In this paper, we consider the problem of maximizing a monotone submodular function subject to knapsack constraint in streaming setting. such setting, elements arrive sequentially and at any point time, algorithm can store only small fraction that have arrived so far. For special case all unit sizes (i.e., cardinality-constraint case), one find $$(0.5-\varepsilon )$$ -approximate solution $$O(K...

Abstract A bipartite graph $H = \left (V_1, V_2; E \right )$ with $\lvert V_1\rvert + \lvert V_2\rvert n$ is semilinear if $V_i \subseteq \mathbb {R}^{d_i}$ for some $d_i$ and the edge relation consists of pairs points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination s linear equalities inequalities in $d_1 d_2$ variables . We show that k , number edges $K_{k,k}$ -free H a...

For the Landau-de Gennes functional on 3D domains, $$\begin{aligned} I_\varepsilon (Q,\Omega ):=\int _{\Omega }\left\{ \frac{1}{2}|\nabla Q|^2+\frac{1}{\varepsilon ^2}\left( -\frac{a^2}{2}\mathrm {tr}(Q^2)-\frac{b^2}{3}\mathrm {tr}(Q^3)+\frac{c^2}{4}[\mathrm {tr}(Q^2)]^2 \right) \right\} \,dx, \end{aligned}$$ it is well-known that under suitable boundary conditions, global minimizer $$Q_\vareps...

We study the renormalisation of non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric scalar field theory with interaction $\phi^2(i\phi)^\varepsilon$ using Wilsonian approach and without any expansion in $\varepsilon$. Specifically, we solve Wetterich equation local potential approximation, both ultraviolet regime loop expansion. calculate scale-dependent effective its infrared limit. The is found ...

This article investigates the nonlinear, steady boundary layer flow and heat transfer of an incompressible Eyring-Powell non-Newtonian fluid from an isothermal sphere with Biot number effects. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller Box technique. The influ...

For Fr$acute{mathbf{text{e}}}$chet algebras $(A, (p_n))$ and $(B, (q_n))$, a linear map $T:Arightarrow B$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(Tab - Ta Tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{N}$, $a, b in A$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$...

For any prime $p$, let $y(p)$ denote the smallest integer $y$ such that every reduced residue class $\pmod p$ is represented by product of some subset $\{1,\dots,y\}$. It easy to see at least as large quadratic nonresidue p$; we prove $y(p) \ll_\varepsilon p^{1/(4 \sqrt e)+\varepsilon}$, thus strengthening Burgess's classical result. This result intermediate strength between two other results, ...

Let \((X, d)\) be a semimetric space and let \(G\) graph. We say that is the diametrical graph of if \(X\) vertex set adjacency vertices \(x\) \(y\) equivalent to equality \(\diam X = d(x, y)\). It shown with diameter \(d^*\) ultrametric iff d_{\varepsilon})\) \(d_{\varepsilon}(x, y) \min\{d(x, y), \varepsilon\}\) complete multipartite for every \(\varepsilon \in (0, d^*)\). A refinement last r...