Quantitative Weighted Bounds for Littlewood-Paley Functions Generated by Fractional Heat Semigroups Related with Schrödinger Operators by Li Yang, Pengtao Li

“…Let L=−Δ+V be a Schrödinger operator on ℝn, where Δ denotes the Laplace operator ∑i=1n∂2/∂xi2 and V is a nonnegative potential belonging to a certain reverse Hölder class RHqℝn with q>n/2. …”
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Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence by Zhao Xiaoqing, Yi Yuan

“…Let $ q $ be a sufficiently large odd integer, and let $ c \in\left(1, \frac{4}{3}\right) $. We denote $ R(c; q) $ as the count of square-free numbers in the intersection of the Lehmer set and the Piatetski-Shapiro sequence. …”
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The Number of Blocks of a Graph with Given Minimum Degree by Lei Li, Baoyindureng Wu

“…A block of a graph is a nonseparable maximal subgraph of the graph. We denote by bG the number of block of a graph G. We show that, for a connected graph G of order n with minimum degree k≥1, bG<2k−3/k2−k−1n. …”
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Witt group of Hermitian forms over a noncommutative discrete valuation ring by L. Oukhtite

“…Moreover, for a noncommutative discrete-valued division algebraD with valuation ring R and residual division algebra D¯, we prove that W(D¯)≅WT(R), where WT(R) denotes the Witt group of regular Hermitian forms on finitely generated torsion R-modules.…”
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A closer look at some new identities by Geoffrey B. Campbell

“…Among these is ∏k=3∞(1−Z)φ,(k)/k=11−Zexp(Z32(1−Z)2−12Z−12Z(1−Z)),  |Z|<1, in which φ3(k) denotes for fixed k, the number of positive integer solutions of (a,b,k)=1 where a<b<k, assuming (a,b,k) is the gcd(a,b,k).…”
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A weak ergodic theorem for infinite products of Lipschitzian mappings by Simeon Reich, Alexander J. Zaslavski

“…For a Lipschitzian self-mapping A of K, we denote by Lip(A) its Lipschitz constant. In this paper, we establish a convergence property of infinite products of Lipschitzian self-mappings of K. …”
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Exact Solutions of the 3D Fractional Helmholtz Equation by Fractional Differential Transform Method by Saleh Alshammari, Salah Abuasad

“…Different values of fractional derivatives are signified explicitly in three dimensions. Outcomes are denoted in the tables for nonfractional exact and approximate solutions and fractional approximate solutions for different fractional orders. …”
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International integration in higher educaton by the case of French universities by L. D. Efanova, V. L. Polyakova

“…The role of international organizations, in particular UNESCO, in the process of integration in higher education has been revealed. The article denotes economic benefit of foreign students, state budget revenues from their education have been considered. …”
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Efficient Two-Derivative Runge-Kutta-Nyström Methods for Solving General Second-Order Ordinary Differential Equations y′′(x)=f(x,y,y′) by T. S. Mohamed, N. Senu, Z. B. Ibrahim, N. M. A. Nik Long

“…This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form y′′(x)=f(x,y,y′) including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. …”
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Some Properties of Subclasses of Multivalent Functions by Muhammet Kamali, Fatma Sağsöz

“…The authors introduce two new subclasses denoted by ℑ∗(Ω,𝜆,𝑝,𝛼) and ℑ∗𝜀(Ω,𝜆,𝑝,𝛼) of the class 𝐴(𝑝,𝑛) of 𝑝-valent analytic functions. …”
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A generalization of a theorem by Cheo and Yien concerning digital sums by Curtis N. Cooper, Robert E. Kennedy

“…For a non-negative integer n, let s(n) denote the digital sum of n. Cheo and Yien proved that for a positive integer x, the sum of the terms of the sequence{s(n):n=0,1,2,…,(x−1)}is (4.5)xlogx+0(x). …”
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Conditional Matching Preclusion Number of Graphs by Yalan Li, Shumin Zhang, Chengfu Ye

“…The conditional matching preclusion number of a graph G, denoted by mp1G, is the minimum number of edges whose deletion results in the graph with no isolated vertices that has neither perfect matching nor almost-perfect matching. …”
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Equivalence classes of matrices over a finite field by Gary L. Mullen

“…Let Fq=GF(q) denote the finite field of order q and F(m,q) the ring of m×m matrices over Fq. …”
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Further Properties of Trees with Minimal Atom-Bond Connectivity Index by Jianping Liu, Jinsong Chen

“…Let G=(V,E) be a graph the atom-bond connectivity (ABC) index is defined as the sum of weights ((du+dv-2)/dudv)1/2 over all edges uv of G, where du denotes the degree of a vertex u of G. In this paper, we determined a few structural features of the trees with minimal ABC index also we characterized the trees with dia[T]=2 and minimal ABC index, where [T] is induced by the vertices of degree greater than 2 in T and dia[T] is the diameter of [T].…”
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Sharp Power Mean Bounds for the Combination of Seiffert and Geometric Means by Yu-Ming Chu, Ye-Fang Qiu, Miao-Kun Wang

“…Here, Mp(a,b), P(a,b), and G(a,b) denote the power of order p, Seiffert, and geometric means of two positive numbers a and b, respectively.…”
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The Atom-Bond Connectivity Index of Catacondensed Polyomino Graphs by Jinsong Chen, Jianping Liu, Qiaoliang Li

“…The atom-bond connectivity (ABC) index is defined as the sum of weights ((du+dv−2)/dudv)1/2 over all edges uv of G, where du denotes the degree of a vertex u of G. In this paper, we give the atom-bond connectivity index of the zigzag chain polyomino graphs. …”
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Innovative zero-valent cobalt decoration on MIL-88 A(Fe)@β-CD for high-efficiency and reusable cr(VI) removal by Eman M. Abd El-Monaem, Nouf Al Harby, Mervette El Batouti, Abdelazeem S. Eltaweil

“…The experimental findings denoted that the ZVCo decoration boosted the adsorption capability of MIL-88 A(Fe)@β-CD, where the adsorption % of Cr(VI) improved from 73.07 to 94.02% after its decoration with 10 wt% of ZVCo. …”
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Formulas for Generalized Two-Qubit Separability Probabilities by Paul B. Slater

“…These yield that part of the total separability probability, P(k,α), for generalized (real, complex, quaternionic, etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality ρPT>ρ holds. Here ρ denotes a 4×4 density matrix, obtained by tracing over the pure states in 4×(4+k)-dimensions, and ρPT denotes its partial transpose. …”
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The combinational structure of non-homogeneous Markov chains with countable states by A. Mukherjea, A. Nakassis

“…Let P(s,t) denote a non-homogeneous continuous parameter Markov chain with countable state space E and parameter space [a,b], −∞<a<b<∞. …”
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Explicit Bounds and Sharp Results for the Composition Operators Preserving the Exponential Class by Fernando Farroni, Raffaella Giova

“…Let f:Ω⊂Rn→Rn be a quasiconformal mapping whose Jacobian is denoted by Jf and let EXP(Ω) be the space of exponentially integrable functions on Ω. …”
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