Explicit Family Research Articles

A Denseness Property of Stochastic Processes

AbstractIn this work, we show the following general property of stochastic processes: Denseness of probability distributions extends to denseness of stochastic processes. In particular, for every $$d\in \mathbb {N}$$ d ∈ N , we consider any class of probability distributions which is dense in the space of probability distributions on $$\mathbb {R}^{d}$$ R d with respect to convergence in distribution. Using these classes, we construct various explicit families of times series, continuous processes and càdlàg processes that are dense in their respective spaces with respect to convergence in distribution. This is particularly interesting because, for processes, finite-dimensional distributions convergence does not automatically imply convergence in distribution.

Explicit families of congruences for the overpartition function

AbstractIn this article we exhibit new explicit families of congruences for the overpartition function, making effective the existence results given previously by Treneer. We give infinite families of congruences modulo m for $$m = 3, 5, 7, 11$$ m = 3 , 5 , 7 , 11 , and finite families for $$m = 13, 17, 19$$ m = 13 , 17 , 19 .

Conjectural Criteria for the Most Singular Points of the Hilbert Schemes of Points

We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the original problem to a problem in convex geometry. Proving either of the two conjectural statements will in particular resolve a long-standing conjecture by Briançon and Iarrobino back in the ’70s for the case of the powers of the maximal ideal. Furthermore, for specific classes of lengths, we conjecturally classify points satisfying the conjectural sufficient conditions. This in particular (conjecturally) provides many new explicit families of examples of maximum dimension tangent space at a point of the Hilbert schemes of points of lengths strictly between two consecutive tetrahedral numbers ( 3 + k 3 ) .

Interpolating Between Rényi Entanglement Entropies for Arbitrary Bipartitions via Operator Geometric Means

AbstractThe asymptotic restriction problem for tensors can be reduced to finding all parameters that are normalized, monotone under restrictions, additive under direct sums and multiplicative under tensor products, the simplest of which are the flattening ranks. Over the complex numbers, a refinement of this problem, originating in the theory of quantum entanglement, is to find the optimal rate of entanglement transformations as a function of the error exponent. This trade-off can also be characterized in terms of the set of normalized, additive, multiplicative functionals that are monotone in a suitable sense, which includes the restriction-monotones as well. For example, the flattening ranks generalize to the (exponentiated) Rényi entanglement entropies of order $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] . More complicated parameters of this type are known, which interpolate between the flattening ranks or Rényi entropies for special bipartitions, with one of the parts being a single tensor factor. We introduce a new construction of subadditive and submultiplicative monotones in terms of a regularized Rényi divergence between many copies of the pure state represented by the tensor and a suitable sequence of positive operators. We give explicit families of operators that correspond to the flattening-based functionals, and show that they can be combined in a nontrivial way using weighted operator geometric means. This leads to a new characterization of the previously known additive and multiplicative monotones, and gives new submultiplicative and subadditive monotones that interpolate between the Rényi entropies for all bipartitions. We show that for each such monotone there exist pointwise smaller multiplicative and additive ones as well. In addition, we find lower bounds on the new functionals that are superadditive and supermultiplicative.

Classical Dynamical r-matrices for the Chern–Simons Formulation of Generalized 3d Gravity

AbstractClassical dynamical r-matrices arise naturally in the combinatorial description of the phase space of Chern–Simons theories, either through the inclusion of dynamical sources or through a gauge fixing procedure involving two punctures. Here we consider classical dynamical r-matrices for the family of Lie algebras which arise in the Chern–Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical r-matrices in this case and show that they can be viewed as generalized complexifications, in a sense which we define, of the equations governing dynamical r-matrices for $$\mathfrak {su}(2)$$ su ( 2 ) and $$\mathfrak {sl}(2,{\mathbb {R}})$$ sl ( 2 , R ) . We obtain explicit families of solutions and relate them, via Weierstrass factorization, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models.

New examples of G2-instantons on [formula omitted]

We study the existence of SU(2)2-invariant G2-instantons on R4×S3 with the coclosed G2-structures found on [1]. We find an explicit 1-parameter family of SU(2)3-invariant G2-instantons on the trivial bundle on R4×S3 and study its “bubbling” behaviour. We prove the existence a 1-parameter family on the identity bundle. We also provide existence results for locally defined SU(2)2-invariant G2-instantons.

