Assume that p∈(1,∞] and u=Ph[ϕ], where ϕ∈Lp(Sn−1,Rn). Then for any x∈Bn, we obtain the sharp inequalities |u(x)|≤Cq1q(x)(1−|x|2)n−1p‖ϕ‖Lpand|u(x)|≤Cq1q(1−|x|2)n−1p‖ϕ‖Lp for some function Cq(x) and constant Cq in terms of Gauss hypergeometric and Gamma functions, where q is the conjugate of p. This result generalizes and extends some known results from harmonic mapping theory (Kalaj and Marković (2012

Backbone curves are usually obtained by analyzing nonlinear oscillators (e.g., the Duffing equation) whereas the backbones of linear oscillators are incidental by-products of such studies. Nevertheless, we focus on the harmonic oscillator, a linear, homogeneous constant-coefficient second-order ordinary differential equation. The backbone of the sinusoidally forced harmonic oscillator is well known

Unifying approaches by amongst others Archimedes, Kepler, Goldberg, Caspar and Klug, Coxeter, and Conway, and extending on a previous formalization of the concept of local symmetry preserving (lsp) operations, we introduce a formal definition of local operations on plane graphs that preserve orientation-preserving symmetries, but not necessarily orientation-reversing symmetries. This operations include

(2020). The Eightieth William Lowell Putnam Mathematical Competition. The American Mathematical Monthly: Vol. 127, No. 8, pp. 675-687.

A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse generalizing a classic problem about circles. We give a brief history of the circle problem an account of Price’s ellipse proof and a reorganized proof with some new ideas designed to situate the result within a dense web of connections to classical mathematics. It is inspired

In Conway’s game, Sylver Coinage, the set of legal plays forms the complement of a numerical semigroup after a finite number of turns. Our goal is to show how the tools and techniques of numerical semigroups can be brought to bear on questions related to Sylver Coinage. We begin by formally connecting the definitions and concepts related to the game of Sylver Coinage with those of numerical semigroups

(2020). Wald’s Identity and Geometric Expectation. The American Mathematical Monthly: Vol. 127, No. 8, pp. 716-716.

Zero divided by zero is arguably the single most important concept underlying calculus. For functions of more than one variable, methods of proof for indeterminate limits are not as familiar as for functions of a single variable. We present a l’Hôpital’s rule that provides a way to simplify and resolve a wide variety of zero-over-zero limits in terms of quotients of their derivatives.

We solve a very classical problem: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms of Yetter, or equivalently of the pseudofaces of McConnell. In particular, we derive expressions for the dihedral angles, face angles, and (an) edge length, the remaining parts being derivable by symmetry or

We give a simple proof of the Siebeck–Marden theorem. We use only basic facts of geometry and complex numbers.

We provide an elementary derivation of the Green function for Poisson’s equation with Neumann boundary data on balls of arbitrary dimension. Surprisingly, until very recently this Green function was only known in dimensions up to three, and an explicit construction (even in low dimensions) on the level of an undergraduate PDE class was lacking. This changes if one derives the Green function for Poisson’s

(2020). Null Subsets of All Sizes Inside Vitali Sets. The American Mathematical Monthly: Vol. 127, No. 8, pp. 743-743.

A numerical semigroup S is an additive subsemigroup of the nonnegative integers, containing zero, with finite complement. Its multiplicity m is its smallest nonzero element. The Apéry set of S is the set of elements Ap(S)={n∈S:n−m∉S}. Fixing a numerical semigroup, we ask how many elements of its Apéry set have nonunique factorization and define several new invariants.

Using the group consisting of the eight Möbius transformations x, – x, 1/x,−1/x, (x−1)/(x+1),(x+1)/(1−x), (x+1)/(x−1), and (1−x)/(x+1), we present an enumerative proof of the classical result for when the element 2 is a quadratic residue in the finite field Fq .

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by February 28, 2021, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions

(2020). A Formula for the Stirling Numbers of the Second Kind. The American Mathematical Monthly: Vol. 127, No. 8, pp. 762-762.

