Classification of bm-Nambu structures of top degree

In this paper, we classify $b^m$-Nambu structures via $b^m$-cohomology. The complex of $b^m$-forms is an extension of the De Rham complex, which allows us to consider singular forms. $b^m$-Cohomology is well understood thanks to Scott (2016) [12], and it can be expressed in terms of the De Rham cohomology of the manifold and of the critical hypersurface using a Mazzeo–Melrose-type formula. Each of the terms in $b^m$-Mazzeo–Melrose formula acquires a geometrical interpretation in this classification. We also give equivariant versions of this classification scheme.     

Generating regular elements

Let R be a prime right Goldie ring. A useful fact is that, if $a,b\in R$ are such that $aR+bR$ contains a regular element, then there exists $\lambda \in R$ such that $a+b\lambda$ is regular. We show that the analogous result holds for $n?1$ pairs of elements: if R contains a field of cardinality at least $n+1$, and if $a_i,b_i\in R$ are such that $a_{i}R+b_{i}R$ contains a regular element for $1?i?n$, then there exists a single element $\lambda \in R$ such that $a_i+b_i\lambda$ is regular for each i.     

The intermediate disorder regime for a directed polymer model on a hierarchical lattice

We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number $b\in \mathbb{N}$ and a segment number $s\in \mathbb{N}$. When $b\le s$ it is known that the model exhibits strong disorder for all positive values of the inverse temperature $\beta$, and thus weak disorder reigns only for $\beta =0$ (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature $\beta \equiv {\beta }_n$ vanishes at an appropriate rate as the size $n$ of the system grows. Our analysis requires separate treatment for the cases $b<s$ and $b=s$. In the case $b<s$ we prove that when the inverse temperature is taken to be of the form ${\beta }_n=\stackrel{?}{\beta }{(b/s)}^{n/2}$ for $\stackrel{?}{\beta }>0$, the normalized partition function of the system converges weakly as $n\to \infty$ to a distribution $L\left(\stackrel{?}{\beta }\right)$ and does so universally with respect to the initial weight distribution. We prove the convergence using renormalization group type ideas rather than the standard Wiener chaos analysis. In the case $b=s$ we find a critical point in the behavior of the model when the inverse temperature is scaled as ${\beta }_n=\stackrel{?}{\beta }/n$; for an explicitly computable critical value ${\kappa }_b>0$ the variance of the normalized partition function converges to zero with large $n$ when $\stackrel{?}{\beta }\le {\kappa }_b$ and grows without bound when $\stackrel{?}{\beta }>{\kappa }_b$. Finally, we prove a central limit theorem for the normalized partition function when $\stackrel{?}{\beta }\le {\kappa }_b$.     

Almost split sequences for polynomial GrT-modules and polynomial parts of Auslander–Reiten components

In [8], Doty, Nakano and Peters defined infinitesimal Schur algebras, combining the approach via polynomial representations with the approach via $G_{r}T$-modules to representations of the algebraic group $G={\mathrm{GL}}_n$. We study analogues of these algebras and their Auslander–Reiten theory for reductive algebraic groups G and Borel subgroups B by considering the categories of polynomial representations of $G_{r}T$ and $B_{r}T$ as full subcategories of $\mathrm{mod}{G}_{r}T$ and $\mathrm{mod}{B}_{r}T$, respectively. We show that every component Θ of the stable Auslander–Reiten quiver ${\mathrm{\Gamma }}_s(G_{r}T)$ of $\mathrm{mod}{G}_{r}T$ whose constituents have complexity 1 contains only finitely many polynomial modules. For $G={\mathrm{GL}}_2,r=1$ and $T?G$ the torus of diagonal matrices, we identify the polynomial part of the stable Auslander–Reiten quiver of $G_{r}T$ and use this to determine the Auslander–Reiten quiver of the infinitesimal Schur algebras in this situation. For the Borel subgroup B of lower triangular matrices of ${\mathrm{GL}}_2$, the category of $B_{r}T$-modules is related to representations of elementary abelian groups of rank r. In this case, we can extend our results about modules of complexity 1 to modules of higher Frobenius kernels arising as outer tensor products.     

Characterization of Lipschitz spaces via commutators of the Hardy–Littlewood maximal function

Let M be the Hardy–Littlewood maximal function and b be a locally integrable function. Denote by $M_b$ and $[b,M]$ the maximal commutator and the (nonlinear) commutator of M with b. In this paper, the author considers the boundedness of $M_b$ and $[b,M]$ on Lebesgue spaces and Morrey spaces when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.     

The F-pure threshold of quasi-homogeneous polynomials

Inspired by the work of Bhatt and Singh [3] we compute the F-pure threshold of quasi-homogeneous polynomials. We first consider the case of a curve given by a quasi-homogeneous polynomial f in three variables $x,y,z$ of degree equal to the degree of xyz and then we proceed with the general case of a Calabi–Yau hypersurface, i.e. a hypersurface given by a quasi-homogeneous polynomial f in $n+1$ variables $x_0,\dots ,x_n$ of degree equal to the degree of $x_0?x_n$.     

A way to construct independence equivalent graphs

Let us denote the independence polynomial of a graph by $I_G\left(x\right)$. If $I_G\left(x\right)=I_H\left(x\right)$ implies that $G?H$ then we say $G$ is independence unique. For graph $G$ and $H$ if $I_G\left(x\right)=I_H\left(x\right)$ but $G$ and $H$ are not isomorphic, then we say $G$ and $H$ are independence equivalent. In [7], Brown and Hoshino gave a way to construct independent equivalent graphs for circulant graphs. In this work we give a way to construct the independence equivalent graphs for general simple graphs and obtain some properties of the independence polynomial of paths and cycles.     

Quadratic forms and congruences for ℓ-regular partitions modulo 3, 5 and 7

Let $b_{\ell }(n)$ be the number of ℓ-regular partitions of n. We show that the generating functions of $b_{\ell }(n)$ with $\ell =3,5,6,7$ and 10 are congruent to the products of two items of Ramanujan's theta functions $\psi (q)$, $f(-q)$ and ${(q;q)}_{\infty }^3$ modulo 3, 5 and 7. So we can express these generating functions as double summations in q. Based on the properties of binary quadratic forms, we obtain vanishing properties of the coefficients of these series. This leads to several infinite families of congruences for $b_{\ell }(n)$ modulo 3, 5 and 7.     

On vector fields with simply connected trajectories and one invariant line

We study polynomial vector fields X on ${\mathbb{C}}^2$ which have simply connected trajectories and satisfy $dP(X)=a?P$, for a constant $a\in {\mathbb{C}}^{?}$ and a primitive polynomial $P\in \mathbb{C}[x,y]$. We determine X, up to an algebraic change of coordinates. In particular, we obtain that X is complete.