Eighth-Order Numerov-Type Methods Using Varying Step Length

This work explores a well-established eighth-algebraic-order numerical method belonging to the explicit Numerov-type family. To enhance its efficiency, we integrated a cost-effective algorithm for adjusting the step size. After each step, the algorithm either maintains the current step length, halves it, or doubles it. Any off-step points required by this technique are calculated using a local interpolation function. Numerical tests involving diverse problems demonstrate the significant efficiency improvements achieved through this approach. The method is particularly effective for solving differential equations with oscillatory behavior, showcasing its ability to maintain high accuracy with fewer function evaluations. This advancement is crucial for applications requiring precise solutions over long intervals, such as in physics and engineering. Additionally, the paper provides a comprehensive MATLAB-R2018a implementation, facilitating ease of use and further research in the field. By addressing both computational efficiency and accuracy, this study contributes a valuable tool for the numerical analysis community.

Variety Evasive Subspace Families

We introduce the problem of constructing explicit variety evasive subspace families. Given a family F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{F}$$\\end{document} of subvarieties of a projective or affine space, a collection H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{H}$$\\end{document} of projective or affine k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document}-subspaces is (F,ϵ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathcal{F},\\epsilon)$$\\end{document}-evasive iffor every V∈F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{V}\\in\\mathcal{F}$$\\end{document}, all but at most ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon$$\\end{document}-fraction of W∈H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W\\in\\mathcal{H}$$\\end{document} intersect every irreducible component of V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{V}$$\\end{document} with (at most) the expected dimension.The problem of constructing such an explicit subspace family generalizes both deterministic black-box polynomial identity testing (PIT) and the problem of constructing explicit (weak) lossless rank condensers.Using Chow forms, we construct explicit k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document}-subspace families of polynomial size that are evasive for all varieties of bounded degree in a projective or affine n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n$$\\end{document}-space. As one application, we obtain a complete derandomization of Noether’s normalization lemma for varieties of low degree in a projective or affine n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n$$\\end{document}-space. In another application, we obtain a simple polynomial-time black-box PIT algorithm for depth-4 arithmetic circuits with bounded top fan-in and bottom fan-in that are not in the Sylvester–Gallai configuration, improving and simplifying a result of Gupta (ECCC TR 14-130).As a complement of our explicit construction, we prove a tight lower bound for the size of k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document}-subspace families that are evasive for degree-d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d$$\\end{document} varieties in a projective n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n$$\\end{document}-space. When n-k=nΩ(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-k=n^{\\Omega(1)}$$\\end{document}, the lower bound is superpolynomial unless d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d$$\\end{document} is bounded. The proof uses a dimension counting argument on Chow varieties that parametrize projective subvarieties.

Family linguistic ecology in Chinese American diasporas: migrants’ heritage language planning and discursive ethnolinguistic identity in-between

ABSTRACT Heritage language maintenance is a challenge faced by many immigrant parents. Previous studies on Chinese immigrant families and inheritance of the mother tongue privileged a pedagogical perspective focusing on second-language acquisition and literacy developments. Little attention has been paid to the nuanced understanding of how Chinese immigrant parents help their children construct language resilience of (un)belongingness across home and host contexts and how children negotiate fluid hybrid identity construction. Through questionnaires, semi-constructed interviews, and home visits with four Chinese immigrant families in the United States, this linguistic ethnography study probes the linguistic ecology, (in)visible language planning, and discursive identity construction in Chinese American diaspora families. The findings suggest migrant parents’ congruence and conflict in explicit and implicit family language ideology, management, and practice. Children negotiate fluid discursive identity construction and mediate hybrid culture characterised by the diversity of pathways of language learning as a (dis)continuum. The findings highlight the significance of a linguistic ecological niche in emergent bilingual immigrants’ heritage language maintenance and development. This research illuminates how parents can foster a greater sense of possibility for transitional children’s journeys of being and becoming bilinguals or multilinguals.

