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International audience; In this article, we give an in-depth analysis of the problem of optimising the total population size for a standard logistic-diffusive model. This optimisation problem stems from the study of spatial ecology and amounts to the following question: assuming a species evolves in a domain, what is the best way to spread resources in order to ensure a maximal population size at equilibrium? {In recent years, many authors contributed to this topic.} We settle here the proof of two fundamental properties of optimisers: the bang-bang one which had so far only been proved under several strong assumptions, and the other one is the fragmentation of maximisers. Here, we prove the bang-bang property in all generality using a new spectral method. The technique introduced to demonstrate the bang-bang character of optimizers can be adapted and generalized to many optimization problems with other classes of bilinear optimal control problems where the state equation is semilinear and elliptic. We comment on it in a conclusion section.Regarding the geometry of maximisers, we exhibit a blow-up rate for the $BV$-norm of maximisers as the diffusivity gets smaller: if $\Omega$ is an orthotope and if $m_\mu$ is an optimal control, then $\Vert m_\mu\Vert_{BV}\gtrsim \sqrt{\mu}$. The proof of this results relies on a very fine energy argument.

International audience; This paper has two parts. The mathematical part provides generalized versions of the robust Ekeland variational principle in terms of set-valued EVP with variable preferences, uncertain parameters and changing weights given to vectorial perturbation functions. The behavioural part that motivates our findings models the formation and stability of a partnership in a changing, uncertain and complex environment in the context of the variational rationality approach of stop, continue and go human dynamics. Our generalizations allow us to consider two very important psychological effects relative to ego depletion and goal gradient hypothesis.

Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form $$L_A(X)=I-A_1X_1-\dots-A_g X_g\succeq0.$$ Here the $X_j$ are real numbers and the set of solutions is called a spectrahedron. These inequalities make sense when the $X_i$ are symmetric matrices of any size, $n\times n$, and enter the formula though tensor product $A_i\otimes X_i$: The solution set of $L_A(X)\succeq0$ is called a free spectrahedron since it contains matrices of all sizes and the defining ``linear pencil" is ``free" of the sizes of the matrices. In this article, we report on empirically observed properties of optimizers obtained from optimizing linear functionals over free spectrahedra restricted to matrices $X_i$ of fixed size $n\times n$. The optimizers we find are always classical extreme points. Surprisingly, in many reasonable parameter ranges, over 99.9\% are also free extreme points. Moreover, the dimension of the active constraint, $\ker(L_A(X^\ell))$, is about twice what we expected. Another distinctive pattern regards reducibility of optimizing tuples $(X_1^\ell,\dots,X_g^\ell)$. We give an algorithm for representing elements of a free spectrahedron as matrix convex combinations of free extreme points; these representations satisfy a very low bound on the number of free extreme points neede, 46 pages body. Includes table of contents and index

In 1964, Michael Edelstein presented an amazing affine isometry acting on the space of square-summable sequences. This operator has no fixed points, but a suborbit that converges to 0 while another...

In Dunfield's catalog of the hyperbolic manifolds in the SnapPy census which are complements of L-space knots in $S^3$, we determine that $22$ have tunnel number $2$ while the remaining all have tunnel number $1$. Notably, these $22$ manifolds contain $9$ asymmetric L-space knot complements. Furthermore, using SnapPy and KLO we find presentations of these $22$ knots as closures of positive braids that realize the Morton-Franks-Williams bound on braid index. The smallest of these has genus $12$ and braid index $4$., This is a significant expansion/rewrite of the original version. The title and abstract have both changed to reflect this. Relevant code is included in the body of the article itself. Originally, this was a note that collected the results of an "office hour" session on the use of SnapPy and KLO held at the ICERM workshop Perspectives on Dehn Surgery, July 15--19, 2019

Let $\mu$ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power $\mu^{\boxplus k}$ for any real $k \geq 1$. The purpose of this short note is to give an elementary description of $\mu^{\boxplus k}$ in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.

