The set of integer partitions of n, wherein the parts are either divisible by 4 or congruent to $\pm 1 \pmod {6}$, arose in a natural way in work of Dombos [‘A new grounded partition identity of type $D_4^{(3)}$’, SIGMA 22 (2026), Article no. 042]. Denote the function that counts the number of such partitions of n by $\mathit {dp}(n)$. Using elementary generating function manipulations and classical q-series results, we prove several congruences satisfied by $\mathit {dp}(n)$. As an example, we prove that, for all $\alpha \geq 1$ and all $n \geq 0$,

$$ \begin{align*} {\it dp} \bigg( 3^{2\alpha + 1}n + \frac{7 \cdot 9^\alpha + 1}{4} \bigg) \equiv 0 \pmod{3}. \end{align*} $$

There are only three known strongly invertible hyperbolic L-space knots with braid index four and tunnel number two. They are t09284, t10496 and o9_34409 in the SnapPy census. In this paper, we give the first infinite family of strongly invertible hyperbolic L-space knots with braid index four and tunnel number two that includes t10496 and o9_34409.

We construct explicit group-algebra elements corresponding to the elementary matrices in the Wedderburn decomposition of a finite group algebra. These elements yield an explicit algebra isomorphism that coincides with the classical Wedderburn isomorphism. As applications, we apply our construction to the symmetric group to compute the transition matrix between two natural bases of the centre and reinterpret the Frobenius characteristic map through the Wedderburn decomposition. Consequently, we show that the Schur basis is naturally compatible with the identity-matrix basis arising from the Wedderburn decomposition.

Let M be a closed oriented $3$-manifold generically embedded in a complex $3$-manifold X. Its CR singular set is an oriented link $L\subset M$. We prove that if a sublink $L'\subset L$ bounds an oriented Seifert surface $S\subset M$ in the complement of $L\setminus L'$, then the CR singularities along $L'$ can be cancelled by an arbitrarily $\mathcal {C}^0$-small isotopy supported in an arbitrarily small neighbourhood of S.

We investigate meromorphic solutions of the difference equation

$$ \begin{align*} f(z)^2+f(z+c)=h(z), \quad c\in\mathbb C\setminus\{0\}, \end{align*} $$

$ N(r,h)+N(r,{1}/{h})=S(r,h). $

$1$

$$ \begin{align*} f(z)=e^{az+b}-1, \quad h(z)=e^{2az+2b}, \end{align*} $$

$a,b\in \mathbb C$

$e^{ac}=2$

$1$

We show that an autoregressive time series on Banach space can be canonically identified with a finite direct product of self-contained and self-determined natural components defined on separate subspaces by the spectral projections for each spectral point of the characteristic linear pencil. We assume the resolvent of the pencil is analytic everywhere except for the spectral points which may be poles or isolated essential singularities. Each natural component is represented as a forward, backward or outward flow fixed by an upstream boundary condition in the far distant past, the far distant future or at a nominated initial time – depending on whether the corresponding spectral point is outside, inside or on the unit circle.

Let B be a ball in $\mathbb {R}^{N}$, $N\geq 1$, let $m\in C(\overline {B}) $ be a radial function that changes sign in B and let $0<q<1$. We consider the existence and nonexistence of (nontrivial) nonnegative solutions and positive solutions for semilinear elliptic problems of the form $-\Delta u=m( x) u^{q}$ in B, $u=\partial _{\nu }u=0$ on $\partial B$, where $\partial _{\nu }$ is the exterior normal derivative.

For a nonempty subset B of a group G, let $\mu _{G,B}(n,c)$ be the least possible value of $\max _{\varnothing \ne S\subseteq B,\ |S|\le c}|AS|$ over all n-element subsets $A\subseteq G$. Given $b_0\in B$, set $D=b_0^{-1}B$ and $H=\langle D\rangle $. We prove that $\mu _{G,B}(n,c)$ is determined exactly by the based affine-core $(H,D)$. The proof uses a simultaneous block compression that controls all admissible multiplier sets at once. As a consequence of the exact determination, $\mu _{\mathbb Z,\lambda B}(n,c)=\mu _{\mathbb Z,B}(n,c)$ for every nonempty $B\subseteq \mathbb Z$ and every nonzero integer $\lambda $. This proves, in the stronger form of an exact identity, the natural nonzero-dilation version of a conjecture of Bollobás et al. [‘Large sumsets from small subsets’, Israel J. Math. 268(1) (2025), 253–314].

The game domination numbers $\gamma _{\mathrm {g}}$ and $\gamma _{\mathrm {g}}^\prime $, and the game total domination numbers $\gamma _{\mathrm {{tg}}}$ and $\gamma _{\mathrm {{tg}}}^\prime $ are investigated on Sierpiński graphs $S_p^n$. For the domination game, the trivial bounds arising from the known domination number and the Grundy domination number of $S_p^n$ are significantly improved by proving that $(2p-3)p^{n-2}\leq \gamma _{\mathrm {g}}(S_p^n), \gamma _{\mathrm {g}}^\prime (S_p^n) \leq (2p-2)p^{n-2}$. For the total domination game, the bounds $\gamma _{\mathrm {{tg}}}(S_p^n), \gamma _{\mathrm {{tg}}}^\prime (S_p^n) \geq (2p-2)p^{n-2}$ are established.

