Constant Multiples Research Articles

On the monotonicity of weighted perimeters of convex bodies

AbstractWe prove that, among weighted isotropic perimeters, only constant multiples of the Euclidean perimeter satisfy the monotonicity property on nested convex bodies. Although the analogous result fails for general weighted anisotropic perimeters, a similar characterization holds for radially‐weighted anisotropic densities.

Integral Formulas for Almost Product Manifolds and Foliations

Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with k=2,3 distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds.

Additive functions in short intervals, gaps and a conjecture of Erdős

With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki–Radziwiłł theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of a corresponding long average. As part of this treatment, we use a variant of the Matomäki–Radziwiłł theorem for divisor-bounded multiplicative functions recently proven in Mangerel (Divisor-bounded multiplicative functions in short intervals. arXiv: 2108.11401). We consider two sets of applications of these methods. Our first application shows that for an additive function {varvec{g:}} mathbb {N} rightarrow mathbb {C} any non-trivial savings in the size of the average gap |{varvec{g}}{} {textbf {(}}{varvec{n}}{} {textbf {)}}-{varvec{g}}{} {textbf {(}}{varvec{n}}-{textbf {1}}{} {textbf {)}} | implies that {varvec{g}} must have a small first centred moment i.e. the discrepancy of {varvec{g}}{} {textbf {(}}{varvec{n}}{} {textbf {)}} from its mean is small on average. We also obtain a variant of such a result for the second moment of the gaps. This complements results of Elliott and of Hildebrand. As a second application, we make partial progress on an old question of Erdős relating to characterizing constant multiples of {{textbf {log}}} ,{varvec{n}} as the only almost everywhere increasing additive functions. We show that if an additive function is almost everywhere non-decreasing then it is almost everywhere well approximated by a constant times a logarithm. We also show that if the set {{varvec{n}} in mathbb {N} : {varvec{g}}{} {textbf {(}}{varvec{n}}{} {textbf {)}} < {varvec{g}}{} {textbf {(}}{varvec{n}}-{textbf {1}}{} {textbf {)}}} is sufficiently sparse, and if {varvec{g}} is not extremely large too often on the primes (in a precise sense), then {varvec{g}} is identically equal to a constant times a logarithm.

Generic existence of multiplicity-1 minmax minimal hypersurfaces via Allen–Cahn

In Guaraco (J. Differential Geom. 108(1):91–133, 2018) a new proof was given of the existence of a closed minimal hypersurface in a compact Riemannian manifold N^{n+1} with nge 2. This was achieved by employing an Allen–Cahn approximation scheme and a one-parameter minmax for the Allen–Cahn energy (relying on works by Hutchinson, Tonegawa, Wickramasekera to pass to the limit as the Allen-Cahn parameter tends to 0). The minimal hypersurface obtained may a priori carry a locally constant integer multiplicity. Here we modify the minmax construction of Guaraco (J. Differential Geom. 108(1):91–133, 2018), by allowing an initial freedom on the choice of the valley points between which the mountain pass construction is carried out, and then optimising over said choice. We then prove that, when 2le nle 6 and the metric is bumpy, this minmax leads to a (smooth closed) minimal hypersurface with multiplicity 1. (When n=2 this conclusion also follows from Chodosh and Mantoulidis (Ann. Math. 191(1):213–328, 2020).) As immediate corollary we obtain that every compact Riemannian manifold of dimension n+1, 2le nle 6, endowed with a bumpy metric, admits a two-sided smooth closed minimal hypersurface (this existence conclusion also follows from Zhou X (Ann. Math. (2), 192(3):767–820, 2020) for minmax constructions via Almgren–Pitts theory).

Non-existence of Yamabe Minimizers on Singular Spheres

We prove that a minimizer of the Yamabe functional does not exist for a sphere \(\mathbb S^n\) of dimension \(n\ge 3\), endowed with a standard edge-cone spherical metric of cone angle greater than or equal to \(4\pi \), along a great circle of codimension two. When the cone angle along the singularity is smaller than \(2\pi \), the corresponding metric is known to be a Yamabe metric, and we show that all Yamabe metrics in its conformal class are obtained from it by constant multiples and conformal diffeomorphisms preserving the singular set.

