Let Pn be the vector space of polynomials of degree at most n, equipped with the inner product 〈f,g〉=∫−11f(x)g(x)dx. Let D:Pn⟶Pn−1 be the differentiation operator and I:Pn−1⟶Pn be the integration operator. In this paper the generalized (Moore–Penrose) inverses of D and I are computed. The answer for D is straightforward, though perhaps somewhat unexpected. The answer for I requires more work, and involves

Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) f′′+P(z)f=0, where P(z) is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration

We investigate various groupoids associated to an arbitrary inverse semigroup with zero. We show that the groupoid of filters with respect to the natural partial order is isomorphic to the groupoid of germs arising from the standard action of the inverse semigroup on the space of idempotent filters. We also investigate the restriction of this isomorphism to the groupoid of tight filters and to the

A subset X of a finite group G is said to be prime-power-independent if each element in X has prime power order and there is no proper subset Y of X with 〈Y,Φ(G)〉=〈X,Φ(G)〉, where Φ(G) is the Frattini subgroup of G. A group G is Bpp if all prime-power-independent generating sets for G have the same cardinality. We prove that, if G is Bpp, then G is solvable. Pivoting on some recent results of Krempa

We compare two notions of G-fiber bundles and G-principal bundles in the literature, with an aim to clarify early results in equivariant bundle theory that are needed in current work of equivariant algebraic topology. We also give proofs of some equivariant generalizations of well-known non-equivariant results involving the classifying space.

In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More concretely, it satisfies several simple separated variables ODE, a first order generalized Abel ODE of degree n−1 and an (n−1)-th order linear ODE. Although some

Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set C. It is shown that, for each m≥3, every real number in the unit interval [0,1] is the sum x1m+x2m+⋯+xnm with each xj in C and some n≤6m. Furthermore, every real number x in the interval [0,8] can be written as x=x13+x23+⋯+x83, the sum of eight cubic powers with each xj in C. Another

We discuss several results in electrostatics: Onsager’s inequality, an extension of Earnshaw’s theorem, and a result stemming from the celebrated conjecture of Maxwell on the number of points of electrostatic equilibrium. Whenever possible, we try to provide a brief historical context and references.

This is an elementary introduction to the representation theory of finite semigroups. We illustrate the Clifford–Munn correspondence between the representations of a semigroup and the representations of its maximal subgroups. The emphasis throughout is on naturally occurring examples.

The networks of this – primarily (but not exclusively) expository – compendium are strongly connected, finite directed graphs X, where each oriented edge (x,y) is equipped with a positive weight (conductance) a(x,y). We are not assuming symmetry of this function, and in general we do not require that along with (x,y), also (y,x) is an edge. The weights give rise to a difference operator, the normalised

We generalize Moore’s nonstandard proof of the Spectral theorem for bounded self-adjoint operators to the case of unbounded operators. The key step is to use a definition of the nonstandard hull of an internally bounded self-adjoint operator due to Raab.

We discuss the geography problem of closed oriented 4-manifolds that admit a Riemannian metric of positive scalar curvature, and use it to survey mathematical work employed to address Gromov’s observation that manifolds with positive scalar curvature tend to be inessential by focusing on the four-dimensional case. We also point out an strengthening of a result of Carr and its extension to the non-orientable

Chase’s lemma provides a powerful tool for translating properties of (co)products in abelian categories into chain conditions. This note discusses the context in which the lemma is used, making explicit what is often neglected in the literature because of its technical nature.

The Chevalley-Warning Theorem is a result on the solution set of a system of polynomial equations f1,…,fr in n variables over a finite field Fq in the low degree case d≔∑j=1rdeg(fj)

For a given finite commutative ring R with 1≠0, one may associate a graph which is called the total graph of R and it is denoted by T(R). This graph has R as the vertex set and its two distinct vertices x and y are adjacent exactly whenever x+y is a zero-divisor of R. In this note, we prove that T(R) is well-covered if and only if either R is local or 2 is a zero-divisor.

This is a survey on some results obtained recently in the classification of compact quantum groups associated to partitions, with a focus on the non-crossing case. We take a global look at the main results in the subject and highlight some key features of the methods used. We conclude by several suggestions for pushing further the classification.

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist

Although the Chabauty compactification of parahoric subgroups is well studied in a general setting, for the particular case of SL(n,Qp) we give a different and more geometric proof of some of the results in Guivarc’h and Rémy (2006) using the Levi decompositions of various parabolic subgroups of SL(n,Qp).

The Morse-Sard theorem gives conditions under which the set of critical values of a function between Euclidean spaces has Lebesgue measure zero. Over the years the result has been extended and strengthened in various ways. We present a result, along with a simple proof, that subsumes many of these generalizations. We also review methods of constructing examples showing that differentiability hypotheses

In this paper we recast the Serret theorem about a characterization of palindromic continued fractions in the context of polynomial continued fractions. Then, using the relation between symmetric tridiagonal matrices and polynomial continued fractions we give a quick exposition of the mathematical aspect of the perfect quantum state transfer problem.