(2020). Discrete Morse Theory by Nicholas Scoville. The American Mathematical Monthly: Vol. 127, No. 8, pp. 763-768.

Abstract:The explicit upper Bellman function is found for the natural dyadic maximal operator acting from into . As a consequence, it is shown that the norm of the natural operator equals for all , and so does the norm of the classical dyadic maximal operator. The main result is a partial consequence of a theorem for the so-called -trees, which generalize dyadic lattices. The Bellman function in this

Abstract:The classical Grauert and Ramspott theorems constitute the foundation of the Oka principle on Stein spaces. In this paper, similar results are established on the maximal ideal space of the Banach algebra  of bounded holomorphic functions on the open unit disk . The results are illustrated by some examples and applications to the theory of operator-valued functions.

Abstract:Let be a continuous-path martingale and let be a stochastic integral, with respect to , of some predictable process with values in . We provide an explicit formula for Burkholder's function associated with the weighted bound

Abstract:We present several examples of compactifications of such that their remainders are nonseparable and carry strictly positive measures.

Abstract:Let be Banach spaces, and let be an -isometry with for some . In this paper, we show that for every , there exists with such that Moreover, the constant ``3'' is the best one in general.

Abstract:Let be a prime and let be a finite group. We prove that is -solvable of -length at most if there are at most two distinct -character degrees in the principal -block of . This generalizes a theorem of Isaacs-Smith as well as a recent result of three of the present authors.

Here, we shed light on the fractional linear and nonlinear Klein–Gorden partial differential equations via Fractional Shifted Legendre Tau Method. With this objective, the operational matrices of fractional-order shifted Legendre functions (FSLFs) are derived and combined with the Tau method to convert the fractional-order differential equations to a system of solvable algebraic equations. The validity

This paper aims to introduce the spacetimes admitting the Z-symmetric tensor. We investigate the properties of these spacetimes for a perfect fluid and a pressureless perfect fluid (a dust). Some theorems about the properties are proved.

In this paper, we establish the link between continuous lattices and closure spaces. By generalizing the notion of algebraic closure space to continuous closure space, we show that continuous lattices can be represented by continuous closure spaces, just as algebraic lattices can be represented by algebraic closure spaces. We also introduce the notion of approximable mappings between continuous closure

This paper aims to construct a new family of numbers and polynomials which are related to the Bell numbers and polynomials by means of the confluent hypergeometric function. We give various properties of these numbers and polynomials (generating functions, explicit formulas, integral representations, recurrence relations, probabilistic representation, …). We also derive some combinatorial sums including

The Collatz conjecture establishes that for every natural number n ∈ ℕ, there exists an r ∈ ℕ such that κr (n) = 1, where κ : ℕ → ℕ is the function defined as n/2 if n es even and as 3n + 1 if n is odd. The map κ induces a topology τκ on ℕ. We prove that the Collatz conjecture and connectedness of the space (ℕ, τκ ) imply that the space is simply connected. Furthermore, we prove that the Collatz conjecture

We first give a summary of the history of transcendental numbers, and then we use a nice technique by G. Dresden to find a new transcendental number. In particular, while previous work looked at the last non-zero digit of nn , we consider the digit immediately before its last non-zero digit and show that the infinite decimal built from these digits is transcendental.

Let R be a commutative ring and Xi.c (R) denotes the set of all integrally closed maximal subrings of R. It is shown that if R is a non-field G-domain, then there exists S ∈ Xi.c (R) with (S : R) = 0. If K is an algebraically closed field which is not absolutely algebraic, then we prove that the polynomial ring K[X] has an integrally closed maximal subring with zero conductor too; a characterization

In this work we prove that the Riesz tensor product of two archimedean d-algebras is again a d-algebra.

We provide an overview of technics that lead to an Euclidean upper bound on the volume of geodesic balls.

We extend the domain of applicability of the concept of (1, 𝛼)-derivations in 3-prime near-rings by analyzing the structure and commutativity of the near-rings admitting (1, 𝛼)-derivations satisfying certain differential identities.