Foliated corona decompositions

We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in $\mathbb{H}$. In previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces. Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and Lafforgue-Naor (2012), they have several noteworthy implications, including the fact that the $L_1$ distortion of a word-ball of radius $n\ge 2$ in the discrete $3$-dimensional Heisenberg group is bounded above and below by universal constant multiples of $\sqrt[4]{\log n}$; this is in contrast to higher dimensional Heisenberg groups, where our previous work showed that the distortion of a word-ball of radius $n\ge 2$ is of order $\sqrt{\log n}$.

ChemSpectra: a web-based spectra editor for analytical data

ChemSpectra, a web-based software to visualize and analyze spectroscopic data, integrating solutions for infrared spectroscopy (IR), mass spectrometry (MS), and one-dimensional 1H and 13C NMR (proton and carbon nuclear magnetic resonance) spectroscopy, is described. ChemSpectra serves as web-based tool for the analysis of the most often used types of one-dimensional spectroscopic data in synthetic (organic) chemistry research. It was developed to support in particular processes for the use of open file formats which enable the work according to the FAIR data principles. The software can deal with the open file formats JCAMP-DX (IR, MS, NMR) and mzML (MS) proposing these data file types to gain interoperable data. ChemSpectra can be extended to read also other formats as exemplified by selected proprietary mass spectrometry data files of type RAW and NMR spectra files of type FID. The JavaScript-based editor can be integrated with other software, as demonstrated by integration into the Chemotion electronic lab notebook (ELN) and Chemotion repository, demonstrating the implementation into a digital work environment that offers additional functionality and sustainable research data management options. ChemSpectra supports different functions for working with spectroscopic data such as zoom functions, peak picking and automatic peak detection according to a default or manually defined threshold. NMR specific functions include the definition of a reference signal, the integration of signals, coupling constant calculation and multiplicity assignment. Embedded into a web application such as an ELN or a repository, the editor can also be used to generate an association of spectra to a sample and a file management. The file management supports the storage of the original spectra along with the last edited version and an automatically generated image of the spectra in png format. To maximize the benefit of the spectra editor for e.g. ELN users, an automated procedure for the transfer of the detected or manually chosen signals to the ELN was implemented. ChemSpectra is released under the AGPL license to encourage its re-use and further developments by the community.

Compact Dupin hypersurfaces

A hypersurface $M$ in ${\bf R}^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is constant on $M$, i.e., each continuous principal curvature function has constant multiplicity on $M$. These conditions are preserved by stereographic projection, so this theory is essentially the same for hypersurfaces in ${\bf R}^n$ or $S^n$. The theory of compact proper Dupin hypersurfaces in $S^n$ is closely related to the theory of isoparametric hypersurfaces in $S^n$, and many important results in this field concern relations between these two classes of hypersurfaces. In 1985, Cecil and Ryan conjectured on p. 184 of the book, Tight and Taut Immersions of Manifolds, that every compact, connected proper Dupin hypersurface $M \subset S^n$ is equivalent to an isoparametric hypersurface in $S^n$ by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments.

A Novel n-Decimal Reversible Radix Binary-Coded Decimal Multiplier Using Radix Encoding Scheme

Reversible logic has been emerging as a replacement for conventional digital computers and has been found to be a promising technology for the future that can improve the quality of circuits in terms of power, speed, heat dissipation, life span, and input traceability. The new generation of quantum processors demands the efficient implementation of decimal data path elements in many commercial, financial, and Internet-based applications. Reversible binary-coded decimal multipliers are among the important circuits in quantum computers. In this approach, we propose a low-power reversible radix binary-coded decimal multiplier. The proposed methodology uses a novel 4221 reversible recorder gate to select precomputed constant multiples and binary to excess-six conversion reversible decimal adder for partial product compression. The proposed multiplier is designed with 180-nm application-specific integrated circuit technology using the Cadence EDA tool and is compared with state-of-the-art BCD algorithms designed using reversible gates. Experimental evaluations revealed that the proposed design demonstrates power and power-delay product reductions of 28% and 37%, respectively, compared to those of the best BCD algorithms implemented in reversible logic.