The study of operator algebras on Hilbert spaces, and C∗-algebras in particular, is one of the most active areas within Functional Analysis. A natural generalization of these is to replace Hilbert spaces (which are L2-spaces) with Lp-spaces, for p∈[1,∞). The study of such algebras of operators is notoriously more challenging, due to the very complicated geometry of Lp-spaces by comparison with Hilbert

In a recent paper, the authors proved that no spin foliation on a compact enlargeable manifold with Hausdorff homotopy graph admits a metric of positive scalar curvature on its leaves. This result extends groundbreaking results of Lichnerowicz, Gromov and Lawson, and Connes on the non-existence of metrics of positive scalar curvature. In this paper we review in more detail the material needed for the

We give a new proof establishing the automorphism groups of the symmetric groups inspired by the analogous result of Ivanov for the extended mapping class group. As a key tool, we consider the actions on the Kneser graphs.

We provide an explicit description of the Poincaré duals of each generator of the rational cohomology ring of the SU(2) character variety for a genus g surface with central extension — equivalently, that of the moduli space of stable holomorphic bundles of rank 2 and odd degree.

We discuss basic properties of holomorphic functions of one complex variable from the point of view of the theory of several complex variables.

We review basic facts on the structure of nearly Kähler manifolds, focussing in particular on the six-dimensional case. A self-contained proof that nearly Kähler six-manifolds are Einstein is given by combining different known results. We finally rephrase the definition of nearly Kähler six-manifold in terms of a pair of partial differential equations.

Since solitary subgroups of (infinite) Abelian groups are precisely the strictly invariant subgroups which are co-Hopfian (as groups), and strictly invariant subgroups turn out to be strongly invariant for large classes of Abelian groups we determine the solitary subgroups for these classes of groups.

In this paper, we generalize the arithmetic functions τ and τe in algebraic number fields and we find some properties of these functions.

We give a new purely algebraic proof of the Baker–Campbell–Hausdorff theorem, which states that the homogeneous components of the formal expansion of log(eAeB) are Lie polynomials. Our proof is based on a recurrence formula for these components and a lemma that states that if under certain conditions a commutator of a non-commuting variable and a given polynomial is a Lie polynomial, then the given

von Neumann’s inequality in matrix theory refers to the fact that the Frobenius scalar product of two matrices is less than or equal to the scalar product of the respective singular values. Moreover, equality can only happen if the two matrices share a joint set of singular vectors, and this latter part is hard to find in the literature. We extend these facts to the separable Hilbert space setting

In this expository paper, we discuss several types of independence of points in the Euclidean space Rd such as algebraic independence, strong independence, weak independence, and give a detailed geometric proof of their interrelationship. We also give yet another proof of Reay’s theorem determining the dimension of a Tverberg set.

Based on the assumption that time evolves only in one direction and mechanical systems can be described by Lagrangeans, a dynamical C*-algebra is presented for non-relativistic particles at atomic scales. Without presupposing any quantization scheme, this algebra is inherently non-commutative and comprises a large set of dynamics. In contrast to other approaches, the generating elements of the algebra

We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the 2+1 Levin–Wen model to obtain 1+1 anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as

In the qualitative theory of differential equations in the plane one of the most difficult objects to study is the existence of limit cycles. There are many papers dedicated to this subject. Here we will present a survey mainly dedicated to the algebraic and explicit non-algebraic limit cycles of the polynomial differential systems in R2 and of the discontinuous piecewise differential systems in R2

The nth cyclotomic polynomial Φn(x) is the minimal polynomial of an nth primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers n. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their

We give a new proof of a theorem of Weyl on the continuous part of the spectrum of Sturm–Liouville operators on the half-line with asymptotically constant coefficients. Earlier arguments, due to Weyl and Kodaira, depended on particular features of Green’s functions for linear ordinary differential operators. We use a concept of asymptotic containment of C∗-algebra representations that has geometric

We focus on the distribution of the extinction times of a population whose size N(t) follows a Feller diffusion process with inhomogeneous growth term r(t). Obtaining a precise description of the extinction times and of their dependence with respect to r(t) has important applications in adaptive biology, for understanding “evolutionary rescue” phenomena. A formula for the distribution of the extinction

The Principles of Quantum Mechanics and of Classical General Relativity indicate that Spacetime in the small (Planck scale) ought to be described by a noncommutative C* Algebra, implementing spacetime uncertainty relations. A model C* algebra of Quantum Spacetime and its Quantum Geometry is described. Interacting Quantum Field Theory on such a background is discussed, with open problems and recent

The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is proven that this idempotent is full if and only if the H-depth is 1 (H-separable extension). Similarly, a split extension has a bimodule projection; this idempotent

Classically, one could imagine a completely static space, thus without time. As is known, this picture is unconceivable in quantum physics due to vacuum fluctuations. The fundamental difference between the two frameworks is that classical physics is commutative (simultaneous observables) while quantum physics is intrinsically noncommutative (Heisenberg uncertainty relations). In this sense, we may

For the free probability analogue of Euclidean space endowed with the Gaussian measure we apply the approach of Arnold to derive Euler equations for a Lie algebra of non-commutative vector fields which preserve a certain trace. We extend the equations to vector fields satisfying non-commutative smoothness requirements. We introduce a cyclic vorticity and show that it satisfies vorticity equations and

In the high-energy quantum-physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as compact quantum metric spaces, and convergence is with respect to a quantum Gromov–Hausdorff-type distance. More recently I have dealt with corresponding statements in the

We show that the identity is the sum of two commutators in the algebra of all operators affiliated with a von Neumann algebra of type II1, settling a question, in the negative, that had puzzled a number of us.

