# Mathematics

###### Featured Researches

## 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.

Read more## 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.

Read more## 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In an unpublished work of Fasel-Rao-Swan the notion of the relative Witt group W E (R,I) is defined.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.

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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).

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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.

Read more## 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?? .

Read more## 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.

Read more## 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.

Read more## 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 .

Read more## 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.

Read more## 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.

Read more## Adversarial Resilience for Sampled-Data Systems under High-Relative-Degree Safety Constraints

Control barrier functions (CBFs) have recently become a powerful method for rendering desired safe sets forward invariant in single- and multi-agent systems. In the multi-agent case, prior literature has considered scenarios where all agents cooperate to ensure that the corresponding set remains invariant. However, these works do not consider scenarios where a subset of the agents are behaving adversarially with the intent to violate safety bounds. In addition, prior results on multi-agent CBFs typically assume that control inputs are continuous and do not consider sampled-data dynamics. This paper presents a framework for normally-behaving agents in a multi-agent system with heterogeneous control-affine, sampled-data dynamics to render a safe set forward invariant in the presence of adversarial agents. The proposed approach considers several aspects of practical control systems including input constraints, clock asynchrony and disturbances, and distributed calculation of control inputs. Our approach also considers functions describing safe sets having high relative degree with respect to system dynamics. The efficacy of these results are demonstrated through simulations.

Read more## 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.

Read more## Survey on L^2-invariants and 3-manifolds

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

Read more## 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.

Read more## 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.

Read more###### 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.

More from Logic###### A note on Woodin's HOD dichotomy

A version of Woodin's HOD dichotomy is proved assuming the existence of just one strongly compact cardinal.

More from Logic###### The Baire closure and its logic

The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator satisfying the standard properties of the topological one, resulting in a closure algebra which we denote Baire(X) . We identify the modal logic of such algebras to be the well-known system S5 , and prove soundness and strong completeness for the cases where X is crowded and either completely metrizable and continuum-sized or locally compact Hausdorff. We also show that every extension of S5 is the modal logic of a subalgebra of Baire(X) , and that soundness and strong completeness also holds in the language with the universal modality.

More from Logic###### 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.

More from Statistics Theory###### Prepivoted permutation tests

We present a general approach to constructing permutation tests that are both exact for the null hypothesis of equality of distributions and asymptotically correct for testing equality of parameters of distributions while allowing the distributions themselves to differ. These robust permutation tests transform a given test statistic by a consistent estimator of its limiting distribution function before enumerating its permutation distribution. This transformation, known as prepivoting, aligns the unconditional limiting distribution for the test statistic with the probability limit of its permutation distribution. Through prepivoting, the tests permute one minus an asymptotically valid p -value for testing the null of equality of parameters. We describe two approaches for prepivoting within permutation tests, one directly using asymptotic normality and the other using the bootstrap. We further illustrate that permutation tests using bootstrap prepivoting can provide improvements to the order of the error in rejection probability relative to competing transformations when testing equality of parameters, while maintaining exactness under equality of distributions. Simulation studies highlight the versatility of the proposal, illustrating the restoration of asymptotic validity to a wide range of permutation tests conducted when only the parameters of distributions are equal.

More from Statistics Theory###### Sharp Sensitivity Analysis for Inverse Propensity Weighting via Quantile Balancing

Inverse propensity weighting (IPW) is a popular method for estimating treatment effects from observational data. However, its correctness relies on the untestable (and frequently implausible) assumption that all confounders have been measured. This paper introduces a robust sensitivity analysis for IPW that estimates the range of treatment effects compatible with a given amount of unobserved confounding. The estimated range converges to the narrowest possible interval (under the given assumptions) that must contain the true treatment effect. Our proposal is a refinement of the influential sensitivity analysis by Zhao, Small, and Bhattacharya (2019), which we show gives bounds that are too wide even asymptotically. This analysis is based on new partial identification results for Tan (2006)'s marginal sensitivity model.

More from Statistics Theory###### 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.

More from K Theory and Homology###### On the monoidal invariance of the cohomological dimension of Hopf algebras

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.

More from K Theory and Homology###### Central extensions of some linear cycle sets

For each member A of a family of linear cycle sets whose underlying abelian group is cyclic of order a power of a prime number, we compute all the central extensions of A by an arbitrary abelian group.

More from K Theory and Homology###### 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.

