Symmetric Subset Research Articles

On metric approximate subgroups

Let [Formula: see text] be a group with a metric invariant under left and right translations, and let [Formula: see text] be the ball of radius [Formula: see text] around the identity. A [Formula: see text]-metric approximate subgroup is a symmetric subset [Formula: see text] of [Formula: see text] such that the pairwise product set [Formula: see text] is covered by at most [Formula: see text] translates of [Formula: see text]. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, Combinatorica, 28(5) (2008) 547–594, doi:10.1007/s00493-008-2271-7; T. Tao, Metric entropy analogues of sum set theory (2014), https://terrytao.wordpress.com/2014/03/19/metric-entropy-analogues-of-sum-set-theory/] along with the version for discrete groups (approximate subgroups). In [E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25(1) (2012) 189–243, doi:10.1090/S0894-0347-2011-00708-X], it was shown for the discrete case that, at the asymptotic limit of [Formula: see text] finite but large, the “approximateness” (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on [Formula: see text] replacing finiteness. In particular, if [Formula: see text] has bounded exponent, we show that any [Formula: see text]-metric approximate subgroup is close to a [Formula: see text]-metric approximate subgroup for an appropriate [Formula: see text].

TOPOLOGICAL MODULES GEOMETRY

The extremal structure of zero-neighbourhoods of a topological module is analyzed reaching unexpected conclusions when the module topology is not Hausdorff. These results motivate us to introduce the notion of metric modules, which are modules endowed with a translation-invariant metric, turning them into an (additive) topological group. We study the central and diametral points of additively symmetric subsets and find examples of convex sets which are not symmetric translates (translates of addivitely symmetric subsets). As a consequence of all of these, it seems natural to transport the well-known Bishop-Phelps property from the category of real topological vector spaces to general topological modules over topological rings. Then we stick to particular topological rings, the unital C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras, showing that the subset of positive elements lying below the unity is an effect algebra. We also prove that every continuous linear operator on a Hausdorff locally convex topological vector space that commutes with all continuous linear projections of one-dimensional range is a multiple of the identity. Finally, we discuss how to transport the previous result to C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras.

Metric decomposability theorems on sets of integers

AbstractA set is called additively decomposable (resp., asymptotically additively decomposable) if there exist sets of cardinality at least two each such that (resp., is finite). If none of these properties hold, the set is called totally primitive. We define ‐decomposability analogously with subsets of . Wirsing showed that almost all subsets of are totally primitive. In this paper, in the spirit of Wirsing, we study decomposability from a probabilistic viewpoint. First, we show that almost all symmetric subsets of are ‐decomposable. Then we show that almost all small perturbations of the set of primes yield a totally primitive set. Further, this last result still holds when the set of primes is replaced by the set of sums of two squares, which is by definition decomposable.