Construction of Weights for Positive Integral Operators

Let ( X , M , μ ) be a σ -finite measure space and denote by P ( X ) the μ -measurable functions f : X → [ 0 , ∞ ] , f &lt; ∞ μ ae. Suppose K : X × X → [ 0 , ∞ ) is μ × μ -measurable and define the mutually transposed operators T and T ′ on P ( X ) by ( T f ) ( x ) = ∫ X K ( x , y ) f ( y ) d μ ( y ) and ( T ′ g ) ( y ) = ∫ X K ( x , y ) g ( x ) d μ ( x ) , f , g ∈ P ( X ) , x , y ∈ X . Our interest is in inequalities involving a fixed (weight) function w ∈ P ( X ) and an index p ∈ ( 1 , ∞ ) such that: (*): ∫ X [ w ( x ) ( T f ) ( x ) ] p d μ ( x ) ≲ C ∫ X [ w ( y ) f ( y ) ] p d μ ( y ) . The constant C &gt; 1 is to be independent of f ∈ P ( X ) . We wish to construct all w for which (*) holds. Considerations concerning Schur’s Lemma ensure that every such w is within constant multiples of expressions of the form ϕ 1 1 / p − 1 ϕ 2 1 / p , where ϕ 1 , ϕ 2 ∈ P ( X ) satisfy T ϕ 1 ≤ C 1 ϕ 1 and T ′ ϕ 2 ≤ C 2 ϕ 2 . Our fundamental result shows that the ϕ 1 and ϕ 2 above are within constant multiples of (**): ψ 1 + ∑ j = 1 ∞ E − j T ( j ) ψ 1 and ψ 2 + ∑ j = 1 ∞ E − j T ′ ( j ) ψ 2 respectively; here ψ 1 , ψ 2 ∈ P ( X ) , E &gt; 1 and T ( j ) , T ′ ( j ) are the jth iterates of T and T ′ . This result is explored in the context of Poisson, Bessel and Gauss–Weierstrass means and of Hardy averaging operators. All but the Hardy averaging operators are defined through symmetric kernels K ( x , y ) = K ( y , x ) , so that T ′ = T . This means that only the first series in (**) needs to be studied.

Positive-definite ternary quadratic forms with the same representations over ℤ

Kaplansky conjectured that if two positive-definite ternary quadratic forms have perfectly identical representations over [Formula: see text], they are equivalent over [Formula: see text] or constant multiples of regular forms, or is included in either of two families parameterized by [Formula: see text]. Our results aim to clarify the limitations imposed to such a pair by computational and theoretical approaches. First, the result of an exhaustive search for such pairs of integral quadratic forms is presented in order to provide a concrete version of the Kaplansky conjecture. The obtained list contains a small number of non-regular forms that were confirmed to have the identical representations up to 3,000,000 by computation. However, a strong limitation on the existence of such pairs is still observed, regardless of whether the coefficient field is [Formula: see text] or [Formula: see text]. Second, we prove that if two pairs of ternary quadratic forms have the identical simultaneous representations over [Formula: see text], their constant multiples are equivalent over [Formula: see text]. This was motivated by the question why the other families were not detected in the search. In the proof, the parametrization of quartic rings and their resolvent rings by Bhargava is used to discuss pairs of ternary quadratic forms.

A characterization of the exponential symmetric Sheffer sequences

ABSTRACT A polynomial sequence is symmetric, or self-dual, if for all . In this paper, we give the characterization of the exponential symmetric Sheffer sequences with weight sequences, and show that all the symmetric Sheffer sequences with weight , or are in fact constant multiples of the Charlier polynomial sequence, the Meixner polynomial sequence, or the Krawtchouk polynomial sequence, respectively, where are the rising factorials, and are the falling factorials.

Generalized Bessel functions of dihedral-type: expression as a series of confluent Horn functions and Laplace-type integral representation

In the first part of this paper, we express the generalized Bessel function associated with dihedral systems and a constant multiplicity function as an infinite series of confluent Horn functions. The key ingredient leading to this expression is an extension of an identity involving Gegenbauer polynomials proved in a previous paper by the authors, together with the use of the Poisson kernel for these polynomials. In particular, we derive an integral representation of this generalized Bessel function over the standard simplex. The second part of this paper is concerned with even dihedral systems and boundary values of one of the variables. Still assuming that the multiplicity function is constant, we obtain a Laplace-type integral representation of the corresponding generalized Bessel function, which extends to all even dihedral systems a special instance of the Laplace-type integral representation proved in Amri and Demni (Moscow Math J 17(2):1–15, 2017).