More from Probability###### Exact lower bound on an "exactly one" probability

The exact lower bound on the probability of the occurrence of exactly one of n random events each of probability p is obtained.

More from Probability###### Stationary Distribution Convergence of the Offered Waiting Processes in Heavy Traffic under General Patience Time Scaling

We study a sequence of single server queues with customer abandonment (GI/GI/1+GI) under heavy traffic. The patience time distributions vary with the sequence, which allows for a wider scope of applications. It is known ([20, 18]) that the sequence of scaled offered waiting time processes converges weakly to a reflecting diffusion process with non-linear drift, as the traffic intensity approaches one. In this paper, we further show that the sequence of stationary distributions and moments of the offered waiting times, with diffusion scaling, converge to those of the limit diffusion process. This justifies the stationary performance of the diffusion limit as a valid approximation for the stationary performance of the GI/GI/1+GI queue. Consequently, we also derive the approximation for the abandonment probability for the GI/GI/1+GI queue in the stationary state.

More from Probability###### 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.

More from Representation Theory###### Triple clasp formulas for C 2 webs

Using the light ladder basis for Kuperberg's C 2 webs, we derive triple clasp formulas for idempotents projecting to the top summand in each tensor product of fundamental representations. We then find explicit formulas for the coefficients occurring in the clasps, by computing these coefficients as local intersection forms. Our formulas provide further evidence for Elias's clasp conjecture, which was given for type A webs, and suggests how to generalize the conjecture to non-simply laced types.

More from Representation Theory###### Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B_0)\geq p-1 or else p = 11 and G/O_{p'}(G) =11^2:SL(2,5). These results are useful in the study of principal blocks with a few characters. We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least 2\sqrt{p-1}+1-k_p(G), where k_p(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

More from Representation Theory###### 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.

More from Category Theory###### Autocompact objects of Ab5 categories

The aim of the paper is to describe autocompact objects in Ab5-categories, i.e. objects in cocomplete abelian categories with exactness preserving filtered colimits of exact sequences, whose covariant Hom-functor commutes with copowers of the object itself. A characterization of non-autocompact object is given, a general criterion of autocompactness of an object via the structure of its endomorphism ring is presented and a criterion of autocompactness of products is proven.

More from Category Theory###### Synthetic Spectra via a Monadic and Comonadic Modality

We extend Homotopy Type Theory with a novel modality that is simultaneously a monad and a comonad. Because this modality induces a non-trivial endomap on every type, it requires a more intricate judgemental structure than previous modal extensions of Homotopy Type Theory. We use this theory to develop an synthetic approach to spectra, where spectra are represented by certain types, and constructions on them by type structure: maps of spectra by ordinary functions, loop spaces by the identity type, and so on. We augment the type theory with a pair of axioms, one which implies that the spectra are stable, and the other which relates synthetic spectra to the ordinary definition of spectra in type theory as Ω -spectra. Finally, we show that the type theory is sound and complete for an abstract categorical semantics, in terms of a category-with-families with a weak endomorphism whose functor on contexts is a bireflection, i.e. has a counit an a unit that are a section-retraction pair.

More from Category Theory###### 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.

More from Dynamical Systems###### Caustic-Free Regions for Billiards on Surfaces of Constant Curvature

In this note we study caustic-free regions for convex billiard tables in the hyperbolic plane or the hemisphere. In particular, following a result by Gutkin and Katok in the Euclidean case, we estimate the size of such regions in terms of the geometry of the billiard table. Moreover, we extend to this setting a theorem due to Hubacher which shows that no caustics exist near the boundary of a convex billiard table whose curvature is discontinuous.

More from Dynamical Systems###### Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles

We construct discontinuous point of the Lyapunov exponent of quasiperiodic Schrödinger cocycles in the Gevrey space G s with s>2 . In contrast, the Lyapunov exponent has been proved to be continuous in the Gevrey space G s with s<2 \cite{klein,cgyz}. This shows that G 2 is the transition space for the continuity of the Lyapunov exponent.

More from Dynamical Systems###### 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.

More from Symplectic Geometry###### Chekanov-Eliashberg dg-algebras for singular Legendrians

The Chekanov-Eliashberg dg-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of wrapped Floer cohomology push-out diagrams. It also leads to a proof of a conjectured isomorphism between partially wrapped Floer cohomology and Chekanov-Eliashberg dg-algebras with coefficients in chains on the based loop space.