Hyperelliptic genus 3 curves with involutions and a Prym map

Hyperelliptic genus 3 curves with involutions and a Prym map

Stationary trajectories in Minkowski spacetimes

We determine the conjugacy classes of the Poincaré group ISO+(n, 1) and apply this to classify the stationary trajectories of Minkowski spacetimes in terms of timelike Killing vectors. Stationary trajectories are the orbits of timelike Killing vectors and, equivalently, the solutions to Frenet–Serret equations with constant curvature coefficients. We extend the 3 + 1 Minkowski spacetime Frenet–Serret equations due to Letaw to Minkowski spacetimes of arbitrary dimension. We present the explicit families of stationary trajectories in 4 + 1 Minkowski spacetime.

The Link between Family Financial Socialization in Adulthood and Investment Literacy of P2P Investors

AbstractThis paper examines how family financial socialization in adulthood is linked to the development of investment literacy among individual family members within the context of innovative financial services, specifically peer-to-peer (P2P) lending. Our findings revealed that P2P lending investors engage in a moderate level family financial socialization suggesting that family, as a key financial socialization agent in childhood and adolescence, maintains its role in adulthood. Additionally, such investors possess a high-level investment knowledge, skills, and attitudes. Explicit family financial socialization has a significant and positive effect on the individuals’ investment knowledge, skills, and attitudes, while the effect of implicit financial socialization is significant but negative for knowledge and attitudes. Such findings suggest that family discussion among adult members result in higher, while observations of family members’ investment behavior led to lower investment literacy. Our study found no significant moderating effect of the strength of social ties indicating that dynamics of family relations neither strengthen nor weaken proximal socialization outcomes. The analysis of differences across demographic groups unveiled statistically significant distinctions concerning respondents’ gender, income, and education. These results provide important insights for stakeholders, underscoring the significant role family socialization in adulthood plays in shaping individuals’ investment literacy, particularly of those investing on P2P lending platforms.

Some algebraic questions about the Reed-Muller code

Let Rq(r,n) denote the rth order Reed-Muller code of length qn over Fq. We consider two algebraic questions about the Reed-Muller code. Let Hq(r,n)=Rq(r,n)/Rq(r−1,n). (1) When q=2, it is known that there is a “duality” between the actions of GL(n,F2) on H2(r,n) and on H2(r′,n), where r+r′=n. The result is false for a general q. However, we find that a slightly modified duality statement still holds when q is a prime or r<charFq. (2) Let F(Fqn,Fq) denote the Fq-algebra of all functions from Fqn to Fq. It is known that when q is a prime, the Reed-Muller codes {0}=Rq(−1,n)⊂Rq(0,n)⊂⋯⊂Rq(n(q−1),n)=F(Fqn,Fq) are the only AGL(n,Fq)-submodules of F(Fqn,Fq). In particular, Hq(r,n) is an irreducible GL(n,Fq)-module when q is a prime. For a general q, Hq(r,n) is not necessarily irreducible. We determine all its submodules and the factors in its composition series. The factors of the composition series of Hq(r,n) provide an explicit family of irreducible representations of GL(n,Fq) over Fq.

On the Futaki invariant of Fano threefolds

We study the zero locus of the Futaki invariant on K-polystable Fano threefolds, seen as a map from the Kähler cone to the dual of the Lie algebra of the reduced automorphism group. We show that, apart from families 3.9, 3.13, 3.19, 3.20, 4.2, 4.4, 4.7 and 5.3 of the Iskovskikh–Mori–Mukai classification of Fano threefolds, the Futaki invariant of such manifolds vanishes identically on their Kähler cone. In all cases, when the Picard rank is greater or equal to two, we exhibit explicit 2-dimensional differentiable families of Kähler classes containing the anti-canonical class and on which the Futaki invariant is identically zero. As a corollary, we deduce the existence of non Kähler–Einstein cscK metrics on all such Fano threefolds.

Exact Mobility Edges for Almost-Periodic CMV Matrices via Gauge Symmetries

Abstract We investigate the symmetries of the so-called generalized extended Cantero–Moral–Velázquez (CMV) matrices. It is well-documented that problems involving reflection symmetries of standard extended CMV matrices can be subtle. We show how to deal with this in an elegant fashion by passing to the class of generalized extended CMV matrices via explicit diagonal unitaries in the spirit of Cantero–Grünbaum–Moral–Velázquez. As an application of these ideas, we construct an explicit family of almost-periodic CMV matrices, which we call the mosaic unitary almost-Mathieu operator, and prove the occurrence of exact mobility edges. That is, we show the existence of energies that separate spectral regions with absolutely continuous and pure point spectrum and exactly calculate them.

