Estimación retrospectiva de los casos iniciales de COVID-19 en Santiago Región Metropolitana en Chile by Jenny Márquez, David García-García, María Isabel Vigo, César Bordehore

“…Conclusions: The official records of COVID-19 infections in SRM and Chile underestimated the real number of positives and showed a delay of about a month in the dynamics of infections. …”
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Effects of the Numerical Values of the Parameters in the Gielis Equation on Its Geometries by Lin Wang, David A. Ratkowsky, Johan Gielis, Paolo Emilio Ricci, Peijian Shi

“…We also set <i>n</i>₁ and <i>n</i>₂ to take negative real numbers rather than only taking positive real numbers, then classify the curves based on extremal properties of <i>r</i>(φ) at φ = 0, π/<i>m</i> when <i>n</i>₁ and <i>n</i>₂ are in different intervals, and analyze how <i>n</i>₁, <i>n</i>₂ precisely affect the shapes of Gielis curves.…”
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On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices by Akbar Ali, Shahzad Ahmed, Zhibin Du, Wei Gao, Muhammad Aslam Malik

“…The general sum-connectivity index of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }(G)$ </tex-math></inline-formula>, is defined as <inline-formula> <tex-math notation="LaTeX">$\sum _{uv\in E(G)}(d(u)+d(v))^{\alpha }$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$uv$ </tex-math></inline-formula> is the edge connecting the vertices <inline-formula> <tex-math notation="LaTeX">$u,v\in V(G)$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$d(w)$ </tex-math></inline-formula> denotes the degree of a vertex <inline-formula> <tex-math notation="LaTeX">$w\in V(G)$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> is a non-zero real number. For <inline-formula> <tex-math notation="LaTeX">$\alpha =-1/2$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n\geq 11$ </tex-math></inline-formula>, Wang <italic>et al.…”
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On Trees with Given Independence Numbers with Maximum Gourava Indices by Ying Wang, Adnan Aslam, Nazeran Idrees, Salma Kanwal, Nabeela Iram, Asima Razzaque

“…A topological index of a molecular graph is a real number that is invariant under graph isomorphism conditions and provides information about its size, symmetry, degree of branching, and cyclicity. …”
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Mathematical Properties of Variable Topological Indices by José M. Sigarreta

“…In this paper we study two general topological indices <inline-formula><math display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>B</mi><mi>α</mi></msub></semantics></math></inline-formula>, defined for each graph <inline-formula><math display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>,</mo><mi>E</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> by <inline-formula><math display="inline"><semantics><mrow><msub><mi>A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>i</mi><mi>j</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msub><mi>f</mi><msup><mrow><mo>(</mo><msub><mi>d</mi><mi>i</mi></msub><mo>,</mo><msub><mi>d</mi><mi>j</mi></msub><mo>)</mo></mrow><mi>α</mi></msup></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>B</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msub><mi>h</mi><msup><mrow><mo>(</mo><msub><mi>d</mi><mi>i</mi></msub><mo>)</mo></mrow><mi>α</mi></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><msub><mi>d</mi><mi>i</mi></msub></semantics></math></inline-formula> denotes the degree of the vertex <i>i</i> and <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is any real number. Many important topological indices can be obtained from <inline-formula><math display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>B</mi><mi>α</mi></msub></semantics></math></inline-formula> by choosing appropriate symmetric functions and values of <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. …”
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General Atom-Bond Sum-Connectivity Index of Graphs by Abeer M. Albalahi, Emina Milovanović, Akbar Ali

“…This paper is concerned with the general atom-bond sum-connectivity index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><msub><mi>S</mi><mi>γ</mi></msub></mrow></semantics></math></inline-formula>, which is a generalization of the recently proposed atom-bond sum-connectivity index, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> is any real number. For a connected graph <i>G</i> with more than two vertices, the number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><msub><mi>S</mi><mi>γ</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined as the sum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><msup><mrow><mo>(</mo><msub><mi>d</mi><mi>x</mi></msub><mo>+</mo><msub><mi>d</mi><mi>y</mi></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mi>γ</mi></msup></semantics></math></inline-formula> over all edges <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mi>y</mi></mrow></semantics></math></inline-formula> of the graph <i>G</i>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>x</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>y</mi></msub></semantics></math></inline-formula> represent the degrees of the vertices <i>x</i> and <i>y</i> of <i>G</i>, respectively. …”
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Production-inventory-distribution coordination and performance optimization for integrated multi-stage supply chains by Zhao, Shitao

“…The computational results demonstrate that the difference in the optimal total operational costs between integer and real-number solutions is not significant. In the second part of the research, both the joint consideration of inventory replenishment and an SCOR model are adopted as the coordination mechanism and the framework in an integrated supply chain with constant demand. …”
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Equidistribution of values of linear forms on a cubic hypersurface by Chow, S