More from Symplectic Geometry###### PFH spectral invariants on the two-sphere and the large scale geometry of Hofer's metric

We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer's metric. Namely: (1) we resolve the Kapovich-Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. We also obtain, as a corollary, that the group of area-preserving diffeomorphisms of the open disc, equipped with an area-form of finite area, is not perfect. Central to all of our proofs are new sequences of spectral invariants over the two-sphere, defined via periodic Floer homology.

More from Symplectic Geometry###### 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.

More from General Topology###### Ideals with Smital properties

A ? -ideal I on a Polish group (X,+) has Smital Property if for every dense set D and a Borel I -positive set B the algebraic sum D+B is a complement of a set from I . We consider several variants of this property and study their connections with countable chain condition, maximality and how well they are preserved via Fubini products.

More from General Topology###### Continuous [0,1] -lattices and injective [0,1] -approach spaces

In 1972, Dana Scott proved a fundamental result on the connection between order and topology which says that injective T 0 spaces are precisely continuous lattices endowed with Scott topology. This paper investigates whether this is true in the enriched context, where the enrichment is a quantale obtained by equipping the interval [0,1] with a continuous t-norm. It is shown that for each continuous t-norm, the specialization [0,1] -order of an injective and separated [0,1] -approach space X is a continuous [0,1] -lattice and the [0,1] -approach structure of X coincides with the Scott [0,1] -approach structure of the specialization [0,1] -order; but, unlike in the classical situation, the converse fails in general.

More from General Topology###### 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.

More from Metric Geometry###### Coarse Freundenthal compactification and ends of groups

A coarse compactification of a proper metric space X is any compactification of X that is dominated by its Higson compactification. In this paper we describe the maximal coarse compactification of X whose corona is of dimension 0 . In case of geodesic spaces X , it coincides with the Freundenthal compactification of X . As an application we provide an alternative way of extending the concept of the number of ends from finitely generated groups to arbitrary countable groups. We present a geometric proof of a generalization of Stallings' theorem by showing that any countable group of two ends contains an infinite cyclic subgroup of finite index. Finally, we define ends of arbitrary coarse spaces.

More from Metric Geometry###### A note on the low-dimensional Minkowski-reduction

In this research-expository paper we recall the basic results of reduction theory of positive definite quadratic forms. Using the result of Ryskov on admissible centerings and the result of Tammela about the determination of a Minkowski-reduced form, we prove that the absolute values of coordinates of a minimum vector in a six-dimensional Minkowski-reduced basis are less or equal to three. To get this little sharpening of the result which can be deduced automatically from Tammela's works we combine some elementary geometric reasonings with the mentioned theoretical results.

More from Metric Geometry###### 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.

More from Differential Geometry###### Embedded Delaunay tori and their Willmore energy

A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below 8? . Combining such a strict inequality with previous works by Keller-Mondino-Rivière and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Moreover, we deduce the existence of smoothly embedded tori minimising the Helfrich functional with small spontaneous curvature. Furthermore, because of their symmetry, the Delaunay tori can be used to construct spheres of high isoperimetric ratio, leading to an alternative proof of the known result for the genus zero case.

More from Differential Geometry###### On Einstein hypersurfaces of a remarkable class of Sasakian manifolds

We present a non existence result of complete, Einstein hypersurfaces tangent to the Reeb vector field of a regular Sasakian manifold which fibers onto a complex Stein manifold.

More from Differential Geometry###### 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).

More from Classical Analysis and ODEs###### General Fractional Integrals and Derivatives with the Sonine Kernels

In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the n -fold general fractional integrals and derivatives and study their properties.

More from Classical Analysis and ODEs###### The summation of infinite partial fraction decomposition I: some formulae related to the Hurwitz zeta function

In this paper we establish a new summation method by expanding ??k (1??z a k ) ?? with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when a k is arithmetic sequence. By this summation we obtain many equalities involve Hurwitz zeta function and Gammma function.

More from Classical Analysis and ODEs###### 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.

More from History and Overview###### Applications of Teaching Secondary Mathematics in Undergraduate Mathematics Courses

Robust preparation of future secondary mathematics teachers requires attention to the acquisition of mathematical knowledge for teaching. Many future teachers learn mathematics content primarily through mathematics major courses that are taught by mathematicians who do not specialize in teacher preparation. How can mathematics education researchers assist mathematicians in making explicit connections between the content of undergraduate mathematics courses and the content of secondary mathematics? We present an articulation of five types of connections that can be used in secondary mathematics teacher preparation and give examples of question prompts that mathematicians can use as applications of teaching secondary mathematics in undergraduate mathematics courses.

