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 значения $P(\zeta_1),P(\zeta_2)$ и не зависящие от степени $n$.
Библиография: 11 названий.
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 proximate order $\rho(r)$ and proximate lower order $\lambda(r)$ of a function $f$ meromorphic in $\mathbb{C}$ such that either $\underline{\Delta}(D)=\liminf_{r\to+\infty}T(r,f)/D(r)>0$ and $\overline{\Delta}(H)=\limsup_{r\to+\infty}T(r,f)/H(r)=+\infty$ or $\underline{\Delta}(D)=0$ and $\overline{\Delta}(H)<+\infty,$ where $D(r)=\exp\left\{\displaystyle\int\nolimits_1^r\dfrac{\rho(t)}{t}dt\right\}$ and $H(r)=\exp\left\{\displaystyle\int\nolimits_1^r\dfrac{\lambda(t)}{t}dt\right\}.$ The obtained results reveal the incorrectness of Lemma 1 and Theorem 1 from the paper by S. H. Dwivedi [S. H. Dwivedi, Compos. Math., 22, No. 1, 39–48 (1970)]. We also generalize the statement of Lemma 2 in the cited paper as follows: If a function $\phi(r)$ is such that $r\phi'(r)/\phi(r)\to\phi_0$ as $r \to +\infty,$ then $\displaystyle\int\nolimits_1^r\dfrac{\phi(t)}{t^{1+\alpha}}dt \sim \dfrac{\phi(r)}{(\phi_0-\alpha)r^\alpha}$ for $0 \le \alpha < \phi_0$ and $\displaystyle\int\nolimits_r^{+\infty}\dfrac{\phi(t)}{t^{1+\alpha}}dt \sim \dfrac{\phi(r)}{(\alpha-\phi_0)r^\alpha}$ for $\alpha > \phi_0$ as $r \to +\infty.$
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 $A$, порождающего в $B$ сильно непрерывную группу $e^{tA}$, $t \in \mathbb{R}$, ограниченных операторов; нелинейная часть уравнения в определенном смысле подчинена $A$ и быстро осциллирует по времени. К указанной задаче Коши применен метод усреднения Крылова-Боголюбова и дано его обоснование.
Библиография: 10 названий.
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 duality and the corresponding optimality system under general normal-integrand assumptions. This Fenchel duality framework is used to derive $\textit{a posteriori}$ error identities for subgradient flows. In doing so, we depart from the usual duality-based $\textit{a posteriori}$ error control framework in the unsteady setting, since the Br\'ezis-Ekeland-Nayroles formulation reveals the following unsteady feature: the minimal primal value and the maximal dual value are both prescribed by the initial datum. This allows us to pass from a combined primal-dual gap identity to separate primal and dual gap identities. These identities quantify the primal and dual errors independently and admit representations in terms of generalized Bregman divergences and, under a spatial convex conjugation formula, as non-negative time-space integral quantities suitable for localization. The abstract framework is applied to a number of variational problems of physical interest, including the unsteady heat equation, the unsteady Stokes equations, the unsteady Navier-Lam\'e equations, the unsteady Bingham flow through a pipe, the unsteady obstacle problem, and the unsteady elasto-plastic torsion problem.
H. Antil, Alex Kaltenbach, Keegan L. A. Kirk· 0 citations
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