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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, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.

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 providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.

This is a chiefly expository paper on the subject of the title which, in our view, has not received a detailed treatment in the literature which is commensurate with its importance. We expect the results presented here to be useful in a number of contexts. For example, several of them will be applied in a forthcoming paper by the authors., Comment: 25 pages

Completely multiplicative functions whose sum is zero ($CMO$). The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors. The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem~3). The same idea applies also in a more general context (theorem~4). After general statements of that sort, including examples of $CMO$ (theorem~5), the paper is devoted to ``small'' functions, that is, functions of the form $\frac{f(n)}{n}$, where the $f(n)$ are bounded. Here is a typical result : if $|f(p)|\le 1$ and $Re\, f(p)\le0$ for all $p$, a necessary and sufficient condition for $\big(\frac{f(n)}{n}\big)$ to be $CMO$ is $\sum \, Re\, f(p)/p=-\infty$ (theorem~8). Another necessary and sufficient condition is given under the assumption that $|1+f(p)|\le 1$ and $f(2)\not=-2$ (theorem~7). A third result gives only a sufficient condition (theorem~9). The three results apply to the particular case $f(p)=-1$, the historical example of Euler. Theorems 7 and 8 need auxiliary results, coming either from the existing literature (Hal\'asz, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Ska\l ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the multiplicative convolution (theorems~10~and~11).

The year 2017 marks the 80th anniversary of Witt’s famous paper containing key results, including the Witt cancellation theorem, which form the foundation for the algebraic theory of quadratic forms. We pay homage to this paper by presenting a transparent and algebraic proof of the Witt cancellation theorem, which itself is based on a cancellation. We also present an overview of some recent spectacular work which is still building on Witt’s original creation of the algebraic theory of quadratic forms.

In this expository paper, we give a complete proof of van den Essen’s theorem that the de Rham cohomology spaces of a holonomic D -module are finite-dimensional in the case of a formal power series ring over a field of characteristic zero. This proof requires results from at least five of van den Essen’s papers as well as his unpublished thesis, and until now has not been available in a self-contained document.

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

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., 19 pages

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 x 1 m + x 2 m + ⋯ + x n m with each x j in C and some n ≤ 6 m . Furthermore, every real number x in the interval [ 0 , 8 ] can be written as x = x 1 3 + x 2 3 + ⋯ + x 8 3 , the sum of eight cubic powers with each x j in C . Another Cantor set C × C is also considered. More specifically, when C × C is embedded into the complex plane ℂ , the Waring–Hilbert problem on C × C has a positive answer for powers less than or equal to 4.

In this expository paper, we discuss several types of independence of points in the Euclidean space R d 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.

In this partly expository paper we give two applications of ideas from dynamical systems to the study of the injectivity properties of a polynomial local diffeomorphism F = ( F 1 , F 2 ) : R 2 → R 2 (by the work of Pinchuk, these maps need not be globally injective). I) The Jacobian conjecture claims that all polynomial local biholomorphisms G = ( G 1 , G 2 ) : ℂ 2 → ℂ 2 must be injective. By the Abhyankar–Moh theory in algebraic geometry, G is injective if G 1 has one place at infinity. We prove that this result carries over to the real case, in the following form. It is shown that F is injective if the (possibly singular) complex curve C of { F 1 = 0 } is irreducible, its projectivization C ˜ has only one point at infinity, and the said point is covered only once by a desingularization ℛ → C ˜ . II) In our second application we show that every polynomial local diffeomorphism of R 2 into itself must be injective at least on certain large regions that contain sequences of disjoint discs of arbitrarily large radii.

In this paper, we present two applications of the theory of singular connections developed by Harvey and Lawson (1993). The first one is a version of the Lelong–Poincare formula with estimates for sections of vector bundles over an almost complex manifold. The second one is a convergence theorem for divisors associated to a general family of symplectic submanifolds constructed by Donaldson (1996) (the case of hypersurfaces) and by Auroux in (1997) (for arbitrary dimensional submanifolds).