Locally compact models for approximate rings

By an approximate subring of a ring we mean an additively symmetric subset X such that X·X∪(X+X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X\\cdot X \\cup (X +X)$$\\end{document} is covered by finitely many additive translates of X. We prove that each approximate subring X of a ring has a locally compact model, i.e. a ring homomorphism f:⟨X⟩→S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f :\\langle X \\rangle \\rightarrow S$$\\end{document} for some locally compact ring S such that f[X] is relatively compact in S and there is a neighborhood U of 0 in S with f-1[U]⊆4X+X·4X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f^{-1}[U] \\subseteq 4X + X \\cdot 4X$$\\end{document} (where 4X:=X+X+X+X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4X:=X+X+X+X$$\\end{document}). This S is obtained as the quotient of the ring ⟨X⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle X \\rangle $$\\end{document} interpreted in a sufficiently saturated model by its type-definable ring connected component. The main point is to prove that this component always exists. In order to do that, we extend the basic theory of model-theoretic connected components of definable rings [developed in Gismatullin et al. (J Symb Log First View: 1–35, 2022, https://doi.org/10.1017/jsl.2022.10) and Krupiński et al. (Ann Pure Appl Logic 173.7(July):103119, 2022) to the case of rings generated by definable approximate subrings and we answer a question from Krupiński et al. (2022) in the more general context of approximate subrings. Namely, let X be a definable (in a structure M) approximate subring of a ring and R:=⟨X⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R:=\\langle X \\rangle $$\\end{document}. Let X¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\bar{X}}$$\\end{document} be the interpretation of X in a sufficiently saturated elementary extension and R¯:=⟨X¯⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\bar{R}}:= \\langle {\\bar{X}} \\rangle $$\\end{document}. It follows from Massicot and Wagner (J Éc Polytech Math 2:55–63, 2015) that there exists the smallest M-type-definable subgroup of (R¯,+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\bar{R}},+)$$\\end{document} of bounded index, which is denoted by (R¯,+)M00\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\bar{R}},+)^{00}_M$$\\end{document}. We prove that (R¯,+)M00+R¯·(R¯,+)M00\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\bar{R}},+)^{00}_M + {\\bar{R}} \\cdot ({\\bar{R}},+)^{00}_M$$\\end{document} is the smallest M-type-definable two-sided ideal of R¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\bar{R}}$$\\end{document} of bounded index, which we denote by R¯M00\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\bar{R}}^{00}_M$$\\end{document}. Then S in the first sentence of the abstract is just R¯/R¯M00\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\bar{R}}/{\\bar{R}}^{00}_M$$\\end{document} and f:R→R¯/R¯M00\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f: R \\rightarrow {\\bar{R}}/{\\bar{R}}^{00}_M$$\\end{document} is the quotient map. In fact, f is the universal “definable” (in a suitable sense) locally compact model. The existence of locally compact models can be seen as a general structural result about approximate subrings: every approximate subring X can be recovered up to additive commensurability as the preimage by a locally compact model f:⟨X⟩→S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f :\\langle X \\rangle \\rightarrow S$$\\end{document} of any relatively compact neighborhood of 0 in S. It should also have various applications to get more precise structural or even classification results. For example, in this paper, we deduce that every [definable] approximate subring X of a ring of positive characteristic is additively commensurable with a [definable] subring contained in 4X+X·4X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4X + X \\cdot 4X$$\\end{document}. This easily implies that for any given K,L∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K,L \\in \\mathbb {N}$$\\end{document} there exists a constant C(K, L) such that every K-approximate subring X (i.e. K additive translates of X cover X·X∪(X+X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X \\cdot X \\cup (X+X)$$\\end{document}) of a ring of positive characteristic ≤L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le L$$\\end{document} is additively C(K, L)-commensurable with a subring contained in 4X+X·4X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4X + X \\cdot 4X$$\\end{document}. Another application of the existence of locally compact models is a classification of finite approximate subrings of rings without zero divisors: for every K∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K \\in \\mathbb {N}$$\\end{document} there exists N(K)∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N(K) \\in \\mathbb {N}$$\\end{document} such that for every finite K-approximate subring X of a ring without zero divisors either |X|<N(K)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|X| <N(K)$$\\end{document} or 4X+X·4X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4X + X \\cdot 4X$$\\end{document} is a subring which is additively K11\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^{11}$$\\end{document}-commensurable with X.

Covering almost all the layers of the hypercube with multiplicities

Given a hypercube Qn:={0,1}n in Rn and k∈{0,…,n}, the k-th layer Qkn of Qn denotes the set of all points in Qn whose coordinates contain exactly k many ones. For a fixed t∈N and k∈{0,…,n}, let P∈R[x1,…,xn] be a polynomial that has zeroes of multiplicity at least t at all points of Qn∖Qkn, and P has zeros of multiplicity exactly t−1 at all points of Qkn. In this short note, we show thatdeg(P)≥max⁡{k,n−k}+2t−2. Matching the above lower bound we give an explicit construction of a family of hyperplanes H1,…,Hm in Rn, where m=max⁡{k,n−k}+2t−2, such that every point of Qkn will be covered exactly t−1 times, and every other point of Qn will be covered at least t times. Note that putting k=0 and t=1, we recover the much celebrated covering result of Alon and Füredi (1993) [1]. Using the above family of hyperplanes we disprove a conjecture of Venkitesh (2022) [23] on exactly covering symmetric subsets of hypercube Qn with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest.We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.

