A partition is called a t-core if none of its hook lengths is a multiple of t. Let $a_t(n)$ denote the number of t-core partitions of n. Garvan, Kim and Stanton proved that for any $n\geq1$ and $m\geq1$, $a_t\big(t^mn-(t^2-1)/24\big)\equiv0\pmod{t^m}$, where $t\in\{5,7,11\}$. Let $A_{t,k}(n)$ denote the number of partition k-tuples of n with t-cores. Several scholars have been subsequently investigated congruence properties modulo high powers of 5 for $A_{5,k}(n)$ with $k\in\{2,3,4\}$. In this paper, by utilizing a recurrence related to the modular equation of fifth order, we establish dozens of congruence families modulo high powers of 5 satisfied by $A_{5,k}(n)$, where $4\leq k\leq25$. Moreover, we deduce an infinite family of internal congruences modulo high powers of 5 for $A_{5,4}(n)$. In particular, we generalize greatly a recent result on a congruence family modulo high powers of 5 enjoyed by $A_{5,4}(n)$, which was proved by Saikia, Sarma and Talukdar (Indian J. Pure Appl. Math., 2024). Finally, we conjecture that there exists a similar phenomenon for $A_{5,k}(n)$ with $k\geq26$.

Let X be a smooth threefold over an algebraically closed field of positive characteristic. We prove that an arbitrary flop of X is smooth. To this end, we study Gorenstein curves of genus one and two-dimensional elliptic singularities defined over imperfect fields.

This paper is a continuation of a project to determine which skew polynomial algebras $S = R[\theta; \alpha]$ satisfy property $(\diamond)$, namely that the injective hull of every simple S-module is locally Artinian, where k is a field, R is a commutative Noetherian k-algebra and α is a k-algebra automorphism of R. Earlier work (which we review) and further analysis done here lead us to focus on the case where S is a primitive domain and R has Krull dimension 1 and contains an uncountable field. Then we show first that if $|\mathrm{Spec}(R)|$ is infinite then S does not satisfy $(\diamond)$. Secondly, we show that when $R = k[X]_{ \lt X \gt }$ and $\alpha (X) = qX$ where $q \in k \setminus \{0\}$ is not a root of unity then S does not satisfy $(\diamond)$. This is in complete contrast to our earlier result that, when $R = k[[X]]$ and α is an arbitrary k-algebra automorphism of infinite order, S satisfies $(\diamond)$. A number of open questions are stated.

An element of a group is called strongly reversible or strongly real if it can be expressed as a product of two involutions. We provide necessary and sufficient conditions for an element of $\mathrm{SL}(n,\mathbb{C})$ to be a product of two involutions. In particular, we classify the strongly reversible conjugacy classes in $\mathrm{SL}(n,\mathbb{C})$.

Let f be a non-constant meromorphic function. We define its linear differential polynomial $ L_k[f] $ by

\begin{equation*}L_k[f]=\displaystyle b_{-1}+\sum_{j=0}^{k}b_jf^{(j)}, \text{where}\; b_j (j=0, 1, 2, \ldots, k) \; \text{are constants with}\; b_k\neq 0.\end{equation*}

$ L_k[f] $

$ f^n+g^n=1 $

It is shown that if $\{H_n\}_{n \in \omega}$ is a sequence of groups without involutions, with $1 \lt |H_n| \leq 2^{\aleph_0}$, then the topologist’s product modulo the finite words is (up to isomorphism) independent of the choice of sequence. This contrasts with the abelian setting: if $\{A_n\}_{n \in \omega}$ is a sequence of countably infinite torsion-free abelian groups, then the isomorphism class of the product modulo sum $\prod_{n \in \omega} A_n/\bigoplus_{n \in \omega} A_n$ is dependent on the sequence.

Let S be an orientable, connected surface of finite topological type, with genus $g \leqslant 2$, empty boundary and complexity at least 2; we prove that any graph endomorphism of the curve graph of S is actually an automorphism. Also, as a complement of the author’s previous results, we prove that under mild conditions on the complexity of the underlying surfaces, any graph morphism between curve graphs is induced by a homeomorphism of the surfaces.