Following his discovery that finite metric spaces have injective envelopes naturally admitting a polyhedral structure, Isbell, in his pioneering work on injective metric spaces, attempted a characterization of cellular complexes admitting the structure of an injective metric space. A bit later, Mai and Tang confirmed Isbell's conjecture that a simplicial complex is injectively metrizable if and only if it is collapsible. Considerable advances in the understanding, classification and applications of injective envelopes have since been made by Dress, Huber, Sturmfels and collaborators, and most recently by Lang. Unfortunately a combination theory for injective polyhedra is still unavailable. Here we expose a connection to the duality theory of cubings -- simply connected non-positively curved cubical complexes -- which provides a more principled and accessible approach to Mai and Tang's result, providing one with a powerful tool for systematic construction of locally-compact injective metric spaces: Main Theorem. Any complete pointed Gromov--Hausdorff limit of locally-finite piecewise-$\ell_\infty$ cubings is injective. This result may be construed as a combination theorem for the simplest injective polytopes, $\ell_\infty$-parallelopipeds, where the condition for retaining injectivity is the combinatorial non-positive curvature condition on the complex. Thus it represents a first step towards a more comprehensive combination theory for injective spaces. In addition to setting the earlier work on injectively metrizable complexes within its proper context of non-positively curved geometry, this paper is meant to provide the reader with a systematic review of the results~ ---~ otherwise scattered throughout the geometric group theory literature~ ---~ on the duality theory and the geometry of cubings, which make this connection possible., Significantly expanded and revised. 40 pages, 6 figures, to appear in Expositiones Mathematicae (electronic https://doi.org/10.1016/j.exmath.2018.06.001)

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. To complete and clarify the whole picture, we show which known concepts are equivalent, and add also a new type of curvature. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behavior of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves.

For a prime p, we call a positive integer n a Frobenius p-number if there exists a finite group with exactly n subgroups of order p^a for some $a\ge 0$. Extending previous results on Sylow's theorem, we prove in this paper that every Frobenius p-number $n\equiv 1\pmod{p^2}$ is a Sylow p-number, i.e., the number of Sylow p-subgroups of some finite group. As a consequence, we verify that 46 is a pseudo Frobenius 3-number, that is, no finite group has exactly 46 subgroups of order 3^a for any $a\ge 0$., Comment: 6 pages, expository, to appear in Expo. Math

In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliation in dimension two and their invariant curves. It appears a characterization of second type foliations and generalized curve foliations as well as a description of the G S V -index in terms of polar curves. We also interpret the proofs concerning the Poincare problem with polar invariants.

The metric d ( A , B ) = tr A + tr B − 2 tr ( A 1 ∕ 2 B A 1 ∕ 2 ) 1 ∕ 2 1 ∕ 2 on the manifold of n × n positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several, positive definite matrices with respect to this metric. We explain some recent work on a fixed point iteration for computing this Wasserstein barycentre. Our emphasis is on ideas natural to matrix analysis.

This is an expository paper surveying briefly the evolution of the foundations of quantum theory from its inception to the present day. In particular, it looks at the impact of group theoretic ideas on the foundations.

Composite type functional equations in several variables play a significant role in various branches of mathematics and they have several interesting applications. Therefore their stability properties are of interest. The aim of this paper is to present a survey of some results and methods concerning stability of several of such equations; especially those somehow connected to the well known Goląb–Schinzel equation.

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator B n taking a continuous function f ∈ C [ 0 , 1 ] to a degree- n polynomial when the number of iterations k tends to infinity and n is kept fixed or when n tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright–Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright–Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of B n a number of times k = k ( n ) to a polynomial f when k ( n ) ∕ n tends to a constant.

This note geometrically generalizes the main theorem in Shahgholian’s 1992 paper [2] , from R 3 to R n ≥ 2 via the Minkowski functional of a convex body with the origin in its interior.

