Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen's conjecture

We study the functional codes $C_h(X)$ defined by Lachaud in [G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in: Arithmetic, Geometry, and Coding Theory, Luminy, France, 1993, de Gruyter, Berlin, 1996, pp. 77–104] where $X\subset {\mathbb{P}}^N$ is an algebraic projective variety of degree d and dimension m. When X is a Hermitian surface in $\mathit{PG}(3,q)$, Sørensen in [A.B. Sørensen, Rational points on hypersurfaces, Reed–Muller codes and algebraic-geometric codes, PhD thesis, Aarhus, Denmark, 1991], has conjectured for $h⩽t$ (where $q=t^2$) the following result:$\mathrm{\#}{X}_{Z(f)}({\mathbb{F}}_q)⩽h(t^3+t^2-t)+t+1$ which should give the exact value of the minimum distance of the functional code $C_h(X)$. In this paper we resolve the conjecture of Sørensen in the case of quadrics (i.e. $h=2$), we show the geometrical structure of the minimum weight codewords and their number; we also estimate the second weight and the geometrical structure of the codewords reaching this second weight.     

Markov processes conditioned on their location at large exponential times

Suppose that ${\left(X_t\right)}_{t\ge 0}$ is a one-dimensional Brownian motion with negative drift $?\mu$. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like $X$ conditioned to hit 0, after which time it behaves as $X$ killed at the last time $X$ visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift $+\mu$ when it is negative, like Brownian motion with negative drift $?\mu$ when it is positive, and is killed according to the local time spent at 0.An extension of this result holds in great generality for a Borel right process conditioned to be in some state $a$ at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the “bang–bang” construction for a general Markov process.As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the $h$-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.     

Amorphic association schemes from bent partitions

Bent partitions are partitions of an elementary abelian group, which have similarities to partitions from spreads. In fact, a spread partition is a special case of a bent partition. In particular, bent partitions give rise to a large number of (vectorial) bent functions. Examples of bent partitions, which generalize the Desarguesian spread, have been presented by Anbar, Meidl and Pirsic, 2021, 2022. Bent partitions, which generalize some other classes of (pre)semifield spreads, have been presented by Anbar, Kalaycı, Meidl 2023. In this article, it is shown that these bent partitions induce $(p^k+1)$-class amorphic associations schemes on ${\mathbb{F}}_{p^m}\times {\mathbb{F}}_{p^m}$, where k is a divisor of m with special properties. This implies a construction of amorphic association schemes from some classes of (pre)semifields.     

The nth-order degenerate breather solution for the Kundu–Eckhaus equation

In this paper, we formulate the compact determinant representation of the formula of $n$th-order breather solution for the Kundu–Eckhaus (KE) equation. Then, we obtain the formula of the $n$th-order degenerate breather solution (breather-positon, b-positon for short) for the KE equation by using the Taylor expansion with respect to degenerate eigenvalues ${\lambda }_{2k-1}\to {\lambda }_1(k=1,2,\dots ,n+1)$. B-positon, which is a special kind of breather solution, is recently recognized as a key role being responsible for generating rogue wave. According to the related formula, the exact expression of first-order b-positon is constructed. Furthermore, the dynamics of the first-, second- and third-order b-positons of the KE equation are discussed in detail, and the approximate trajectories and space-dependent ‘phase shift’ of the collision of b-positons are depicted by explicit expressions, respectively, which may be used to predict where rogue wave occurs.     

Multiplicity results for two kinds of equivariant systems

Two equivariant problems of the form $\epsilon \mathrm{\Delta }u=\nabla F\left(u\right)$ are considered, where $F$ is a real function which is invariant under the action of a group $G$, and, using Morse theory, for each problem an arbitrarily great number of orbits of solutions is founded, choosing $\epsilon$ suitably small.The first problem is a $O\left(2\right)$-equivariant system of two equations, which can be seen as a complex Ginzburg-Landau equation, while the second one is a system of $m$ equations which is equivariant for the action of a finite group of real orthogonal matrices $m\times m$.     

The Cauchy problem for a generalized Camassa–Holm equation

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces $H^s$, $s>3/2$ for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in $H^r$-topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.     

Existence of strong minimizers for the Griffith static fracture model in dimension two

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization to the vectorial situation of the decay estimate by De Giorgi, Carriero, and Leaci. This is based on replacing the coarea formula by a method to approximate $SB{D}^p$ functions with small jump set by Sobolev functions, and is restricted to two dimensions. The other two ingredients will appear in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy $GSB{D}^p$ functions by $SB{V}^p$ ones.     

Signed graphs and the freeness of the Weyl subarrangements of type Bℓ

A Weyl arrangement is the hyperplane arrangement defined by a root system. Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type $A_{\ell }$ are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of their graph. In addition, the Weyl subarrangements of type $B_{\ell }$ can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness of the arrangements between type $A_{\ell -1}$ and type $B_{\ell }$. In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type $B_{\ell }$ under certain assumption.     

Solving bivariate systems using Rational Univariate Representations

Given two coprime polynomials $P$ and $Q$ in $\mathbb{Z}[x,y]$ of degree bounded by $d$ and bitsize bounded by $\tau$, we address the problem of solving the system $\{P,Q\}$. We are interested in certified numerical approximations or, more precisely, isolating boxes of the solutions. We are also interested in computing, as intermediate symbolic objects, rational parameterizations of the solutions, and in particular Rational Univariate Representations (RURs), which can easily turn many queries on the system into queries on univariate polynomials. Such representations require the computation of a separating form for the system, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system.We present new algorithms for computing linear separating forms, RUR decompositions and isolating boxes of the solutions. We show that these three algorithms have worst-case bit complexity ${\tilde{O}}_B(d^6+d^5\tau )$, where $\tilde{O}$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity. We also present probabilistic Las Vegas variants of our two first algorithms, which have expected bit complexity ${\tilde{O}}_B(d^5+d^4\tau )$. A key ingredient of our proofs of complexity is an amortized analysis of the triangular decomposition algorithm via subresultants, which is of independent interest.     