We study the problem of approximation of functions (ψ, β)-differentiable (in the Stepanets sense) whose (ψ, β)-derivative belongs to the class H𝛼 by biharmonic Poisson integrals in the uniform metric.

In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell \(F_4\) type. We extend this simple closed-form expression

We show that for a large class of potential functions and big coupling constant \(\lambda \) the Schrödinger cocycle over the expanding map \(x\mapsto bx ~( \text{ mod } 1)\) on \(\mathbb {T}\) has a Lyapunov exponent \(>(\log \lambda )/4\) for all energies, provided that the integer \(b\ge \lambda ^3\).

In this paper, we obtain the transition probability formulas for the asymmetric simple exclusion process and the q-deformed totally asymmetric zero range process on the ring by applying the coordinate Bethe ansatz. We also compute the distribution function for a tagged particle with general initial condition.

We consider a family of strongly-asymmetric unimodal maps \(\{f_t\}_{t\in [0,1]}\) of the form \(f_t=t\cdot f\) where \(f:[0,1]\rightarrow [0,1]\) is unimodal, \(f(0)=f(1)=0\), \(f(c)=1\) is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } xc, \end{array}\right. \end{aligned}$$ where we assume that \(\beta >1\). We show that such a family contains

We study the limiting absorption principle and the well-posedness of Maxwell equations with anisotropic sign-changing coefficients in the time-harmonic domain. The starting point of the analysis is to obtain Cauchy problems associated with two Maxwell systems using a change of variables. We then derive a priori estimates for these Cauchy problems using two different approaches. The Fourier approach

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density \(\rho \in [0, 1]\) of the underlying SSEP. Our first

Cell crawling requires the generation of intracellular forces by the cytoskeleton and their transmission to an extracellular substrate through specific adhesion molecules. Crawling cells show many features of excitable systems, such as spontaneous symmetry breaking and crawling in the absence of external cues, and periodic and propagating waves of activity. Mechanical instabilities in the active cytoskeleton

We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups

Modern practice for training classification deepnets involves a terminal phase of training (TPT), which begins at the epoch where training error first vanishes. During TPT, the training error stays effectively zero, while training loss is pushed toward zero. Direct measurements of TPT, for three prototypical deepnet architectures and across seven canonical classification datasets, expose a pervasive

The deep structure of the continental collision between India and Asia and whether India’s lower crust is underplated beneath Tibet or subducted into the mantle remain controversial. It is also unknown whether the active normal faults that facilitate orogen-parallel extension of Tibetan upper crust continue into the lower crust and upper mantle. Our receiver-function images collected parallel to the

Scientists worldwide are racing to develop effective vaccines against severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the causative agent of the COVID-19 pandemic. An important and perhaps underappreciated aspect of this endeavor is ensuring that the vaccines being developed confer immunity to all viral lineages in the global population. Toward this end, a seminal study published in PNAS

The failure of polypeptides to achieve conformational maturation following biosynthesis can result in the formation of protein aggregates capable of disrupting essential cellular functions. In the secretory pathway, misfolded asparagine (N)-linked glycoproteins are selectively sorted for endoplasmic reticulum-associated degradation (ERAD) in response to the catalytic removal of terminal alpha-linked

The tropical Andes are an important natural laboratory to understand speciation in many taxa. Here we examined the evolutionary history of parasites of the Leishmania braziliensis species complex based on whole-genome sequencing of 67 isolates from 47 localities in Peru. We first show the origin of Andean Leishmania as a clade of near-clonal lineages that diverged from admixed Amazonian ancestors,

Major depressive disorder emerges from the complex interactions of biological systems that span genes and molecules through cells, networks, and behavior. Establishing how neurobiological processes coalesce to contribute to depression requires a multiscale approach, encompassing measures of brain structure and function as well as genetic and cell-specific transcriptional data. Here, we examine anatomical

