Featured Researches

Algebraic Topology

A stratified Kan-Quillen equivalence

We exhibit a Quillen equivalence between two model categories encoding the homotopy theory of stratified spaces : the model category of filtered simplicial sets, and that of filtered spaces. Additionally, we introduce a new class of filtered spaces, that of vertical filtered CW-complexes, providing a nice model for the homotopy category of stratified spaces.

Group Theory

The reverse decomposition of unipotents for bivectors

For the second fundamental representation of the general linear group over a commutative ring R we construct straightforward and uniform polynomial expressions of elementary generators as products of elementary conjugates of an arbitrary matrix and its inverse. Towards the solution we get stabilization theorems for any column of a matrix from G L ( n 2 ) (R) or from the exterior square of G L n (R) , n?? .

General Topology

The strong universality of ANRs with a suitable algebraic structure

Let M be an ANR space and X be a homotopy dense subspace in M . Assume that M admits a continuous binary operation ??M?M?�M such that for every x,y?�M the inclusion x?�y?�X holds if and only if x,y?�X . Assume also that there exist continuous unary operations u,v:M?�M such that x=u(x)?�v(x) for all x?�M . Given a 2 ? -stable ? 0 2 -hereditary weakly Σ 0 2 -additive class of spaces C , we prove that the pair (M,X) is strongly ( ? 0 1 ?�C,C) -universal if and only if for any compact space K?�C , subspace C?�C of K and nonempty open set U?�M there exists a continuous map f:K?�U such that f ?? [X]=C . This characterization is applied to detecting strongly universal Lawson semilattices.

Algebraic Geometry

Diagonal double Kodaira fibrations with minimal signature

We study some special systems of generators on finite groups, introduced in previous work by the first author and called "diagonal double Kodaira structures", in order to investigate non-abelian, finite quotients of the pure braid group on two strands P 2 ( Σ b ) , where Σ b is a closed Riemann surface of genus b . In particular, we prove that, if a finite group G admits a diagonal double Kodaira structure, then |G|??2 , and equality holds if and only if G is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two 3 -dimensional families of double Kodaira fibrations having signature 16 .

Spectral Theory

A lower bound on the spectral gap of one-dimensional Schrödinger operators

In this note we provide an explicit lower bound on the spectral gap of one-dimensional Schrödinger operators with non-negative bounded potentials and subject to Neumann boundary conditions.

Quantum Algebra

Q-W-algebras, Zhelobenko operators and a proof of De Concini-Kac-Procesi conjecture

This monograph, along with a self-consistent presentation of the theory of q-W-algebras including the construction of algebraic group analogues of Slodowy slices, contains a description of q-W-algebras in terms of Zhelobenko type operators introduced in the book. This description is applied to prove the De Concini-Kac-Procesi conjecture on the dimensions of irreducible modules over quantum groups at roots of unity.

Operator Algebras

On the entropy and index of the winding endomorphisms of Q p

For p?? , the p -adic ring C ??-algebra Q p is the universal C ??-algebra generated by a unitary U and an isometry S p such that S p U= U p S p and ??p?? l=0 U l S p S ??p U ?�l =1 . For any k coprime with p we define an endomorphism ? k ?�End( Q p ) by setting ? k (U):= U k and ? k ( S p ):= S p . We then compute the entropy of ? k , which turns out to be log|k| . Finally, for selected values of k we also compute the Watatani index of ? k showing that the entropy is the natural logarithm of the index.

Category Theory

Cartesian Fibrations of Complete Segal Spaces

Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in (??1) -category theory to study presheaves valued in (??1) -categories. In this work we define and study fibrations modeling presheaves valued in simplicial spaces and their localizations. This includes defining a model structure for these fibrations and giving effective tools to recognize its fibrations and weak equivalences. This in particular gives us a new method to construct Cartesian fibrations via complete Segal spaces. In addition to that, it allows us to define and study fibrations modeling presheaves of Segal spaces.

Differential Geometry

Pluripotential Monge-Amp{è}re flows in big cohomology class

We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact K{ä}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential K{ä}hler-Ricci flows on compact K{ä}hler manifolds of general type as well as on K{ä}hler varieties with semi-log canonical singularities.