Let $G$ be a group and $H$ a subgroup of $G$. This note introduces an equivalent definition of hyperbolic embedded subgroup based on Bowditch's approach to relatively hyperbolic groups in terms of fine graphs., Version 4. Version accepted for publication in Communications in Algebra

Globally, measures, such as lockdown, quarantining, and physical distancing, have been implemented to curb the spread of COVID-19. As the vaccines are now available and reintegration into society is beginning, measures such as vaccine certificates are being implemented around the world. We conducted a scoping review to identify the initial digital solutions for COVID-19 vaccine certificates and evaluate them on the basis of purpose and use case, technological architecture, and ethical and legal implications. Articles identified from a Google search and a search of MEDLINE, Ovid and preprint servers were reviewed in duplicate, and data were extracted using a data extraction form. Data were extracted for date, location, type of article, source, companies identified for creating vaccine certificates, technology used, type of evidence provided (article quoting research study or an expert opinion), digital architecture, security and privacy measures, and use cases. Technology emerged as the most dominant theme followed by ethics, travel, legal concerns, public policy, and scientific concerns. Our review identified eight solutions that are working toward COVID-19 vaccine certificates world-wide, all optimizing blockchain technology. COVID-19 vaccine certificates are being considered in 11 countries and are in place in 5 others. Many issues concerning the themes we identified remain to be addressed to facilitate successful implementation.

In this note we study the induced $p$-norm of circulant matrices $A(n,\pm a, b)$, acting as operators on the Euclidean space $\mathbb{R}^n$. For circulant matrices whose entries are nonnegative real numbers, in particular for $A(n,a,b)$, we provide an explicit formula for the $p$-norm, $1 \leq p \leq \infty$. The calculation for $A(n,-a,b)$ is more complex. The 2-norm is precisely determined. As for the other values of $p$, two different categories of upper and lower bounds are obtained. These bounds are optimal at the end points (i.e. $p=1$ and $p = \infty$) as well as at $p=2$., Comment: 14 pages, to be published in Linear and Multilinear Algebra

In this paper we study the problem of decomposing a given tensor into a tensor train such that the tensors at the vertices are orthogonally decomposable. When the tensor train has length two, and the orthogonally decomposable tensors at the two vertices are symmetric, we recover the decomposition by considering random linear combinations of slices. Furthermore, if the tensors at the vertices are symmetric and low-rank but not orthogonally decomposable, we show that a whitening procedure can transform the problem into the orthogonal case. When the tensor network has length three or more and the tensors at the vertices are symmetric and orthogonally decomposable, we provide an algorithm for recovering them subject to some rank conditions. Finally, in the case of tensor trains of length two in which the tensors at the vertices are orthogonally decomposable but not necessarily symmetric, we show that the decomposition problem reduces to the novel problem of decomposing a matrix into an orthogonal matrix multiplied by diagonal matrices on either side. We provide and compare two solutions, one based on Sinkhorn's theorem and one on Procrustes' algorithm. We conclude with a multitude of open problems in linear and multilinear algebra that arose in our study.

We prove unexpected decay of the L2-distance from the solution u(t) of a hyperbolic scalar conservation law, to some convex, flow-invariant target sets.

We consider reflexive Bergman spaces A(p)(Omega) on polygonal domains Omega of the complex plane. With some restrictions to the angles of the boundary of Omega, we show that the boundedness of the Toeplitz operator T-g : A(p)(Omega) -> A(p)(Omega) with a positive symbol g is equivalent to the boundedness of the Berezin transform of g or to g times the area measure being a Carleson measure. The result is also formulated for more general simply connected domains. The main technical tool is a new weighted Forelli-Rudin-type estimate.

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. We introduce commutatively closed graphs and investigate properties of commutatively closed subsets of a ring with the help of graph theory. A particular attention is paid to constructions of subsets of a given diameter. In particular, we compute the diameter of artinian semisimple algebras.

In this paper, we study SDEs with respect to optional semimartingales or optional SDEs. Our leading idea is to explore the concept and technique of local time of optional processes to extend severa...

International audience; The goal of this paper is to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailor-made version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the square intensity of a common noise that is inserted in the mean field game or, equivalently, as the diffusivity parameter in the related parabolic version of the master equation. As established in the recent contribution (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98-162), the randomly forced mean field game becomes indeed uniquely solvable for a relevant choice of a Wright-Fisher common noise, the counterpart of which in the master equation is a Kimura operator on the simplex. We here elaborate on (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98-162) to make the mean field game with common noise both uniquely solvable and potential, meaning that its unique solution is in fact equal to the unique minimizer of a suitable stochastic mean field control problem. Taking the limit as the intensity of the common noise vanishes, we obtain a rigorous proof of the aforementioned selection principle. As a byproduct, we get that the classical solution to the viscous master equation associated with the mean field game with common noise converges to the gradient of the value function of the mean field control problem without common noise. We hence select a particular weak solution of the master equation of the original mean field game. Lastly, we establish an intrinsic uniqueness criterion for this solution within a suitable class of weak solutions to the master equation satisfying a weak one-sided Lipschitz inequality.