More from History and Overview###### Math{é}matiques, musique et cosmologie : A partir du Tim{é}e de Platon

Based on Plato's Timaeus, we present some reflections on music, cosmology and mathematics and their mutual influence.The article is dedicated to the composer Walter Zimmermann. The final version of this article will appear in the volume "Les jeux subtils de la po{é}tique, des nombres et de la philosophie. Autour de la musique de Walter Zimmermann", ed. P. Michel, M. Andreatta and J.-L. Besada, Hermann, Paris, 2021.

More from History and Overview###### 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.

More from Algebraic Topology###### On Axiomatic Characterization of Alexander-Spanier Normal Homology Theory of General Topological Spaces

The Alexandroff-?ech normal cohomology theory [Mor 1 ], [Bar], [Ba 1 ],[Ba 2 ] is the unique continuous extension \cite{Wat} of the additive cohomology theory [Mil], [Ber-Mdz 1 ] from the category of polyhedral pairs K 2 Pol to the category of closed normally embedded, the so called, P -pairs of general topological spaces K 2 Top . In this paper we define the Alexander-Spanier normal cohomology theory based on all normal coverings and show that it is isomorphic to the Alexandroff-?ech normal cohomology. Using this fact and methods developed in [Ber-Mdz 3 ] we construct an exact, the so called, Alexander-Spanier normal homology theory on the category K 2 Top , which is isomorphic to the Steenrod homology theory on the subcategory of compact pairs K 2 C . Moreover, we give an axiomatic characterization of the constructed homology theory.

More from Algebraic Topology###### Self-duality of the lattice of transfer systems via weak factorization systems

For a finite group G , G -transfer systems are combinatorial objects which encode the homotopy category of G - N ??operads, whose algebras in G -spectra are E ??G -spectra with a specified collection of multiplicative norms. For G finite Abelian, we demonstrate a correspondence between G -transfer systems and weak factorization systems on the poset category of subgroups of G . This induces a self-duality on the lattice of G -transfer systems.

More from Algebraic Topology###### 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.

More from Commutative Algebra###### Hilbert polynomial of length functions

Let L be a general length function for modules over a Noetherian ring R. We use L to define Hilbert series and polynomials for R[X]-modules. The leading term of any such polynomial is an invariant of R[X]-modules, which refines the algebraic entropy.

More from Commutative Algebra###### Hilbert coefficients and Buchsbaumness of the associated graded ring of filtration

Let A be a Noetherian local ring with the maximal ideal m and I be an m -primary ideal in A . In this paper, we study a boundary condition of an inequality on Hilbert coefficients of an I -admissible filtration I . When A is a Buchsbaum local ring, the above equality forces Buchsbaumness on the associated graded ring of filtration. Our result provides a positive resolution of a question of Corso in a general set up of filtration.

More from Commutative Algebra###### 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.

More from Complex Variables###### On the moduli space of the standard Cantor set

We consider a generalized Cantor set E(?) for an infinite sequence ?=( q n ) ??n=1 of positive numbers with 0< q n <1 , and examine the quasiconformal equivalence to the standard middle one-third Cantor set E( ? 0 ) . We may give a necessary and sufficient condition for E(?) to be quasiconformally equivalent to E( ? 0 ) in terms of ? .

More from Complex Variables###### On the heterogeneous distortion inequality

We study Sobolev mappings f??W 1,n loc ( R n , R n ) , n?? , that satisfy the heterogeneous distortion inequality |Df(x)| n ?�K J f (x)+ ? n (x) |f(x)| n for almost every x??R n . Here K?�[1,?? is a constant and ??? is a function in L n loc ( R n ) . Although we recover the class of K -quasiregular mappings when ??? , the theory of arbitrary solutions is significantly more complicated, partly due to the unavailability of a robust degree theory for non-quasiregular solutions. Nonetheless, we obtain a Liouville-type theorem and the sharp Hölder continuity estimate for all solutions, provided that ???L n?��?( R n )??L n+ε ( R n ) for some ε>0 . This gives an affirmative answer to a question of Astala, Iwaniec and Martin.

More from Complex Variables###### 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.