Some statistics on Stirling permutations and Stirling derangements

A permutation of the multiset $\{1,1,2,2,\dots ,n,n\}$ is called a Stirling permutation of order $n$ if every entry between the two occurrences of $i$ is greater than $i$ for each $i\in \{1,2,\dots ,n\}$. In this paper, we introduce the definitions of block, even indexed entry, odd indexed entry, Stirling derangement, marked permutation and bicolored increasing binary tree. We first study the joint distribution of ascent plateaux, even indexed entries and left-to-right minima over the set of Stirling permutations of order $n$. We then present an involution on Stirling derangements.     

Interval conjectures for level Hilbert functions

We conjecture that the set of all Hilbert functions of (artinian) level algebras enjoys a very natural form of regularity, which we call the Interval Conjecture (IC): If, for some positive integer α, $(1,h_1,\dots ,h_i,\dots ,h_e)$ and $(1,h_1,\dots ,h_i+\alpha ,\dots ,h_e)$ are both level h-vectors, then $(1,h_1,\dots ,h_i+\beta ,\dots ,h_e)$ is also level for each integer $\beta =0,1,\dots ,\alpha$. In the Gorenstein case, i.e. when $h_e=1$, we also supply the Gorenstein Interval Conjecture (GIC), which naturally generalizes the IC, and basically states that the same property simultaneously holds for any two symmetric entries, say $h_i$ and $h_{e?i}$, of a Gorenstein h-vector.These conjectures are inspired by the research performed in this area over the last few years. A series of recent results seems to indicate that it will be nearly impossible to characterize explicitly the sets of all Gorenstein or of all level Hilbert functions. Therefore, our conjectures would at least provide the existence of a very strong — and natural — form of order in the structure of such important and complicated sets.We are still far from proving the conjectures at this point. However, we will already solve a few interesting cases, especially when it comes to the IC, in this paper. Among them, that of Gorenstein h-vectors of socle degree 4, that of level h-vectors of socle degree 2, and that of non-unimodal level h-vectors of socle degree 3 and any given codimension.     

Hybrid stochastic epidemic SIR models with hidden states

This paper focuses on realistic hybrid SIR models that take into account stochasticity. The proposed systems are applicable to most incidence rates that are used in the literature including the bilinear incidence rate, the Beddington–DeAngelis incidence rate, and a Holling type II functional response. Given that many diseases can lead to asymptomatic infections, this paper looks at a system of stochastic differential equations that also includes a class of hidden state individuals, for which the infection status is unknown. Assuming that the direct observation of the percentage of hidden state individuals being infected, $\alpha \left(t\right)$, is not given and only a noise-corrupted observation process is available. Using nonlinear filtering techniques in conjunction with an invasion type analysis, this paper shows that the long-term behavior of the disease is governed by a threshold $\lambda \in \mathbb{R}$ that depends on the model parameters. It turns out that if $\lambda <0$ the number $I\left(t\right)$ of infected individuals converges to zero exponentially fast (extinction). However, if $\lambda >0$, the infection is endemic and the system is persistent. We showcase our theorems by applying them in some illuminating examples.     

Adaptive estimation for stochastic damping Hamiltonian systems under partial observation

The paper considers a process $Z_t=(X_t,Y_t)$ where $X_t$ is the position of a particle and $Y_t$ its velocity, driven by a hypoelliptic bi-dimensional stochastic differential equation. Under adequate conditions, the process is stationary and geometrically $\beta$-mixing. In this context, we propose an adaptive non-parametric kernel estimator of the stationary density $p$ of $Z$, based on $n$ discrete time observations with time step $\delta$. Two observation schemes are considered: in the first one, $Z$ is the observed process, in the second one, only $X$ is measured. Estimators are proposed in both settings and upper risk bounds of the mean integrated squared error (MISE) are proved and discussed in each case, the second one being more difficult than the first one. We propose a data driven bandwidth selection procedure based on the Goldenshluger and Lespki (2011) method. In both cases of complete and partial observations, we can prove a bound on the MISE asserting the adaptivity of the estimator. In practice, we take advantage of a very recent improvement of the Goldenshluger and Lespki (2011) method provided by Lacour et al. (2016), which is computationally efficient and easy to calibrate. We obtain convincing simulation results in both observation contexts.     

Graphical characterization of positive definite non symmetric quasi-Cartan matrices

It is known that each positive definite quasi-Cartan matrix $A$ is $\mathbb{Z}$-equivalent to a Cartan matrix $A_{\Delta }$ called Dynkin type of $A$, the matrix $A_{\Delta }$ is uniquely determined up to conjugation by permutation matrices. However, in most of the cases, it is not possible to determine the Dynkin type of a given connected quasi-Cartan matrix by simple inspection. In this paper, we give a graph theoretical characterization of non-symmetric connected quasi-Cartan matrices. For this purpose, a special assemblage of blocks is introduced. This result complements the approach proposed by Barot (1999, 2001), for ${\mathbb{A}}_n$, ${\mathbb{D}}_n$ and ${\mathbb{E}}_m$ with $m=6,7,8$.