We consider a varying-coefficient partially linear transformation model with current status data, which extends several semiparametric models for current status data in the literature. Sieve maximu...

Richard Kimberly Heck and Paolo Mancosu have claimed that the possibility of non-Cantorian assignments of cardinalities to infinite concepts shows that Hume's Principle (HP) is not implicit in the ...

In this paper, we study some properties of multi-solitons for the non-linear Schrodinger equations in Rd with general non-linearities. Multi-solitons have already been constructed in H1(Rd) in pape...

In this paper we present a connection between systems of differential equations for the recurrence coefficients of polynomials orthogonal with respect to the generalized Meixner and the deformed La...

We investigate interlacing properties of zeros of Laguerre polynomials $ L_{n}^{(\alpha)}(x)$ and $ L_{n+1}^{(\alpha +k)}(x),$ $ \alpha > -1, $ where $ n \in \mathbb{N}$ and $ k \in {\{ 1,2 }\}$. We prove that, in general, the zeros of these polynomials interlace partially and not fully. The sharp $t-$interval within which the zeros of two equal degree Laguerre polynomials $ L_n^{(\alpha)}(x)$ and $ L_n^{(\alpha +t)}(x)$ are interlacing for every $n \in \mathbb{N}$ and each $ \alpha > -1$ is $ 0 < t \leq 2,$ \cite{DrMu2}, and the sharp $t-$interval within which the zeros of two consecutive degree Laguerre polynomials $ L_n^{(\alpha)}(x)$ and $ L_{n-1}^{(\alpha +t)}(x)$ are interlacing for every $n \in \mathbb{N}$ and each $ \alpha > -1$ is $ 0 \leq t \leq 2,$ \cite{DrMu1}. We derive conditions on $n \in \mathbb{N}$ and $\alpha,$ $ \alpha > -1$ that determine the partial or full interlacing of the zeros of $ L_n^{(\alpha)}(x)$ and the zeros of $ L_n^{(\alpha + 2 + k)}(x),$ $ k \in {\{ 1,2 }\}$. We also prove that partial interlacing holds between the zeros of $ L_n^{(\alpha)}(x)$ and $ L_{n-1}^{(\alpha + 2 +k )}(x)$ when $ k \in {\{ 1,2 }\},$ $n \in \mathbb{N}$ and $ \alpha > -1$. Numerical illustrations of interlacing and its breakdown are provided.

In this paper, we study the sequences of matrix Cauchy biorthogonal polynomials. We will focus on the algebraic aspects of the problem, finding a connection with a kind of matrix mixed-type Hermite...

WKB theoretic transformation to the boosted Airy equation is discussed. The formulae for the discontinuities of the Borel transforms of the WKB solutions to a differential equation which can be tra...

As an extension of the Askey-scheme, recently, it has been introduced the so-called d-Askey-scheme, which is a table describing families of hypergeometric d-orthogonal polynomials. In this paper, w...

In my joint papers with Iwaki and Koike, we found an intriguing relation between the Voros coefficients in the exact WKB analysis and the free energy in the topological recursion introduced by Eyna...

We study a family of type II multiple orthogonal polynomials. We consider orthogonality conditions with respect to a vector measure, in which each component is a q-analogue of the binomial distribu...

The wave equation $\left(\partial_{tt} - c^2 \Delta_x\right) u(x,t) = e^{-t} f(x,t)$ is shown to have a unique solution if $u$ and its partial derivatives in $x$ are in $L^2(e^{-t})$ on the cone, and the solution can be explicit given in the Fourier series of orthogonal polynomials on the cone. This provides a particular solution for the boundary value problems of the non-homogeneous wave equation on the cone, which can be combined with a solution to the homogeneous wave equation in the cone to obtain the full solution.

We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalised sextic Freud weight \[\omega(x;t,\lambda)=|x|^{2\lambda+1}\exp\left(-x^6+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $\lambda>-1$ and $t\in\mathbb{R}$. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalised hypergeometric functions ${}_1F_2(a_1;b_1,b_2;z)$. We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlighting a fascinating connection between generalised quartic, sextic, octic and decic Freud weights when expressing their first moments in terms of generalised hypergeometric functions., Comment: 18 pages, 3 figures

We study the logarithmic asymptotic of multiple orthogonal polynomials arising in a mixed-type Hermite–Pade approximation problem associated with the rational perturbation of a Nikishin system of f...