More from General Mathematics###### A new generalized Wright function and its properties

In this paper we introduce a new multiple-parameters generalization of the Wright function arose from an eigenvalues problem concerning an hyper-Laguerre-type operator involving Caputo derivatives. We show that by giving particular values at the parameters including in the function, it leads right to well known special functions (the classical Wright function, the α -Mittag-Leffler function, the Tricomi function etc...). In addition, we investigate a nonlinear fractional differential equation admitting the new generalization Wright function as solution, and in particular isochronous solutions.

More from General Mathematics###### Some Results on Analysis and number theory

In this paper we obtain bounds for integer solutions of quadratic polynomials in two variables that represent a natural number. Also we get some results on twin prime numbers. In addition, we use linear functionals to prove some results of the mathematical analysis and the Fermat's last theorem.

More from General Mathematics###### 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.

More from Spectral Theory###### Laplace-Beltrami spectrum of ellipsoids that are close to spheres and analytic perturbation theory

We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids sufficiently close to the sphere, the first N eigenvalues have multiplicity at most two, and characterize those that are simple. For the triaxial ellipsoids sufficiently close to the sphere that are not biaxial, we show that at least the first sixteen eigenvalues are all simple. We also give the results of various numerical experiments, including comparisons to our results from the analytic perturbation theory, and approximations for the eigenvalues of ellipsoids that degenerate into infinite cylinders or two-dimensional disks. We propose a conjecture on the exact number of nodal domains of near-sphere ellipsoids.

More from Spectral Theory###### Perturbative diagonalisation for Maryland-type quasiperiodic operators with flat pieces

We consider quasiperiodic operators on Z d with unbounded monotone sampling functions ("Maryland-type"), which are not required to be strictly monotone and are allowed to have flat segments. Under several geometric conditions on the frequencies, lengths of the segments, and their positions, we show that these operators enjoy Anderson localization at large disorder.

More from Spectral Theory###### 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.

More from Quantum Algebra###### On a necessary condition for unitary categorification of fusion rings

In arXiv:1910.12059 Liu, Palcoux and Wu proved a remarkable necessary condition for a fusion ring to admit a unitary categorification, by constructing invariants of the fusion ring that have to be positive if it is unitarily categorifiable. The main goal of this note is to provide a somewhat more direct proof of this result. In the last subsection we discuss integrality properties of the Liu-Palcoux-Wu invariants.

More from Quantum Algebra###### Quantum decorated character stacks

We initiate the study of decorated character stacks and their quantizations using the framework of stratified factorization homology. We thereby extend the construction by Fock and Goncharov of (quantum) decorated character varieties to encompass also the stacky points, in a way that is both compatible with cutting and gluing and equivariant with respect to canonical actions of the modular group of the surface. In the cases G=S L 2 ,PG L 2 we construct a system of categorical charts and flips on the quantum decorated character stacks which generalize the well--known cluster structures on the Fock--Goncharov moduli spaces.

More from Quantum Algebra###### 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?? .

More from Group Theory###### Higher braid groups and regular semigroups from polyadic-binary correspondence

In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R -matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n -simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.

More from Group Theory###### On the cokernel of the Baumslag rationalization

We prove that for the free group of rank two F the cokernel of the homomorphism to its Baumslag rationalization F?�Bau(F) is not abelian. Moreover, this cokernel contains a free subgroup of countable rank. This answers a question of Emmanuel Farjoun.

More from Group Theory###### 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.

More from Number Theory###### A geometric linear Chabauty comparison theorem

The Chabauty-Coleman method is a p -adic method for finding all rational points on curves of genus g whose Jacobians have Mordell-Weil rank r<g . Recently, Edixhoven and Lido developed a geometric quadratic Chabauty method that was adapted by Spelier to cover the case of geometric linear Chabauty. We compare the geometric linear Chabauty method and the Chabauty-Coleman method and show that geometric linear Chabauty can outperform Chabauty-Coleman in certain cases. However, as Chabauty-Coleman remains more practical for general computations, we discuss how to strengthen Chabauty-Coleman to make it theoretically equivalent to geometric linear Chabauty.

More from Number Theory###### Hyperbolic Heron Triangles and Elliptic Curves

We define hyperbolic Heron triangles (hyperbolic triangles with "rational" side-lengths and area) and parametrize them in two ways as rational points of certain elliptic curves. We show that there are infinitely many hyperbolic Heron triangles with one angle α and area A for any (admissible) choice of α and A ; in particular, the congruent number problem has always infinitely many solutions in the hyperbolic setting. We also explore the question of hyperbolic triangles with a rational median and a rational area bisector (median splitting the triangle in half).