K Theory and Homology

A note on relative Vaserstein symbol

In an unpublished work of Fasel-Rao-Swan the notion of the relative Witt group W E (R,I) is defined. In this article we will give the details of this construction. Then we studied the injectivity of the relative Vaserstein symbol V R,I :U m 3 (R,I)/ E 3 (R,I)??W E (R,I) . We established injectivity of this symbol if R is an affine non-singular algebra of dimension 3 over a perfect C 1 -field and I is a local complete intersection ideal of R . It is believed that for a 3 -dimensional affine algebra non-singularity is not necessary for establishing injectivity of the Vaserstein symbol . At the end of the article we will give an example of a singular 3 -dimensional algebra over a perfect C 1 -field for which the Vaserstein symbol is injective.

Commutative Algebra

On weakly 1 -absorbing prime ideals of commutative rings

Let R be a commutative ring with identity. In this paper, we introduce the concept of weakly 1 -absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal I of R is called weakly 1 -absorbing prime if for all nonunit elements a,b,c?�R such that 0?�abc?�I , then either ab?�I or c?�I . A number of results concerning weakly 1 -absorbing prime ideals and examples of weakly 1 -absorbing prime ideals are given. It is proved that if I is a weakly 1 -absorbing prime ideal of a ring R and 0??I 1 I 2 I 3 ?�I for some ideals I 1 , I 2 , I 3 of R such that I is free triple-zero with respect to I 1 I 2 I 3 , then I 1 I 2 ?�I or I 3 ?�I . Among other things, it is shown that if I is a weakly 1 -absorbing prime ideal of R that is not 1 -absorbing prime, then I 3 =0 . Moreover, weakly 1 -absorbing prime ideals of PID's and Dedekind domains are characterized. Finally, we investigate commutative rings with the property that all proper ideals are weakly 1 -absorbing primes.

General Mathematics

Repeated Sums and Binomial Coefficients

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial coefficient identities will be shown in order to prove this definition. Using this new definition, we simplify some particular sums such as the repeated Harmonic sum and the repeated Binomial-Harmonic sum. We derive formulae for simplifying general repeated sums as well as a variant containing binomial coefficients. Additionally, we study the m -th difference of a sequence and show how sequences whose m -th difference is constant can be related to binomial coefficients.

Representation Theory

The braid group action on exceptional sequences for weighted projective lines

We give a new and intrinsic proof of the transitivity of the braid group action on the set of full exceptional sequences of coherent sheaves on a weighted projective line. We do not use here the corresponding result of Crawley-Boevey for modules over hereditary algebras. As an application we prove that the strongest global dimension of the category of coherent sheaves on a weighted projective line $\XX$ does not depend on the parameters of $\XX$. Finally we prove that the determinant of the matrix obtained by taking the values of n $\ZZ$-linear functions defined on the Grothendieck group $K_0(\XX) \simeq \ZZ^n$ of the elements of a full exceptional sequence is an invariant, up to sign.

Metric Geometry

C 1,α -rectifiability in low codimension in Heisenberg groups

A natural notion of higher order rectifiability is introduced for subsets of Heisenberg groups H n in terms of covering a set almost everywhere by a countable union of ( C 1,α H ,H) -regular surfaces, for some 0<α?? . We prove that a sufficient condition for C 1,α -rectifiability of low-codimensional subsets in Heisenberg groups is the almost everywhere existence of suitable approximate tangent paraboloids.

Functional Analysis

Binet's convergent factorial series in the theory of the Gamma function

We investigate a generalization of Binet's factorial series in the parameter α μ(z)= ??m=1 ??b m (α) ??m?? k=0 (z+α+k) for the Binet function μ(z)=log?(z)??z??1 2)logz+z??1 2 log(2?) After a brief review of the Binet function μ(z) , several properties of the Binet polynomials b m (α) are presented. We compute the corresponding factorial series for the derivatives of the Binet function and apply those series to the digamma and polygamma functions. Finally, we compare Binet's generalized factorial series with Stirling's \emph{asymptotic} expansion and demonstrate by a numerical example that, with a same number of terms evaluated, the Binet generalized factorial series with an optimized value of α can beat the best possible accuracy of Stirling's expansion.