Targeted treatments for advanced gastric cancer (GC) are needed, particularly for HER2-negative GC, which represents the majority of cases (80 to 88%). In this study, in silico analyses of the lysine histone demethylases (KDMs) involved in diverse biological processes and diseases revealed that PHD finger protein 8 (PHF8, KDM7B) was significantly associated with poor clinical outcome in HER2-negative

Human decisions can be biased by irrelevant information. For example, choices between two preferred alternatives can be swayed by a third option that is inferior or unavailable. Previous work has identified three classic biases, known as the attraction, similarity, and compromise effects, which arise during choices between economic alternatives defined by two attributes. However, the reliability, interrelationship

While near-cognate codons are frequently used for translation initiation in eukaryotes, their efficiencies are usually low (<10% compared to an AUG in optimal context). Here, we describe a rare case of highly efficient near-cognate initiation. A CUG triplet located in the 5′ leader of POLG messenger RNA (mRNA) initiates almost as efficiently (∼60 to 70%) as an AUG in optimal context. This CUG directs

Whereas the gill chambers of jawless vertebrates open directly into the environment, jawed vertebrates evolved skeletal appendages that drive oxygenated water unidirectionally over the gills. A major anatomical difference between the two jawed vertebrate lineages is the presence of a single large gill cover in bony fishes versus separate covers for each gill chamber in cartilaginous fishes. Here, we

In the high spin–orbit-coupled Sr2IrO4, the high sensitivity of the ground state to the details of the local lattice structure shows a large potential for the manipulation of the functional properties by inducing local lattice distortions. We use epitaxial strain to modify the Ir–O bond geometry in Sr2IrO4 and perform momentum-dependent resonant inelastic X-ray scattering (RIXS) at the metal and at

Midlatitude Asia (MLA), strongly influenced by westerlies-controlled climate, is a key source of global atmospheric dust, and plays a significant role in Earth’s climate system . However, it remains unclear how the westerlies, MLA aridity, and dust flux from this region evolved over time. Here, we report a unique high-resolution eolian dust record covering the past 3.6 Ma, retrieved from the thickest

There is an expectation that, on average, pain will increase with age, through accumulated injury, physical wear and tear, and an increasing burden of disease. Consistent with that expectation, pain rises with age into old age in other wealthy countries. However, in America today, the elderly report less pain than those in midlife. This is the mystery of American pain. Using multiple datasets and definitions

The oligoadenylate synthetase (OAS)–RNase L system is an IFN-inducible antiviral pathway activated by viral infection. Viral double-stranded (ds) RNA activates OAS isoforms that synthesize the second messenger 2-5A, which binds and activates the pseudokinase-endoribonuclease RNase L. In cells, OAS activation is tamped down by ADAR1, an adenosine deaminase that destabilizes dsRNA. Mutation of ADAR1

Influenza A virus (IAV) infection during pregnancy causes severe maternal and perinatal complications, despite a lack of vertical transmission of IAV across the placenta. Here, we demonstrate a significant alteration in the maternal vascular landscape that underpins the maternal and downstream fetal pathology to IAV infection in mice. In IAV infection of nonpregnant mice, the local lung inflammatory

Drought alters carbon (C) allocation within trees, thereby impairing tree growth. Recovery of root and leaf functioning and prioritized C supply to sink tissues after drought may compensate for drought-induced reduction of assimilation and growth. It remains unclear if C allocation to sink tissues during and following drought is controlled by altered sink metabolic activities or by the availability

It has proven difficult to identify the underlying genes in complex autoimmune diseases. Here, we use forward genetics to identify polymorphisms in the vitamin D receptor gene (Vdr) promoter, controlling Vdr expression and T cell activation. We isolated these polymorphisms in a congenic mouse line, allowing us to study the immunomodulatory properties of VDR in a physiological context. Congenic mice

Fishers with individual catch quota, but limited control over the mix of species caught, depend on trade and catch–quota balancing allowances to fully utilize their quota without discarding. However, these allowances can theoretically lead to overfishing if total allowable catches (TACs) are consistently exceeded. This study investigates usage of balancing allowances by the Icelandic demersal fleet