More from Number Theory###### 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.

More from Analysis of PDEs###### Quantum and Semiquantum Pseudometrics and Applications

We establish a Kantorovich duality for the pseudometric E ??introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. 223 (2017), 57--94], obtained from the usual Monge-Kantorovich distance d MK,2 between classical densities by quantization of one of the two densities involved. We show several type of inequalities comparing d MK,2 , E ??and M K ??, a full quantum analogue of d MK,2 introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165--205], including an up to ??triangle inequality for M K ??. Finally, we show that, when nice optimal Kantorovich potentials exist for E ??, optimal couplings induce classical/quantum optimal transports and the potentials are linked by a semiquantum Legendre type transform.

More from Analysis of PDEs###### Time Dependent Quantum Perturbations Uniform in the Semiclassical Regime

We present a time dependent quantum perturbation result, uniform in the Planck constant, for perturbations of potentials whose gradients are Lipschitz continuous by potentials whose gradients are only bounded a.e.. Though this low regularity of the full potential is not enough to provide the existence of the classical underlying dynamics, at variance with the quantum one, our result shows that the classical limit of the perturbed quantum dynamics remains in a tubular neighbourhood of the classical unperturbed one of size of order of the square root of the size of the perturbation. We treat both Schrödinger and von Neumann-Heisenberg equations.

More from Analysis of PDEs###### 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 .

More from Algebraic Geometry###### Connections and L ??liftings of semiregularity maps

Let E ??be a finite complex of locally free sheaves on a complex manifold X . We prove that to every connection of type (1,0) on E ??it is canonically associated an L ??morphism g: A 0,??X (Ho m ??O X ( E ??, E ??))??A ????X A ??,??X [2] that lifts the 1-component of Buchweitz-Flenner semiregularity map. An application to deformations of coherent sheaves on projective manifolds is given.

More from Algebraic Geometry###### Moduli of Bridgeland semistable holomorphic triples

We prove that the moduli stack of Bridgeland semistable holomorphic triples over a curve of g(C)?? with a fixed numerical class and phase is an algebraic stack of finite type over C and admits a proper good moduli space. We prove that this also holds for a class of Bridgeland stability conditions on the category of holomorphic chains T C,n . In the process, we construct an explicit geometric realisation of T C,n and prove the open heart property for noetherian hearts in admissible categories of D b (X) , where X is a smooth projective variety over C , whose orthogonal complements are geometric triangulated categories.

More from Algebraic Geometry###### 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.

More from Rings and Algebras###### The Allison-Faulkner construction of E 8

We show that the Tits index E 133 8 cannot be obtained by means of the Tits construction over a field with no odd degree extensions. We construct two cohomological invariants, in degrees 6 and 8, of the Tits construction and the more symmetric Allison-Faulkner construction of Lie algebras of type E 8 and show that these invariants can be used to detect the isotropy rank.

More from Rings and Algebras###### Duo property for rings by the quasinilpotent perspective

In this paper, we focus on the duo ring property via quasinilpotent elements which gives a new kind of generalizations of commutativity. We call this kind of ring qnil-duo. Firstly, some properties of quasinilpotents in a ring are provided. Then the set of quasinilpotents is applied to the duo property of rings, in this perspective, we introduce and study right (resp., left) qnil-duo rings. We show that this concept is not left-right symmetric. Among others it is proved that if the Hurwitz series ring H(R;α) is right qnil-duo, then R is right qnil-duo. Every right qnil-duo ring is abelian. A right qnil-duo exchange ring has stable range 1.

More from Rings and Algebras###### 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.

More from Functional Analysis###### Quantitative translations for viscosity approximation methods in hyperbolic spaces

In the setting of hyperbolic spaces, we show that the convergence of Browder-type sequences and Halpern iterations respectively entail the convergence of their viscosity version with a Rakotch map. We also show that the convergence of a hybrid viscosity version of the Krasnoselskii-Mann iteration follows from the convergence of the Browder type sequence. Our results follow from proof-theoretic techniques (proof mining). From an analysis of theorems due to T. Suzuki, we extract a transformation of rates for the original Browder type and Halpern iterations into rates for the corresponding viscosity versions. We show that these transformations can be applied to earlier quantitative studies of these iterations. From an analysis of a theorem due to H.-K. Xu, N. Altwaijry and S. Chebbi, we obtain similar results. Finally, in uniformly convex Banach spaces we study a strong notion of accretive operator due to Brezis and Sibony and extract an uniform modulus of uniqueness for the property of being a zero point. In this context, we show that it is possible to obtain Cauchy rates for the Browder type and the Halpern iterations (and hence also for their viscosity versions).