We investigate asymptotic behavior of polynomials $p^{\omega}_n(z)$ satisfying varying non-Hermitian orthogonality relations $$ \int_{-1}^{1} x^kp^{\omega}_n(x)h(x) e^{\mathrm{i} \omega x}\mathrm{d} x =0, \quad k\in\{0,\ldots,n-1\}, $$ where $h(x) = h^*(x) (1 - x)^{\alpha} (1 + x)^{\beta}, \ \omega = \lambda n, \ \lambda \geq 0 $ and $h(x)$ is holomorphic and non-vanishing in a certain neighborhood in the plane. These polynomials are an extension of so-called kissing polynomials ($\alpha = \beta = 0$) introduced in connection with complex Gaussian quadrature rules with uniform good properties in $\omega$.

The Laguerre-Sobolev polynomials form an orthogonal polynomial system on the positive half-line with respect to the classical Laguerre measure with parameter α>−1 and, in general, two point masses ...

The Voros coefficient of the Gauss hypergeometric differential equation with a large parameter is defined for the origin and its explicit form and the details of derivation are given.

We consider several special functions defined by matrix integrals on Hermitian matrix space, namely, analogues of Gauss hypergeomeric, Kummer's confluent hypergeometric, Bessel, Hermite-Weber and A...

In this paper, we obtain a probabilistic relationship between the exponential Bell polynomials and the weighted sums of independent Poisson random variables. A recently established probabilistic co...

The main task of this paper is to model the dependency and effects of the Lehman Brothers financial collapse event using a superposed and coupled Ornstein-Uhlenbeck type system of stochastic differ...

We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint., 26 pages. arXiv admin note: text overlap with arXiv:1910.13117

Let [n]:={1,2,…,n} and let a k-set denote a set of cardinality k. A family of sets is union-closed (UC) if the union of any two sets in the family is also in the family. Frankl’s conjecture states ...

Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were...

We extend the computations from our previous paper arXiv:2005.07054 to determine the maximum number of rational points on a curve over $\mathbb{F}_3$ and $\mathbb{F}_4$ with fixed gonality and small genus. We find, for example, that there is no curve of genus 5 and gonality 6 over a finite field. We propose two conjectures based on our data. First, an optimal curve of genus $g$ has gonality at most $\lfloor \frac{g+3}{2} \rfloor$. Second, a curve of gonality $\gamma$ and large genus over $\mathbb{F}_q$ has $\gamma(q+1)$ rational points., Comment: Final published version as it will appear in Experimental Mathematics; 19 pages; code available at https://github.com/RationalPoint/gonality [Note that an error was corrected since the previous version: up to isomorphism, there is exactly 1 curve of genus 4 and gonality 5 over GF(3).]

We present an explicit minimal set of generators for the defining ideal of the family of Backelin semigroups and find its Betti numbers. In particular, we compute the type of the semigroup as well,...

In this paper, we describe all the hyperbolic 24-cell 4-manifolds with exactly one cusp. There are four of these manifolds up to isometry. These manifolds are the first examples of one-cusped hyperbolic 4-manifolds of minimum volume.

In the present article, we extend our study of Kantorovich type exponential sampling operators introduced in [4]. We derive the Voronovskaya type theorem and its quantitative estimates for these op...

We classify all decompositions of $M_3(\mathbb{C})$ into a direct vector-space sum of two subalgebras such that one of the subalgebras contains the identity matrix., 29 p

Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. For a character $\chi \in \rm{Irr}(G)$, the number $\rm{cod}(\chi):=|G:\rm{ker}\chi|/\chi(1)$ is called the co-degree of $\chi$. The set of co-degrees of all irreducible characters of $G$ is denoted by $\rm{cod}(G)$. In this paper, we show that for a non-trivial finite group $G$, $|\rm{Irr}(G)|=|\rm{cod}(G)|$ if and only if $G$ is isomorphic to the cyclic group $\mathbb{Z}_2$ or the symmetric group $S_3$.

Our first result provides a new characterization of Auslander algebras using a property of hereditary torsion pairs. The second result shows an Auslander algebra $\Lambda$ is left or right glued if and only if $\Lambda$ is representation-finite. Finally, our third result shows the module category of any Auslander algebra contains a tilting module with a particular property, which we call the hereditary property. Applications of this property are investigated.

Let $\Lambda$ be an artin algebra and let $\mathcal{P}^{, Comment: 4 pages

We consider the problem {−Δu=λK(x)f(u)in B1c,u(x)=0on ∂B1,u(x)→0as |x|→∞, where B1c={x∈Rn:|x|>1}, n>2, λ is a positive parameter, K:B1c→R+ belongs to a class of continuous functions which satisfy c...