To prove these results, we construct a finite subgraph whose union of iterated rigid expansions is the curve graph $\mathcal{C}(S)$. The sets constructed, and the method of rigid expansion, are closely related to Aramayona and Leininger’s finite rigid sets. We prove as a consequence that Aramayona and Leininger’s rigid set also exhausts $\mathcal{C}(S)$ via rigid expansions. The combinatorial rigidity results follow as an immediate consequence, based on the author’s previous results.

We show that the loop space of a moment-angle complex associated to a two-dimensional simplicial complex decomposes as a finite type product of spheres, loops on spheres and certain indecomposable spaces which appear in the loop space decomposition of Moore spaces. We also give conditions on certain subcomplexes under which, localised away from sufficiently many primes, the loop space of a moment-angle complex decomposes as a finite type product of spheres and loops on spheres.

Let (W, S) be a Coxeter system of rank n, and let $p_{(W, S)}(t)$ be its growth function. It is known that $p_{(W, S)}(q^{-1}) \lt \infty$ holds for all $n \leq q \in \mathbb{N}$. In this paper, we will show that this still holds for $q = n-1$, if (W, S) is 2-spherical. Moreover, we will prove that $p_{(W, S)}(q^{-1}) = \infty$ holds for $q = n-2$, if the Coxeter diagram of (W, S) is the complete graph. These two results provide a complete characterization of the finiteness of the growth function in the case of 2-spherical Coxeter systems with a complete Coxeter diagram.

Chan and Seceleanu have shown that if a weighted shift operator on $\ell^p(\mathbb{Z})$, $1\leq p \lt \infty$, admits an orbit with a non-zero limit point then it is hypercyclic. We present a new proof of this result that allows to extend it to very general sequence spaces. In a similar vein, we show that, in many sequence spaces, a weighted shift with a non-zero weakly sequentially recurrent vector has a dense set of such vectors, but an example on $c_0(\mathbb{Z})$ shows that such an operator is not necessarily hypercyclic. On the other hand, we obtain that weakly sequentially hypercyclic weighted shifts are hypercyclic. Chan and Seceleanu have, moreover, shown that if an adjoint multiplication operator on a Bergman space admits an orbit with a non-zero limit point then it is hypercyclic. We extend this result to very general spaces of analytic functions, including the Hardy spaces.

We study the problem of conjugating a diffeomorphism of the interval to (positive) powers of itself. Although this is always possible for homeomorphisms, the smooth setting is rather interesting. Besides the obvious obstruction given by hyperbolic fixed points, several other aspects need to be considered. As concrete results we show that, in class C1, if we restrict to the (closed) subset of diffeomorphisms having only parabolic fixed points, the set of diffeomorphisms that are conjugate to their powers is dense, but its complement is generic. In higher regularity, however, the complementary set contains an open and dense set. The text is complemented with several remarks and results concerning distortion elements of the group of diffeomorphisms of the interval in several regularities.

Let $(A,\mathfrak{m} )$ be a hypersurface local ring of dimension $d \geq 1$ and let I be an $\mathfrak{m} $-primary ideal. We show that there is a integer rI$\geq\;-1$ (depending only on I) such that if M is any non-free maximal Cohen–Macaulay (= MCM) A-module the function $n \rightarrow \ell(\operatorname{Tor}^A_1(M, A/I^{n+1}))$ (which is of polynomial type) has degree rI. Analogous results hold for Hilbert polynomials associated to Ext-functors. Surprisingly, a key ingredient is the classification of thick subcategories of the stable category of MCM A-modules (obtained by Takahashi, see [11, 6.6]).

A classical result of Reinhold Baer states that a group G = XN, which is the product of two normal supersoluble subgroups X and N, is supersoluble if and only if Gʹ is nilpotent. This result has been weakened in [6] for a finite group G: in fact, we do not need that both X and N are normal, but only that N is normal and X permutes with every maximal subgroup of each Sylow subgroup of N.