We establish an effective form of Benford’s law for a class of strictly increasing and log-concave sequences $(f_n)_{n\geq 0}$. We apply our result to the partition and plane partition functions, obtaining improvements on bounds due to Iyer for the first occurrence of prescribed leading digit strings.

The return probability in the simple symmetric random walk on the four-dimensional lattice $\mathbb {Z}^4$ is evaluated in closed form in terms of the generalised hypergeometric function.

Hodges and Yong [‘Coxeter combinatorics and spherical Schubert geometry’, J. Lie Theory 32 (2022), 447–474] proposed a reduced word criterion for Levi spherical Schubert varieties. It was proved in type A and in rank two, while counterexamples were found in types $D_4$ and $F_4$. A later standard Coxeter element criterion settled Levi sphericity in every finite Lie type, but whether the original criterion remained valid in types B and C was still open. We prove that the two criteria agree in both families. These are the last unresolved infinite Weyl families. Consequently, the original reduced word criterion characterises Levi sphericity for every Weyl group element and every allowed Levi subgroup precisely in types A, B, C and $G_2$. The paper completes the classification of irreducible finite Weyl types for which the original criterion is valid.

Let $\mathbb {F}$ be either $\mathbb {R}$ or $\mathbb {C}$, and put $d_{\mathbb {F}}=\dim _{\mathbb {R}}\mathbb {F}$. Let $A_m\in \mathbb {F}^{N_m\times m}$ have independent standard Gaussian entries and assume $N_m/m\to \gamma>1$. We determine the exponential scale of the smallest least singular value among all square row submatrices:

$$ \begin{align*} M_m^{\mathbb{F}}=\min_{T\subset [N_m],\, |T|=m}\sigma_{\mathrm{min}}(A_{m,T}). \end{align*} $$

With $c_\gamma =\gamma ^\gamma /(\gamma -1)^{\gamma -1}$, we prove $m^{-1}\log M_m^{\mathbb {F}}\xrightarrow {P}-d_{\mathbb {F}}^{-1}\log c_\gamma $ and, when $N_m=\gamma m+O(1)$, the convergence holds at rate $O(m^{-1})$. As an application, for real phase retrieval at the critical number $N=2m-1$ of measurements, the Balan–Wang stability parameter satisfies $\omega (A_m)=4^{-m+o_P(m)}$, so the sharp exponential base in the Gaussian Balan–Wang problem is $1/4$. The lower estimate extends to all bounded-density ensembles and continuity is necessary.

The theory of Liouville numbers has played a fundamental role in transcendence theory since Liouville’s pioneering construction of explicit transcendental numbers. A classical result of Adams [‘On the algebraic independence of certain Liouville numbers’, J. Pure Appl. Algebra 13(1) (1978), 41–47] established the algebraic independence of certain Liouville series associated with multiplicatively independent integer bases. We prove non-Archimedean analogues of Adam’s theorem in two distinct settings: the field of p-adic numbers and function fields. We establish algebraic independence results for families of Liouville-type series defined in these settings, extending Adams’ theorem from the classical real case to the non-Archimedean framework.

In this paper, we establish lower bounds for extreme values of derivatives of Dirichlet L-functions in the range $1/2<\sigma <1$. Compared with the work of Aistleitner et al. [‘On large values of $L(\sigma , \chi )$’, Q. J. Math. 70 (2019), 831–848], our result shows that derivatives of Dirichlet L-functions can attain extreme values of the same order of magnitude as the original L-functions in the interval ${1/2<\sigma <1}$.

Let A be an $n\times n$ doubly stochastic matrix. The Marcus–Ree inequality asserts that $\|A\|_{\mathrm F}^2\leq \mathrm {maxtrace}(A)$. Matrices attaining equality are called Erdős matrices. Recent work of Karmakar et al. [‘Characterization of Erdős matrices by their zero entries’, Linear Algebra Appl. 739 (2026), 154–169] shows that every Erdős matrix is a restricted common diagonal sum matrix. By the structural theory of Brualdi and Dahl [‘Diagonal sums of doubly stochastic matrices’, Linear Multilinear Algebra 70 (2022), 4946–4972], a restricted common diagonal sum matrix with fully indecomposable skeleton $S=(s_{ij})$ admits additive potentials satisfying $a_{ij}=(u_i+v_j)s_{ij}$. Karmakar et al. observed that $\min _i u_i+\min _jv_j\geq 0$ is sufficient for Erdős-ness. We prove that it is also necessary. Thus, recognition reduces to solving one linear system per fully indecomposable block and checking signs, with no residual maxtrace computation.