“…Let <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi> <mo class="MathClass-rel">∈</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mi>r</mi></mrow></msup></math>, and let η be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi> <mo class="MathClass-rel">∈</mo> <msup><mrow><mrow><mo class="MathClass-open">[</mo><mrow><mo class="MathClass-bin">−</mo><mi>P</mi><mo class="MathClass-punc">,</mo><mi>P</mi></mrow><mo class="MathClass-close">]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></math> to the system <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>x</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">=</mo> <mn>0</mn></math>, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo class="MathClass-rel">|</mo><mi>L</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>x</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-bin">−</mo><mi>τ</mi><mo class="MathClass-rel">|</mo> <mo class="MathClass-rel">&lt;</mo> <mi>η</mi></math>. …”

Sharp Bounds on the Generalized Multiplicative First Zagreb Index of Graphs with Application to QSPR Modeling by Sakander Hayat, Farwa Asmat

“…In the context of graph theory, the generalized multiplicative first Zagreb index of a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> is defined as the product of the sum of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>α</mo></semantics></math></inline-formula>th powers of the vertex degrees of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>α</mo></semantics></math></inline-formula> is a real number such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>α</mo><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>α</mo><mo>≠</mo><mn>1</mn></mrow></semantics></math></inline-formula>. …”
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Degree-Based Graph Entropy in Structure–Property Modeling by Sourav Mondal, Kinkar Chandra Das

“…Now, the <i>k</i>-th degree-based graph entropy for <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mrow><mi>d</mi><mo>,</mo><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><mfenced separators="" open="(" close=")"><mfrac><mrow><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow><mi>k</mi></msup></mrow><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo>(</mo><msub><mi>v</mi><mi>j</mi></msub><mo>)</mo></mrow><mi>k</mi></msup></mrow></mfrac><mspace width="0.166667em"></mspace><mi>l</mi><mi>o</mi><mi>g</mi><mspace width="0.166667em"></mspace><mfrac><mrow><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow><mi>k</mi></msup></mrow><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo>(</mo><msub><mi>v</mi><mi>j</mi></msub><mo>)</mo></mrow><mi>k</mi></msup></mrow></mfrac></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> where <i>k</i> is real number. The first-degree-based entropy is generated for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, which has been well nurtured in last few years. …”
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Standards of specialized diabetes care. Edited by Dedov I.I., Shestakova M.V., Mayorov A.Yu. 9th edition by Ivan I. Dedov, Marina V. Shestakova, Aleksandr Y. Mayorov, Olga K. Vikulova, Gagik R. Galstyan, Tamara L. Kuraeva, Valentina A. Peterkova, Olga M. Smirnova, Elena G. Starostina, Elena V. Surkova, Olga Y. Sukhareva, Alla Y. Tokmakova, Minara S. Shamkhalova, Ivona Renata Jarek-Martynova, Ekaterina V. Artemova, Diana D. Beshlieva, Olga N. Bondarenko, Natalya N. Volevodz, Olga R. Grigoryan, Irina S. Gomova, Zera N. Dzhemilova, Roza M. Esayan, Liudmila I. Ibragimova, Viktor Y. Kalashnikov, Irina V. Kononenko, Dmitry N. Laptev, Dmitry V. Lipatov, Oleg G. Motovilin, Tatiana V. Nikonova, Roman V. Rozhivanov, Ekaterina A. Shestakova

“…Results of Russian epidemiological study (NATION) con- firmed that only 54% of Type 2 DM are diagnosed. So real number of patients with DM in Russia is 9 million patients (about 6% of population). …”
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Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function by Wen-Xiu Ma

“…Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> </semantics> </math> </inline-formula> be two integers with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>l</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, <i>c</i> a real number greater than or equal to 1, and <i>f</i> a multivariable function satisfying <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>3</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>w</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> when <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. …”
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A Class of Bounded Iterative Sequences of Integers by Artūras Dubickas

“…In this note, we show that, for any real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>∈</mo><mo>[</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, any finite set of positive integers <i>K</i> and any integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, the sequence of integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><msub><mi>s</mi><mn>3</mn></msub><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula> satisfying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>s</mi><mi>i</mi></msub><mo>∈</mo><mi>K</mi></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mi>i</mi></msub></semantics></math></inline-formula> is a prime number, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>≤</mo><msub><mi>s</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≤</mo><mi>τ</mi><msub><mi>s</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mi>i</mi></msub></semantics></math></inline-formula> is a composite number, is bounded from above. …”
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On General Reduced Second Zagreb Index of Graphs by Lkhagva Buyantogtokh, Batmend Horoldagva, Kinkar Chandra Das