Variational Problems in Conformal Geometry

We study the Euler-Lagrange equation for several natural functionals defined on a conformal class of almost Hermitian metrics, whose expression involves the Lee form $\theta$ of the metric. We show that the Gauduchon metrics are the unique extremal metrics of the functional corresponding to the norm of the codifferential of the Lee form. We prove that on compact complex surfaces, in every conformal class there exists a unique metric, up to multiplication by a constant, which is extremal for the functional given by the $L^2$-norm of $dJ\theta$, where $J$ denotes the complex structure. These extremal metrics are not the Gauduchon metrics in general, hence we extend their definition to any dimension and show that they give unique representatives, up to constant multiples, of any conformal class of almost Hermitian metrics.

Four-manifolds with harmonic 2-forms of constant length

It was shown by Seaman that if a compact, connected, oriented, riemannian 4-manifold (M, g) of positive sectional curvature admits a harmonic 2-form of constant length, then M has definite intersection form and such a harmonic form is unique up to constant multiples. In this paper, we show that such a manifold is diffeomorphic to $$\mathbb {CP}_{2}$$ with a slightly weaker curvature hypothesis and there is an infinite dimensional moduli space of such metrics near the Fubini-Study metric on $$\mathbb {CP}_{2}$$ .

Freeness of multi-reflection arrangements via primitive vector fields

In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. We then extend our results further to all imprimitive irreducible unitary reflection groups. In this case the exponents turn out to depend on the exponents of a certain Galois twist of the dual reflection representation that comes from a Beynon-Lusztig type semi-palindromicity of the fake degrees.

Note on the mean values of derivatives of quadratic Dirichlet L-functions in function fields

We study the mean values of the first and the second derivative of quadratic Dirichlet L-functions L(s,χD) over the rational function field. We show that the moments of first derivatives L′(12,χD) are just constant multiples of the moments of L(12,χD). For the second derivatives, we improve the error term by q12(1+ϵ) and show that there is an extra term of size g3q2g+13 in the asymptotic formula of Andrade and Rajagopal for the first moment of L″(12,χD).

Structure of eigenvectors of random regular digraphs

Let $n$ be a large integer, let $d$ satisfy $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in {\mathcal C}$. Further, denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this paper, we study structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and sloping with many levels. As a corollary of this theorem, we show that every eigenvector of $M$, except for constant multiples of $(1,1,\dots,1)$, possesses a weak delocalization property: all subsets of coordinates with the same value have cardinality less than $Cn\ln^2 d/\ln n$. For a large constant $d$ this provides a principally new structural information on eigenvectors, implying that the number of their level sets grows to infinity with $n$, and -- combined with additional arguments -- that the rank of the adjacency matrix $M$ is at least $n-1$, with probability going to 1 as $n\to\infty$. Moreover, our main result for $d$ slowly growing to infinity with $n$ contributes a crucial step towards establishing the circular law for the empirical spectral distribution of sparse random $d$-regular directed graphs. As a key technical device we introduce a decomposition of ${\mathcal C}^n$ into vectors of different degree of structuredness, which is an alternative to the decomposition based on the least common denominator in the regime when the underlying random matrix is very sparse.

Rational growth and degree of commutativity of graph products

Let G be an infinite group and let X be a finite generating set for G such that the growth series of G with respect to X is a rational function; in this case G is said to have rational growth with respect to X. In this paper a result on sizes of spheres (or balls) in the Cayley graph Γ(G,X) is obtained: namely, the size of the sphere of radius n is bounded above and below by positive constant multiples of nαλn for some integer α≥0 and some λ≥1.As an application of this result, a calculation of degree of commutativity (d. c.) is provided: for a finite group F, its d. c. is defined as the probability that two randomly chosen elements in F commute, and Antolín, Martino and Ventura have recently generalised this concept to all finitely generated groups. It has been conjectured that the d. c. of a group G of exponential growth is zero. This paper verifies the conjecture (for certain generating sets) when G is a right-angled Artin group or, more generally, a graph product of groups of rational growth in which centralisers of non-trivial elements are “uniformly small”.

具常重特征的拟线性双曲组的经典解的奇性形成

具常重特征的拟线性双曲组的经典解的奇性形成