For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[ \min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, equality for $s^+$ holds exactly for trees, whereas equality for $s^-$ holds exactly for trees and complete graphs. The proof combines the $P_3$-removal lemma in the no-cut-vertex case with a detailed equality analysis of the underlying doubly nonnegative matrix inequality. Every block is forced to be complete, and a minimal-counterexample argument gives an exact rank-one decomposition of the folded matrix $M^c$. The resulting non-edge vanishings, together with $AX=XA$, rule out an interface between a bridge and a nontrivial block.
We prove that the bracket width of the Lie algebra of vector fields on any smooth affine algebraic variety of dimension $n$ is at most $(n+1)^2$. We give improved bounds for some families of $\mathbb{C}^*$-varieties, in particular for $\mathrm{SL}_n(\mathbb{C})$ and for the Koras--Russell cubic threefold.
In this paper, we provide an alternative proof of Chandee and Li's result on the second moment of GL4×GL2$ \mathrm{GL}_4 \times \mathrm{GL}_2$ special L$L$ ‐values. Our method is conceptually more direct as it neither detects the “Eisenstein–Kloosterman” cancelation nor uses the Poisson summation formula.
Z. Qi, Rui-Hua Qiao· Bulletin of the London Mathe...· 0 citations
Пусть $P$ - полином степени $n\ge 3$ с вещественными критическими точками, и пусть $\zeta_1$, $\zeta_2$ - произвольные соседние критические точки этого полинома. Устанавливаются точные неравенства для значений $P$ и его производной на интервале $(\zeta_1,\zeta_2)$, включающие точки $\zeta_1$, $\zeta_2$, критические зна...
V. N. Dubinin· Математический сборник· 0 citations
For a function $\Gamma(r)=\exp\left\{\displaystyle\int\nolimits_1^r\dfrac{\gamma(t)}{t}dt\right\},$ $\gamma(r)$ is a proximate order, we deduce the expression $\Gamma(r)=r^{\gamma(r)}L(r),$ where $L(r)$ is a slowly varying function on $[1,+\infty),$ i.e., $rL'(r)/L(r)\to 0$ as $r\to+\infty.$ We define the notions of pr...
Микола Заболоцький, Тарас Заболоцький, Мар'яна Мостова· Ukrains'kyi Matematychnyi Zh...· 0 citations
В банаховом пространстве $B$ рассматриваются дифференциальные уравнения второго порядка, являющиеся абстрактными обобщениями гиперболических уравнений, с начальными условиями (задача Коши). Главный стационарный линейный оператор уравнения представлен квадратом, вообще говоря, неограниченного замкнутого оператора $A$, п...
V. Levenshtam, M. R. Yavaeva· Математические заметки· 0 citations
We derive duality-based $\textit{a posteriori}$ error identities for a broad class of subgradient flows induced by time-dependent convex integral functionals. Starting from the Br\'ezis-Ekeland-Nayroles principle, we identify an unsteady primal energy functional and derive its Fenchel dual formulation, including strong...
H. Antil, Alex Kaltenbach, Keegan L. A. Kirk· 0 citations
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