More from Functional Analysis###### D(X)<1 or D ^ (X)<1 imply Property( K )

Two new Banach space moduli, that involve weak convergent sequences, are introduced. It is shown that if either one of these moduli are strictly less than 1 then the Banach space has Property( K )

More from Functional Analysis###### Optimization and Control

###### Adversarial Resilience for Sampled-Data Systems under High-Relative-Degree Safety Constraints

Control barrier functions (CBFs) have recently become a powerful method for rendering desired safe sets forward invariant in single- and multi-agent systems. In the multi-agent case, prior literature has considered scenarios where all agents cooperate to ensure that the corresponding set remains invariant. However, these works do not consider scenarios where a subset of the agents are behaving adversarially with the intent to violate safety bounds. In addition, prior results on multi-agent CBFs typically assume that control inputs are continuous and do not consider sampled-data dynamics. This paper presents a framework for normally-behaving agents in a multi-agent system with heterogeneous control-affine, sampled-data dynamics to render a safe set forward invariant in the presence of adversarial agents. The proposed approach considers several aspects of practical control systems including input constraints, clock asynchrony and disturbances, and distributed calculation of control inputs. Our approach also considers functions describing safe sets having high relative degree with respect to system dynamics. The efficacy of these results are demonstrated through simulations.

More from Optimization and Control###### Tightness and Equivalence of Semidefinite Relaxations for MIMO Detection

The multiple-input multiple-output (MIMO) detection problem, a fundamental problem in modern digital communications, is to detect a vector of transmitted symbols from the noisy outputs of a fading MIMO channel. The maximum likelihood detector can be formulated as a complex least-squares problem with discrete variables, which is NP-hard in general. Various semidefinite relaxation (SDR) methods have been proposed in the literature to solve the problem due to their polynomial-time worst-case complexity and good detection error rate performance. In this paper, we consider two popular classes of SDR-based detectors and study the conditions under which the SDRs are tight and the relationship between different SDR models. For the enhanced complex and real SDRs proposed recently by Lu et al., we refine their analysis and derive the necessary and sufficient condition for the complex SDR to be tight, as well as a necessary condition for the real SDR to be tight. In contrast, we also show that another SDR proposed by Mobasher et al. is not tight with high probability under mild conditions. Moreover, we establish a general theorem that shows the equivalence between two subsets of positive semidefinite matrices in different dimensions by exploiting a special "separable" structure in the constraints. Our theorem recovers two existing equivalence results of SDRs defined in different settings and has the potential to find other applications due to its generality.

More from Optimization and Control###### Over-approximating reachable tubes of linear time-varying systems

We present a method to over-approximate reachable tubes over compact time-intervals, for linear continuous-time, time-varying control systems whose initial states and inputs are subject to compact convex uncertainty. The method uses numerical approximations of transition matrices, is convergent of first order, and assumes the ability to compute with compact convex sets in finite dimension. We also present a variant that applies to the case of zonotopic uncertainties, uses only linear algebraic operations, and yields zonotopic over-approximations. The performance of the latter variant is demonstrated on an example.

More from Optimization and Control###### 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.

More from Operator Algebras###### Projector Matrix Product Operators, Anyons and Higher Relative Commutants of Subfactors

A bi-unitary connection in subfactor theory of Jones producing a subfactor of finite depth gives a 4-tensor appearing in a recent work of Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete on 2-dimensional topological order and anyons. In their work, they have a special projection called a projector matrix product operator. We prove that the range of this projection of length k is naturally identified with the k-th higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between 2-dimensional topological order and subfactor theory.

More from Operator Algebras###### Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori

Fuzzy tori are finite dimensional C*-algebras endowed with an appropriate notion of noncommutative geometry inherited from an ergodic action of a finite closed subgroup of the torus, which are meant as finite dimensional approximations of tori and more generally, quantum tori. A mean to specify the geometry of a noncommutative space is by constructing over it a spectral triple. We prove in this paper that we can construct spectral triples on fuzzy tori which, as the dimension grow to infinity and under other natural conditions, converge to a natural spectral triple on quantum tori, in the sense of the spectral propinquity. This provides a formal assertion that indeed, fuzzy tori approximate quantum tori, not only as quantum metric spaces, but as noncommutative differentiable manifolds -- including convergence of the state spaces as metric spaces and of the quantum dynamics generated by the Dirac operators of the spectral triples, in an appropriate sense.