Number Theory

p-Adic distribution of CM points and Hecke orbits. II: Linnik equidistribution on the supersingular locus

For a prime number p , we study the asymptotic distribution of CM points on the moduli space of elliptic curves over C p . In stark contrast to the complex case, in the p -adic setting there are infinitely many different measures describing the asymptotic distribution of CM points. In this paper we identify all of these measures. A key insight is to translate this problem into a p -adic version of Linnik's classical problem on the asymptotic distribution of integer points on spheres. To do this translation, we use the close relationship between the deformation theories of elliptic curves and formal modules and then apply results of Gross and Hopkins. We solve this p -adic Linnik problem using a deviation estimate extracted from the bounds for the Fourier coefficients of cuspidal modular forms of Deligne, Iwaniec and Duke. We also identify all accumulation measures of an arbitrary Hecke orbit.

History and Overview

The Cognition of Counterexample in Mathematics Students

Studying Mathematics requires a synthesis of skills from a multitude of academic disciplines; logical reasoning being chief among them. This paper explores mathematical logical preparedness of students entering first year university mathematics courses and also the effectiveness of using logical facility to predict successful course outcomes. We analyze data collected from students enrolled at the University of Winnipeg in a pre-service course for high school teachers. We do find that, being able to successfully answer logical questions, both before and after intervention, are significant in relation to improved student outcomes.

Rings and Algebras

Determining when an algebra is an evolution algebra

Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper we obtain necessary and sufficient conditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalent to the so-called SDC problem , that is, the simultaneous diagonalisation via congruence of a given set of matrices. More precisely we show that an n -dimensional algebra A is an evolution algebra if, and only if, a certain set of n symmetric n?n matrices { M 1 ,?? M n } describing the product of A are SDC . We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intriguing as evolution algebras model asexual reproduction unlike the classical ones.

Combinatorics

Turán problems for k -geodetic digraphs

Turán-type problems have been widely investigated in the context of undirected simple graphs. Turán problems for paths and cycles in directed graphs have been treated by Bermond, Heydemann et al. Recently k -geodetic digraphs have received great attention in the study of extremal problems, particularly in a directed analogue of the degree/girth problem. Ustimenko et al studied the problem of the largest possible size of a diregular k -geodetic digraph with order n and gave asymptotic estimates. In this paper we consider the maximal size of a k -geodetic digraph with given order with no assumption of diregularity. We solve this problem for all k and classify the extremal digraphs. We then provide a partial solution for the more difficult problem of the largest number of arcs in a strongly-connected k -geodetic digraph with given order and provide lower bounds for these numbers that we conjecture to be extremal for sufficiently large n . We close with some results on generalised Turán problems for k -geodetic digraphs.

Dynamical Systems

A genus 4 origami with minimal hitting time and an intersection property

In a minimal flow, the hitting time is the exponent of the power law, as r goes to zero, for the time needed by orbits to become r-dense. We show that on the so-called Ornithorynque origami the hitting time of the flow in an irrational slope equals the diophantine type of the slope. We give a general criterion for such equality. In general, for genus at least two, hitting time is strictly bigger than diophantine type.

Classical Analysis and ODEs

The Newton Polyhedron and positivity of 2 F 3 hypergeometric functions

As for the 2 F 3 hypergeometric function of the form 2 F 3 [ a 1 , a 2 b 1 , b 2 , b 3 ????????x 2 ](x>0), where all of parameters are assumed to be positive, we give sufficient conditions on ( b 1 , b 2 , b 3 ) for its positivity in terms of Newton polyhedra with vertices consisting of permutations of ( a 2 , a 1 +1/2,2 a 1 ) or ( a 1 , a 2 +1/2,2 a 2 ). As an application, we obtain an extensive validity region of (α,λ,μ) for the inequality ??x 0 (x?�t ) λ t μ J α (t)dt??(x>0).