Let α ∈ ( 0 , 1 ) , h ∈ R and let f α , h be an even function with the properties: f α , h ( x ) = 0 for x ∈ ( 1 , + ∞ ) , f α , h is linear over the intervals [ 0 , α ] and [ α , 1 ] , f α , h ( 0 ) = 1 , f α , h ( α ) = h , and f α , h ( 1 ) = 0 . In this paper we prove that f α , h is positive definite on R ⟺ m ( α ) ≤ h ≤ 1 − α , where m ( α ) = 0 if 1 / α ∉ N , and m ( α ) = − α if 1 / α ∈ N .

In this paper we consider certain dynamical local–global principle for Mordell–Weil type groups over number fields like S -units, abelian varieties with trivial ring of endomorphisms and odd algebraic K -theory groups.

In this paper we characterize the Birkhoff–James orthogonality for elements of K ( X ; Y ) . In this way we extend the Bhatia–Semrl theorem. As an application, we consider the approximate orthogonality preserving property. Moreover, we give a new characterization of inner product spaces.

An algebra A is said to be zero product determined if every bilinear map f from A × A into an arbitrary vector space X with the property that f ( x , y ) = 0 whenever x y = 0 is of the form f ( x , y ) = Φ ( x y ) for some linear map Φ : A → X . It is known, and easy to see, that an algebra generated by idempotents is zero product determined. The main new result of this partially expository paper states that for finite dimensional (unital) algebras the converse is also true. Thus, if such an algebra is zero product determined, then it is generated by idempotents.

A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G and is denoted by σ ( G ) . It is an interesting problem in group theory to determine integers n such that there exists a group G with σ ( G ) = n and n − 1 > 1 is not a power of a prime. It is known that there exist integers n > 2 , which are not covering numbers of a group. So far it has been shown that for n 27 the integers 7, 11, 19, 21, 22 and 25 are not covering numbers of a group. It is an open question if the set of integers with this property is finite or infinite. In this paper an idempotent quasigroup of order n > 2 with the property that every two distinct elements generate the entire quasigroup is obtained. With this it is shown that for any integer n > 2 there exists a loop whose covering number is n .

In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G 2 . Some of these results are new. We give self-contained proofs here. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. For the sake of completeness we decided to collect them here in a self-contained way to be easily accessible for future usage in calibrated geometry. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of G 2 manifolds with special properties, which appear in the first named author’s work with S. Salur about G 2 dualities.

The conic sections, as well as the solids obtained by revolving these curves, and many of their surprising properties, were already studied by Greek mathematicians since at least the fourth century B.C. Some of these properties come to the light, or are rediscovered, from time to time. In this paper we characterize the conic sections as the plane curves whose tangent lines cut off from a certain similar curve segments of constant area. We also characterize some quadrics as the surfaces whose tangent planes cut off from a certain similar surface compact sets of constant volume. Our work is developed in the most general multidimensional case., Comment: 23 pages, 9 figures. To appear in Expositiones Mathematicae

One of the purposes of this paper is to clarify the strong analogy between potential theory on the open unit disk and the homogeneous tree, to which we dedicate an introductory section. We then exemplify this analogy by a study of Riesz measures. Starting from interesting work by Favorov and Golinskii [A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators. Linear and complex analysis, pp. 37-47, Amer. Math. Soc. Transl. (2) 226, Amer. Math. Soc., Providence, RI, 2009], we consider subharmonic functions on the open unit disk, resp. on the homogenous tree. Supposing that we can control the way how those functions may tend to infinity at the boundary, we derive moment type conditions for the Riesz measures. One one hand, we generalise the previous results for the disk, and on the other hand, we show how to obtain analogous results in the discrete setting of the tree.

International audience; In this paper, we present a unified framework for our previous constructions of martingales with the same one-dimensional marginals as particular cases of processes increasing in the convex order. This framework encompasses our former uses of Levy sheets, Sato sheets and self-decomposable laws. New examples of processes increasing in the convex order are also exhibited, but we do not know how to associate to them martingales with the same one-dimensional marginals.