More from Operator Algebras###### Geometric Topology

###### Survey on L^2-invariants and 3-manifolds

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

More from Geometric Topology###### Minimal volume entropy and fiber growth

This article deals with topological assumptions under which the minimal volume entropy of a closed manifold~ M , and more generally of a finite simplicial complex~ X , vanishes or is positive. These topological conditions are expressed in terms of the growth of the fundamental group of the fibers of maps from a given finite simplicial complex~ X to lower dimensional simplicial complexes~ P . We also give examples of finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy.

More from Geometric Topology###### From zero surgeries to candidates for exotic definite four-manifolds

One strategy for distinguishing smooth structures on closed 4 -manifolds is to produce a knot K in S 3 that is slice in one smooth filling W of S 3 but not slice in some homeomorphic smooth filling W ??. In this paper we explore how 0 -surgery homeomorphisms can be used to potentially construct exotic pairs of this form. In order to systematically generate a plethora of candidates for exotic pairs, we give a fully general construction of pairs of knots with the same zero surgeries. By computer experimentation, we find 5 topologically slice knots such that, if any of them were slice, we would obtain an exotic four-sphere. We also investigate the possibility of constructing exotic smooth structures on # n C P 2 in a similar fashion.

More from Geometric Topology###### 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.

More from Numerical Analysis###### A study on a feedforward neural network to solve partial differential equations in hyperbolic-transport problems

In this work we present an application of modern deep learning methodologies to the numerical solution of partial differential equations in transport models. More specifically, we employ a supervised deep neural network that takes into account the equation and initial conditions of the model. We apply it to the Riemann problems over the inviscid nonlinear Burger's equation, whose solutions might develop discontinuity (shock wave) and rarefaction, as well as to the classical one-dimensional Buckley-Leverett two-phase problem. The Buckley-Leverett case is slightly more complex and interesting because it has a non-convex flux function with one inflection point. Our results suggest that a relatively simple deep learning model was capable of achieving promising results in such challenging tasks, providing numerical approximation of entropy solutions with very good precision and consistent to classical as well as to recently novel numerical methods in these particular scenarios.

More from Numerical Analysis###### Solving time-fractional differential equation via rational approximation

Fractional differential equations (FDEs) describe subdiffusion behavior of dynamical systems. Its non-local structure requires to take into account the whole evolution history during the time integration, which then possibly causes additional memory use to store the history, growing in time. An alternative to a quadrature of the history integral is to approximate the fractional kernel with the sum of exponentials, which is equivalent to consider the FDE solution as a sum of solutions to a system of ODEs. One possibility to construct this system is to approximate the Laplace spectrum of the fractional kernel with a rational function. In this paper, we use the adaptive Antoulas--Anderson (AAA) algorithm for the rational approximation of the kernel spectrum which yields only a small number of real valued poles. We propose a numerical scheme based on this idea and study its stability and convergence properties. Moreover, we apply the algorithm to a time-fractional Cahn-Hilliard problem.

More from Numerical Analysis###### 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.

More from Combinatorics###### The Non-Existence of Block-Transitive Subspace Designs

Let q be a prime power and V??F n q . A t - (n,k,λ ) q design, or simply a subspace design, is a pair D=(V,B) , where B is a subset of the set of all k -dimensional subspaces of V , with the property that each t -dimensional subspace of V is contained in precisely λ elements of B . Subspace designs are the q -analogues of balanced incomplete block designs. Such a design is called block-transitive if its automorphism group Aut(D) acts transitively on B . It is shown here that if t?? and D=(V,B) is a block-transitive t - (n,k,λ ) q design then D is trivial, that is, B is the set of all k -dimensional subspaces of V .

More from Combinatorics###### Generalized Catalan numbers from hypergraphs

The Catalan numbers (C_n)_{n >= 0} = 1,1,2,5,14,42,... form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we define a generalization of the Catalan numbers. In fact we define an infinite collection of generalizations C_n^(m), m >= 1, with m=1 giving the usual Catalans. The sequence C_n^(m) comes from studying certain matrix models attached to hypergraphs. We also give some combinatorial interpretations of these numbers, and conjecture some asymptotics.

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