On a Runge theorem over $${\mathbb {R}}_3$$

In this paper, we investigate a topological characterization of the Runge theorem in the Clifford algebra \( {\mathbb {R}}_3\) via the description of the homology groups of axially symmetric open subsets of the quadratic cone in \({\mathbb {R}}_3\).

On extension of the data driven ROM inverse scattering framework to partially nonreciprocal arrays

Data-driven reduced order models (ROMs) recently emerged as powerful tool for the solution of inverse scattering problems. The main drawback of this approach is that it was limited to measurement arrays with reciprocally collocated transmitters and receivers, that is, square symmetric matrix (data) transfer functions. To relax this limitation, we use our previous work Druskin et al (2021 Inverse Problems 37 075003), where the ROMs were combined with the Lippmann–Schwinger integral equation to produce a direct nonlinear inversion method. In this work we extend this approach to more general transfer functions, including those that are non-symmetric, e.g., obtained by adding only receivers or sources. The ROM is constructed based on the symmetric subset of the data and is used to construct all internal solutions. Remaining receivers are then used directly in the Lippmann–Schwinger equation. We demonstrate the new approach on a number of 1D and 2D examples with non-reciprocal arrays, including a single input/multiple outputs inverse problem, where the data is given by just a single-row matrix transfer function. This allows us to approach the flexibility of the Born approximation in terms of acceptable measurement arrays; at the same time significantly improving the quality of the inversion compared to the latter for strongly nonlinear scattering effects.

Optimal robust mean and location estimation via convex programs with respect to any pseudo-norms

We consider the problem of robust mean and location estimation with respect to any pseudo-norm of the form\(x\in {{\mathbb {R}}}^d\mapsto \left\| x\right\| _S = \sup _{v\in S}\bigl < v,x \bigr>\) where S is any symmetric subset of \({{\mathbb {R}}}^d\). We show that the deviation-optimal minimax sub-Gaussian rate for confidence \(1-\delta \) is $$\begin{aligned} \max \left( \frac{\ell ^*(\Sigma ^{1/2}S)}{\sqrt{N}}, \sup _{v\in S}\left\| \Sigma ^{1/2}v\right\| _2\sqrt{\frac{\log (1/\delta )}{N}}\right) \end{aligned}$$where \(\ell ^*(\Sigma ^{1/2}S)\) is the Gaussian mean width of \(\Sigma ^{1/2}S\) and \(\Sigma \) the covariance of the data. This improves the entropic minimax lower bound from Lugosi and Mendelson (Probab Theory Relat Fields 175(3–4):957–973, 2019) and closes the gap characterized by Sudakov’s inequality between the entropy and the Gaussian mean width for this problem. This shows that the right statistical complexity measure for the mean estimation problem is the Gaussian mean width. We also show that this rate can be achieved by a solution to a convex optimization problem in the adversarial and \(L_2\) heavy-tailed setup by considering minimum of some Fenchel–Legendre transforms constructed using the median-of-means principle. We finally show that this rate may also be achieved in situations where there is not even a first moment but a location parameter exists.

Arnold’s potentials and quantum catastrophes II

In paper I (Znojil, 2020) it has been demonstrated that besides the well known use of the Arnold’s one-dimensional polynomial potentials V(k)(x)=xk+1+c1xk−1+… in the classical Thom’s catastrophe theory, some of these potentials (viz., the confining ones, with k=2N+1) could also play an analogous role of genuine benchmark models in quantum mechanics, especially in the dynamical regime in which N+1 valleys are separated by N barriers. For technical reasons, just the ground states in the spatially symmetric subset of V(k)(x)=V(k)(−x) have been considered. In the present paper II we will show that and how both of these constraints can be relaxed. Thus, even the knowledge of the trivial leading-order form of the excited states will be shown sufficient to provide a new, truly rich level-avoiding spectral pattern. Secondly, the fully general asymmetric-potential scenarios will be shown tractable perturbatively.