In our paper, we improve the result mentioned above by showing that we only need X to permute with the maximal subgroups of the non-cyclic Sylow subgroups of N. Also, we extend this result (and several others) to relevant classes of infinite groups.

The central idea behind our results stems from grasping the key aspects of what happens in [6]. It turns out that tensor products play a very crucial role, and it is precisely this shift of perspective that makes it possible not only to improve those results but also extend them to infinite groups.

We show that for all real biquadratic fields not containing $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7}$ and $\sqrt{13}$, the Pythagoras number of the ring of algebraic integers is at least 6. We also provide an upper bound on the norm and the minimal (codifferent) trace of additively indecomposable integers in some families of these fields.

In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain $\Omega \subset \mathbb{R}^n$ in order to have the following Hardy inequality:

\begin{equation*}\int_{\Omega} W(x) |\nabla u|_A^2 dx \geq \int_{\Omega} |\nabla d|^2_AH(x)|u|^2 dx, \,\,\, u \in C^{1}_0(\Omega),\end{equation*}

where d(x) is a suitable quasi-norm (gauge), $|\xi|^2_A = \langle A(x)\xi, \xi \rangle$ for $\xi \in \mathbb{R}^n$ and A(x) is an n × n symmetric, uniformly positive definite matrix defined on a bounded domain $\Omega \subset \mathbb{R}^n$. We also give its Lp analogue. As a consequence, we present examples for a standard Laplacian on $\mathbb{R}^n$, Baouendi–Grushin operator, and sub-Laplacians on the Heisenberg group, the Engel group and the Cartan group. Those kind of characterisations for a pair of functions $(W(x),H(x))$ are obtained also for the Rellich inequality. These results answer the open problems of Ghoussoub-Moradifam [16].

In this paper, we prove a new uncertainty principle for functions with radial symmetry by differentiating a radial version of the Stein–Weiss inequality. The difficulty is to prove the differentiability in the limit of the best constant that unlike the general case it is not known. We provide also an integral alternative formula for the logarithmic weight $(\log|\xi|)$ in Fourier domain.

Let q be a power of a prime p, let $\mathbb F_q$ be the finite field with q elements and, for each nonconstant polynomial $F\in \mathbb F_{q}[X]$ and each integer $n\ge 1$, let $s_F(n)$ be the degree of the splitting field (over $\mathbb F_q$) of the iterated polynomial $F^{(n)}(X)$. In 1999, Odoni proved that $s_A(n)$ grows linearly with respect to n if $A\in \mathbb F_q[X]$ is an additive polynomial not of the form $aX^{p^h}$; moreover, if q = p and $B(X)=X^p-X$, he obtained the formula $s_{B}(n)=p^{\lceil \log_p n\rceil}$. In this paper we note that $s_F(n)$ grows at least linearly unless $F\in \mathbb F_q[X]$ has an exceptional form and we obtain a stronger form of Odoni’s result, extending it to affine polynomials. In particular, we prove that if A is additive, then $s_A(n)$ resembles the step function $p^{\lceil \log_p n\rceil}$ and we indeed have the identity $s_A(n)=\alpha p^{\lceil \log_p \beta n\rceil}$ for some $\alpha, \beta\in \mathbb Q$, unless A presents a special irregularity of dynamical flavour. As applications of our main result, we obtain statistics for periodic points of linear maps over $\mathbb F_{q^i}$ as $i\to +\infty$ and for the factorization of iterates of affine polynomials over finite fields.

We prove that any compact R-analytic group is linear when R is a pro-p domain of characteristic zero.

We show that dualising transfer maps in Hochschild cohomology of symmetric algebras over complete discrete valuations rings commutes with Tate duality. This is analogous to a similar result for Tate cohomology of symmetric algebras over fields. We interpret both results in the broader context of Calabi–Yau triangulated categories.