The original approach of Hutchinson to fractals considers the defining equation as a fixed point problem, and then applies the Banach Contraction Principle. To do this, the Blaschke Completeness Theorem is essential. Avoiding Blaschke’s result, this note presents an alternative way to fractals via the Kuratowski noncompactness measure. Moreover, our technique extends the existence part of Hutchinson’s

We completely characterize extreme contractions between two-dimensional strictly convex and smooth real Banach spaces, perhaps for the very first time. In order to obtain the desired characterization, we introduce the notions of (weakly) compatible point pair (CPP) and μ-compatible point pair (μ-CPP) in the geometry of Banach spaces. As a concrete application of our abstract results, we describe all

Let F be a field, and fix a q∈F. The q-deformed Heisenberg algebra H(q) is the unital associative algebra over F with generators A, B and a relation which asserts that AB−qBA is the multiplicative identity in H(q). We extend H(q) into an algebra R(q) defined by generators A, B and a relation which asserts that AB−qBA is central in R(q). We identify all elements of R(q) that are Lie polynomials in A

An introduction to Joyal’s theory of combinatorial species is given and through it an alternative view of Rota’s twelvefold way emerges.

We study fixed point properties of the automorphism group of the universal Coxeter group Aut(Wn). In particular, we prove that whenever Aut(Wn) acts by isometries on complete d-dimensional CAT(0) space with d<⌊n2⌋, then it must fix a point. We also prove that Aut(Wn) does not have Kazhdan’s property (T). Further, strong restrictions are obtained on homomorphisms of Aut(Wn) to groups that do not contain

In this paper we describe the subject of automorphism groups of domains in complex space. This has been an active area of research for fifty years or more, and continues to be dynamic and developing today. We discuss noncompact automorphism groups, the Bun Wong/Rosay theorem, the Greene/Krantz conjecture, semicontinuity of automorphism groups, the method of scaling, and other current topics. Contributions

We give an exposition of two fundamental results of the theory of crossed products. One of these states that every regular representation of a reduced crossed product is faithful whenever the underlying Hilbert space representation of the C∗-algebra that together with an automorphism group gives rise to the crossed product is faithful. The other result states that a full and a reduced crossed products

In this note we show that every Kirchberg algebra in the UCT class is the C∗-algebra of a Hausdorff, ample, amenable and locally contracting groupoid. The non-unital case follows from Spielberg’s graph-based models for Kirchberg algebras. Our contribution is the unital case and our proof builds on Spielberg’s construction.

Let S be a closed and oriented surface of genus g at least 2. In this (mostly expository) article, the object of study is the space P(S) of marked isomorphism classes of projective structures on S. We show that P(S), endowed with the canonical complex structure, carries exotic hermitian structures that extend the classical ones on the Teichmüller space T(S) of S. We shall notice also that the Kobayashi

Let S be a finitely generated semigroup of isometries of a compact Riemannian manifold. We present a simple argument to prove that, if there exists a point x whose orbit Sx is dense, then the orbit of any point is equidistributed.

We prove the quantitative equivalence of two important geometrical conditions, pertaining to the regularity of a domain Ω⊂RN. These are: (i) the uniform two-sided supporting sphere condition, and (ii) the Lipschitz continuity of the outward unit normal vector. In particular, the answer to the question posed in our title is: “Those domains whose unit normal is well defined and has Lipschitz constant

We provide proofs for the fact that certain orders have no infinite descending chains and no infinite antichains.

Fibrations over a category B, introduced to category theory by Grothendieck, encode pseudo-functors Bop⇝Cat, while the special case of discrete fibrations encodes presheaves Bop→Set. A two-sided discrete variation encodes functors Bop×A→Set, which are also known as profunctors from A to B. By work of Street, all of these fibration notions can be defined internally to an arbitrary 2-category or bicategory

The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure.

The Bateman–Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green–Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau’s conjecture. We discuss the Bateman–Horn conjecture, its applications, and its origins.

We study Birkhoff–James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the

Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite set S⊂P such that the statistics of the period of the continued fraction expansions along the sequence px:p∈S approach the ‘normal’ statistics given by the Gauss–Kuzmin measure. Under the generalized Riemann hypothesis, we prove that there exist full density subsets S⊂P and T⊂N satisfying the