Toward super‐approximation in positive characteristic

In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is a family of expanders under certain algebraic conditions. Here is a more precise formulation of our main result. For a positive integer $c_0$, we say a square-free polynomial is $c_0$-admissible if degree of irreducible factors of $f$ are distinct integers with prime factors at least $c_0$. Suppose $\Omega$ is a finite symmetric subset of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$, where $p$ is a prime more than $5$. Let $\Gamma$ be the group generated by $\Omega$. Suppose the Zariski-closure of $\Gamma$ is connected, simply-connected, and absolutely almost simple; further assume that the field generated by the traces of ${\rm Ad}(\Gamma)$ is $\mathbb{F}_p(t)$. Then for some positive integer $c_0$ the family of Cayley graphs ${\rm Cay}(\pi_{f(x)}(\Gamma),\pi_{f(x)}(\Omega))$ as $f$ ranges in the set of $c_0$-admissible polynomials is a family of expanders, where $\pi_{f(t)}$ is the quotient map for the congruence modulo $f(t)$.

Classification of Stroke Severity Using Clinically Relevant Symmetric Gait Features Based on Recursive Feature Elimination With Cross-Validation

Stroke is a leading cause of disability among elderly individuals, and gait impairment is a typical characteristic related to the stroke severity experienced by patients. The aim of this study is to propose a novel stroke severity classification method using symmetric gait features with recursive feature elimination with cross-validation (RFECV). An experiment was conducted on data acquired from thirteen chronic stroke patients and eighteen elderly participants. They walked on a treadmill at four different speeds based on their preferred speed. In this study, symmetric gait features representing the ratio between the left- and right-side values were used as inputs along with the general gait features that did not completely contain the patients’ gait characteristics. We used four different machine learning (ML) techniques to determine the optimal subset for differentiating between the elderly and stroke groups according to severity based on RFECV. To verify the performance of RFECV and the symmetric gait features, four different feature sets were used: 1) all forty-five general features, 2) all twenty-one symmetric features, 3) the optimal general feature subset obtained by using RFECV, and 4) the optimal symmetric feature subset obtained by using RFECV. It was confirmed that the classification performance increased when symmetric gait data and the RFECV technique were applied. The best classification result was obtained by RF-RFECV with an RF classifier derived from the symmetric features (accuracy: 96.01%). The findings of this study can help clinicians diagnose the stroke severity experienced by patients based on information obtained using ML technology.

Resistance distance and Kirchhoff index in dihedral Cayley graphs

Let D2n=〈a,b∣an=b2=1,bab=a−1〉={1,a,…,an−1,b,ba,…,ban−1} be the dihedral group of order 2n and let S be a symmetric subset of D2n. The dihedral Cayley graph Cay(D2n,S) is the graph with vertex set VCay(D2n,S)=D2n and edge set ECay(D2n,S)={g,h}∣gh−1∈S,g,h∈D2n. In this paper, the complete information for the eigenvalues and their corresponding orthonormal eigenvectors of Cay(D2n,S) is provided. The closed-form formulae of the Kirchhoff index and the resistance distances between any two vertices of Cay(D2n,S) are then derived.

On the generic family of Cayley graphs of a finite group

The Cayley graphs Gm(G)=Cay(Gm,S) of Cartesian powers of the finite group G with respect to canonical symmetric subsets S=S(m)⊂Gm are investigated. It is proved that for each m>1 the graph Gm(G) determines the group G up to isomorphism. The groups of automorphisms Aut(Gm(G)) are determined. It is shown that if G is a non-abelian group, then Aut(Gm(G))≃(Gm⋊Aut(G))⋊Dm+1, where Dm+1 is the dihedral group of order 2m+2. If G is an abelian group (with some exceptions for m=3), then Aut(Gm(G))≃Gm⋊(Aut(G)×Sm+1), where Sm+1 is the symmetric group of degree m+1. Relations between Cayley graphs Gm(G) and Bergman-Isaacs Theorem on rings with fixed-point-free group actions are discussed.