“…The graph invariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula>, known under the name general reduced second Zagreb index, is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub><mrow><mo>(</mo><mi mathvariant="normal">Γ</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi mathvariant="normal">Γ</mi><mo>)</mo></mrow></msub><mrow><mo>(</mo><msub><mi>d</mi><mi mathvariant="normal">Γ</mi></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>α</mi><mo>)</mo></mrow><mrow><mo>(</mo><msub><mi>d</mi><mi mathvariant="normal">Γ</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>α</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi mathvariant="normal">Γ</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the degree of the vertex <i>v</i> of the graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Γ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is any real number. In this paper, among all trees of order <i>n</i>, and all unicyclic graphs of order <i>n</i> with girth <i>g</i>, we characterize the extremal graphs with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>≥</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></semantics></math></inline-formula>. …”
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Standards of Specialized Diabetes Care / Edited by Dedov I.I., Shestakova M.V., Mayorov A.Yu. 11th Edition by I. Dedov, M. Shestakova, A. Mayorov, N. Mokrysheva, E.  Andreeva, O. Bezlepkina, V. Peterkova, E. Artemova, P. Bardiugov, D. Beshlieva, O. Bondarenko, F. Burumkulova, O. Vikulova, N. Volevodz, G. Galstyan, I. Gomova, O. Grigoryan, Z. Dzhemilova, L. Ibragimova, V. Kalashnikov, I. Kononenko, T. Kuraeva, D. Laptev, D. Lipatov, O. Melnikova, M. Mikhina, M. Michurova, O. Motovilin, T. Nikonova, R. Rozhivanov, O. Smirnova, E. Starostina, E. Surkova, O. Sukhareva, A. Tiselko, A. Tokmakova, M. Shamkhalova, E. Shestakova, I. Jarek-Martynowa, M. Yaroslavceva

“…Results of Russian epidemiological study (NATION) confirmed that 54% of patients with Type 2 DM are undiagnosed. So real number of patients with DM in Russia is 11-12 million patients (about 7% of population). …”
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On Some Properties of the Limit Points of (<i>z</i>(<i>n</i>)/<i>n</i>)_<i>n</i> by Eva Trojovská, Kandasamy Venkatachalam

“…A recent result of Trojovská implies the existence of a positive real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup><mo>∩</mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the empty set. …”
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Standards of specialized diabetes care. Edited by Dedov I.I., Shestakova M.V., Mayorov A.Yu. 10th edition by Ivan I. Dedov, Marina V. Shestakova, Alexander Yu. Mayorov, Natalya G. Mokrysheva, Olga K. Vikulova, Gagik R. Galstyan, Tamara L. Kuraeva, Valentina A. Peterkova, Olga M. Smirnova, Elena G. Starostina, Elena V. Surkova, Olga Y. Sukhareva, Alla Y. Tokmakova, Minara S. Shamkhalova, Ivona Renata Jarek-Martynova, Ekaterina V. Artemova, Diana D. Beshlieva, Olga N. Bondarenko, Natalya N. Volevodz, Irina S. Gomova, Olga R. Grigoryan, Zera N. Dzhemilova, Roza M. Esayan, Liudmila I. Ibragimova, Viktor Y. Kalashnikov, Irina V. Kononenko, Dmitry N. Laptev, Dmitry V. Lipatov, Olga G. Melnikova, Margarina S. Mikhina, Marina S. Michurova, Oleg G. Motovilin, Tatiana V. Nikonova, Roman V. Rozhivanov, Igor A. Sklyanik, Ekaterina A. Shestakova

“…Results of Russian epidemiological study (NATION) confirmed that only 54% of Type 2 DM are diagnosed. So real number of patients with DM in Russia is 10 million patients (about 7% of population). …”
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On the General Sum Distance Spectra of Digraphs by Weige Xi, Lixiang Cai, Wutao Shang, Yidan Su

“…Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>T</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mi>diag</mi><mrow><mo stretchy="false">(</mo><mi>S</mi><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><mi>S</mi><msub><mi>T</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><mi>S</mi><msub><mi>T</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> be the diagonal matrix with the vertex sum transmissions of <i>G</i> in the diagonal and zeroes elsewhere. For any real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the general sum distance matrix of <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><msub><mi>D</mi><mi>α</mi></msub><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mi>α</mi><mi>S</mi><mi>T</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mi>S</mi><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>.…”
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Jordan semi-triple derivations and Jordan centralizers on generalized quaternion algebras by Ai-qun Ma, Lin Chen, Zijie Qin

“…In this paper, we investigate Jordan semi-triple derivations and Jordan centralizers on generalized quaternion algebras over the field of real numbers. We prove that every Jordan semi-triple derivation on generalized quaternion algebras over the field of real numbers is a derivation. …”
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Method for Constructing a Commutative Algebra of Hypercomplex Numbers by Alpamys T. Ibrayev

“…This article demonstrates the following for the first time: (1) the possibility of constructing a normed commutative algebra of quaternions and octonions with division over the field of real numbers; (2) the possibility of constructing a normed commutative algebra of six-dimensional and ten-dimensional hypercomplex numbers with division over the field of real numbers; (3) a method for constructing a normed commutative algebra of N-dimensional hypercomplex numbers with division over the field of real numbers for even values of N; and (4) the possibility of constructing a normed commutative algebra of other N-dimensional hypercomplex numbers with division over the field of real numbers. …”
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