Explicit Harmonic Self-maps of Complex Projective Spaces

We study { textrm{SU}( p + 1 ) times textrm{SU}( n - p ) } -equivariant maps between complex projective spaces. For every { n, p in mathbb {N}} with { 0 le p < n } , we construct two explicit families of uncountable many harmonic self-maps of mathbb{C}mathbb{P}^{n}, one given by holomorphic maps and the other by maps that are neither holomorphic nor antiholomorphic. We prove that each solution is equivariantly weakly stable and explicitly compute the equivariant spectrum for some specific maps in both families.

Smoothing of Binary Codes, Uniform Distributions, and Applications.

The action of a noise operator on a code transforms it into a distribution on the respective space. Some common examples from information theory include Bernoulli noise acting on a code in the Hamming space and Gaussian noise acting on a lattice in the Euclidean space. We aim to characterize the cases when the output distribution is close to the uniform distribution on the space, as measured by the Rényi divergence of order α∈(1,∞]. A version of this question is known as the channel resolvability problem in information theory, and it has implications for security guarantees in wiretap channels, error correction, discrepancy, worst-to-average case complexity reductions, and many other problems. Our work quantifies the requirements for asymptotic uniformity (perfect smoothing) and identifies explicit code families that achieve it under the action of the Bernoulli and ball noise operators on the code. We derive expressions for the minimum rate of codes required to attain asymptotically perfect smoothing. In proving our results, we leverage recent results from harmonic analysis of functions on the Hamming space. Another result pertains to the use of code families in Wyner's transmission scheme on the binary wiretap channel. We identify explicit families that guarantee strong secrecy when applied in this scheme, showing that nested Reed-Muller codes can transmit messages reliably and securely over a binary symmetric wiretap channel with a positive rate. Finally, we establish a connection between smoothing and error correction in the binary symmetric channel.

Nondegeneracy of the entire solution for the n-Laplace Hénon equation of Liouville type

"Motivated by the work of Takahashi [10], we establish nondegeneracy of the explicit family of solutions of the n-Laplace H´enon equation of Liouville type on the whole space."

Tame automorphism groups of polynomial rings with property (T) and infinitely many alternating group quotients

We construct new families of groups with property (T) and infinitely many alternating group quotients. One of those consists of subgroups of A u t ( F p [ x 1 , … , x n ] ) \mathrm {Aut}(\mathbf {F}_{p}[x_1, \dots , x_n]) generated by a suitable set of tame automorphisms. Finite quotients are constructed using the natural action of A u t ( F p [ x 1 , … , x n ] ) \mathrm {Aut}(\mathbf {F}_{p}[x_1, \dots , x_n]) on the n n -dimensional affine spaces over finite extensions of F p \mathbf {F}_p . As a consequence, we obtain explicit presentations of Gromov hyperbolic groups with property (T) and infinitely many alternating group quotients. Our construction also yields an explicit infinite family of expander Cayley graphs of degree 4 4 for alternating groups of degree p 7 − 1 p^7-1 for any odd prime p p .

Families of Galois representations and (𝜑,𝜏)-modules

Let p p be a prime, and let K K be a finite extension of Q p \mathbf {Q}_p , with absolute Galois group G K \mathcal {G}_K . Let π \pi be a uniformizer of K K and let K ∞ K_\infty be the Kummer extension obtained by adjoining to K K a system of compatible p n p^n -th roots of π \pi , for all n n , and let L L be the Galois closure of K ∞ K_\infty . Using these extensions, Caruso has constructed é tale ( ϕ , τ ) (\phi ,\tau ) -modules, which classify p p -adic Galois representations of K K . In this paper, we use locally analytic vectors and theories of families of ϕ \phi -modules over Robba rings to prove the overconvergence of ( ϕ , τ ) (\phi ,\tau ) -modules in families. As examples, we also compute some explicit families of ( ϕ , τ ) (\phi ,\tau ) -modules in some simple cases.