Analysis of PDEs

The spectral gap to torsion problem for some non-convex domains

In this paper we study the following torsion problem { ?�Δu=1 u=0 in Ω, on ?��? Let Ω??R 2 be a bounded, convex domain and u 0 (x) be the solution of above problem with its maximum y 0 ?��?. Steinerberger proved that there are universal constants c 1 , c 2 >0 satisfying λ max ( D 2 u 0 ( y 0 ))?��? c 1 exp(??c 2 diam(Ω) inrad(Ω) ). And he proposed following open problem: "Does above result hold true on domains that are not convex but merely simply connected or perhaps only bounded? The proof uses convexity of the domain Ω in a very essential way and it is not clear to us whether the statement remains valid in other settings." Here by some new idea involving the computations on Green's function, we compute the spectral gap λ max D 2 u( y 0 ) for some non-convex smooth bounded domains, which gives a negative answer to above open problem. Also some extensions are given.

Geometric Topology

Survey on L^2-invariants and 3-manifolds

In this paper give a survey about L^2-invariants focusing on 3-manifolds.

Numerical Analysis

Coupled Torsional and Transverse Vibration Analysis of Panels Partially Supported by Elastic Beam

This study presents torsional and transverse vibration analysis of a solar panel including a rectangular thin plate locally supported by an elastic beam. The plate is totally free in all boundaries, except for the local part attached to the beam. The response of the system, which is subjected to a combination of torsional and transverse vibration, identifies with a couple of PDEs developed by the Euler-Bernoulli assumption and classical plate theory. To calculate the system's natural frequencies, the domain of the solution is discretized by zeroes of the Chebyshev polynomials to apply the Modified Generalized Differential Quadrature method (MGDQ). Furthermore, governing equations along with continuity and boundary conditions are discretized. After obtaining solutions to the eigenvalue problem, several studies are investigated to validate the accuracy of the proposed method. As can be concluded from the tables, MGDQ improves the accuracy of results obtained by GDQ. Results for various case studies reveal that MGDQ is properly devised for the vibration analysis of systems with local boundary and continuity conditions.

Complex Variables

Big Hankel operators on Hardy spaces of strongly pseudoconvex domains

In this article, we investigate the (big) Hankel operators H f on Hardy spaces of strongly pseudoconvex domains with smooth boundaries in C n . We also give a necessary and sufficient condition for boundedness of the Hankel operator H f on the Hardy space of the unit disc, which is new in the setting of one variable.

Statistics Theory

Berry-Esseen bounds of second moment estimators for Gaussian processes observed at high frequency

Let Z:={ Z t ,t??} be a stationary Gaussian process. We study two estimators of E[ Z 2 0 ] , namely f ? T (Z):= 1 T ??T 0 Z 2 t dt , and f ? n (Z):= 1 n ??n i=1 Z 2 t i , where t i =i ? n , i=0,1,??n , ? n ?? and T n :=n ? n ?��? . We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem involving f ? T (Z) and f ? n (Z) . We apply these results to asymptotically stationary Gaussian processes and estimate the drift parameter for Gaussian Ornstein-Uhlenbeck processes.

Logic

Negotiation sets: a general framework

It is well-known fact that there exists 1-1 correspondence between so-called double (or flou) sets and intuitionistic sets (also known as orthopairs). At first glance, these two concepts seem to be irreconcilable. However, one must remember that algebraic operations in these two classes are also defined differently. Hence, the expected compatibility is possible. Contrary to this approach, we combine standard definition of double set with operations which are typical for intuitionistic sets. We show certain advantages and limitations of this viewpoint. Moreover, we suggest an interpretation of our sets and operations in terms of logic, data clustering and multi-criteria decision making. As a result, we obtain a structure of discussion between several participants who propose their "necessary" and "allowable" requirements or propositions.

Symplectic Geometry

Contact three-manifolds with exactly two simple Reeb orbits

It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation; the contact struture is universally tight; and in the case of the three-sphere, the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid.

Probability

A CLT for degenerate diffusions with periodic coefficients, and application to homogenisation of linear PDEs

In this article, we obtain a functional CLT for a class of degenerate diffusion processes with periodic coefficients, thus generalizing the already classical results in the context of uniformly elliptic diffusions. As an application, we also discuss periodic homogenization of a class of linear degenerate elliptic and parabolic PDEs.