On the 3D ideal MHD equations with partial damping

We consider the 3D ideal MHD equations on T3 with partial dampings. The dampings, which act as partial dissipations, are defined by the fractional Laplacian removing Fourier modes in certain symmetric subsets Kj⊂Z3 (j=1,2). This model is motivated by Elgindi et al. (2017) and Kim and Dubrulle (2002) where the phenomenon of energy cascade is analyzed for the 2D inviscid fluid flow with partial dampings. Our main result shows that this model has a unique global-in-time smooth solution under suitable assumptions on Kj and αj.

Integral equienergetic non-isospectral unitary Cayley graphs

We prove that the Cayley graphs X(G,S) and X+(G,S) are equienergetic for any abelian group G and any symmetric subset S. We then focus on the family of unitary Cayley graphs GR=X(R,R⁎), where R is a finite commutative ring with identity. We show that under mild conditions, {GR,GR+} are pairs of integral equienergetic non-isospectral graphs (generically connected and non-bipartite). Then, we obtain conditions such that {GR,G¯R} are equienergetic non-isospectral graphs. Finally, we characterize all integral equienergetic non-isospectral triples {GR,GR+,G¯R} such that all the graphs are also Ramanujan.

Counting the decimation classes of binary vectors with relatively prime length and density

We present a method for determining the number of decimation classes of density $$\delta $$ binary vectors indexed by a finite abelian group G of size $$\ell $$ and exponent $${\ell ^*}$$ such that $$\delta $$ is relatively prime to $${\ell ^*}$$ . This method is the first which is not based on exhaustive vector generation and exploits the subgroup lattice of $${{\mathbb {Z}}_{{\ell ^*}}^{\times }}$$ . Instead, our method is based on our newly developed theory of multipliers for arbitrary subsets of finite abelian groups, our results on orbits under the action of the multiplier group, and finding the number of solutions of a potentially highly symmetric subset sum problem. Implementing our method on vectors indexed by $${\mathbb {Z}}_{\ell }$$ of odd length $$\ell $$ and density $$(\ell +1)/2$$ greatly increased the number of $$\ell $$ for which the number of decimation classes of such vectors is known. Additionally, our newly developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing.

On Runge pairs and topology of axially symmetric domains

We prove a Runge theorem for and describe the homology of axially symmetric open subsets of H.

Symmetric Helmholtz Fermi-surface harmonics for an optimal representation of anisotropic quantities on the Fermi surface: Application to the electron-phonon problem

We outline a numerical procedure to incorporate the crystal symmetries in the Helmholtz Fermi-surface harmonics basis set, which are the solutions of the Helmholtz equation defined on the Fermi surface. This improvement allows for an optimal representation of anisotropic quantities defined on the Fermi surface in terms of few symmetric elements of the set. We demonstrate the general validity of our approach by identifying the fully symmetric Helmholtz Fermi-surface harmonics subset for several representative systems with different crystal structures, namely, FCC-Cu, HEX-MgB$_2$, and BCC-YH$_6$. Furthermore, we illustrate the potential of the method applied to the electron-phonon problem, showing that the anisotropic electron-phonon mass-enhancement parameter $\lambda_{\bf k}$ can be represented to high accuracy by a handful of coefficients. This works as an effective filter, paving the way for a reduction of several orders of magnitude in the computation of superconductivity, impurity problems, or any other Fermi surface dependent property of metals from first principles.

Low Correlation Noise Stability of Symmetric Sets

We study the Gaussian noise stability of subsets A of Euclidean space satisfying \(A=-A\). It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. On the other hand, in dimension two and higher, the ball or its complement does not always maximize noise stability among symmetric sets of fixed Gaussian measure. In summary, we provide the first known positive and negative results for the symmetric Gaussian problem.

Isomorphic and symmetric adult-onset generalized morphea are associated with distinctive clinical features: A retrospective multicenter study

To the Editor: Generalized morphea (GM) is a rare subtype of localized scleroderma characterized by widespread involvement. GM is defined by the presence of at least 4 lesions larger than 3 cm that involve at least 2 anatomic sites.1 Isomorphic and symmetric subsets of GM were recently delineated based on a computerized lesion mapping approach.2 In the isomorphic pattern, the lesions favor areas of skin friction (ie, brassiere band, waistband, inguinal creases), whereas in the symmetric one, they occur with a symmetric distribution on the trunk and extremities.