events shaping family life, telling a story within and between family members. The elderly collected some mementos for children, but never recorded stories related to those objects. In this paper, in order to understand the status quo of memento storytelling and sharing of elderly people, contextual inquiry was conducted, which further helped us to identify design opportunities and requirements. Resulting design was defined after brainstorm and user consultation, which was Slots- Memento, a system consisting a slot machine-like device used by the elderly and a flash drive used by the young. The Slots machine-like device utilizes with the metaphor of slots machine, which integrates functions of memento photo displaying, story recording, and preservation. In the flash disk, the young could copy memento photos to it. The system aims to facilitate memento story sharing and preservation within family members. Preliminary evaluation and user test were conducted in evaluation section, the results showed that Slots-Memento was understood and accepted by the elderly users. Photos of mementos were easy to recall memories. It enabled the elderly people to be aware of
The Gaia Data Release 3 (DR3) contains reflectance spectra at visible wavelengths for 60,518 asteroids over the range between 374-1034 nm, representing a large sample that is well suited to studies of asteroid families. We want to assess the potential of Gaia spectra in identifying asteroid family members. Here, we focus on two L-type families, namely Tirela/Klumpkea and Watsonia. These families are known for their connection to Barbarian asteroids, which are potentially abundant in calcium-aluminum rich inclusions (CAIs). Our method is based (1) on a color taxonomy specifically built on Gaia data and (2) the similarity of spectra of candidate members with the template spectrum of a specific family. We identified objects in the halo of Tirela/Klumpkea, along with possible interlopers. We also found an independent group of eight asteroids erroneously linked to the family by the hierarchical clustering method (HCM). Consequently, the knowledge of the size distribution of the family has been significantly improved, with a more consistent shape at the larger end. The Watsonia family is a more intricate case, mainly due to its smaller size and the less marked difference between the spec
We construct a stable homotopy type invariant for any Legendrian submanifold in a jet bundle equipped with a linear-at-infinity generating family. We show that this spectrum lifts the generating family homology groups. When the generating family extends to a generating family for an embedded Lagrangian filling, we lift the Seidel isomorphism to the spectrum level. As applications, we establish topological constraints on Lagrangian fillings arising from generating families, algebraic constraints on whether generating families admit fillings, and lower bounds on how many fiber dimensions are needed to construct a generating family for a Legendrian.