Analytically Computing the Moments of a Conic Combination of Independent Noncentral Chi-Square Random Variables and Its Application for the Extended Cox–Ingersoll–Ross Process with... by Sanae Rujivan, Athinan Sutchada, Kittisak Chumpong, Napat Rujeerapaiboon

“…This paper focuses mainly on the problem of computing the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>γ</mi><mi>th</mi></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, moment of a random variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Y</mi><mi>n</mi></msub><mo>:</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>α</mi><mi>i</mi></msub><msub><mi>X</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> in which the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>α</mi><mi>i</mi></msub></semantics></math></inline-formula>’s are positive real numbers and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mi>i</mi></msub></semantics></math></inline-formula>’s are independent and distributed according to noncentral chi-square distributions. …”
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Turán and Ramsey problems for alternating multilinear maps by Youming Qiao

“…A simple example of this is the statement that every sequence of real numbers has a subsequence that is either increasing or decreasing (though that can be proved directly with a simpler argument). …”
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The Inverse and General Inverse of Trapezoidal Fuzzy Numbers with Modified Elementary Row Operations by Mashadi, Yuliana Safitri, Sukono, Igif Gimin Prihanto, Muhamad Deni Johansyah, Moch Panji Agung Saputra

“…Based on these conditions, we show that various properties that apply to real numbers also apply to any trapezoidal fuzzy number. …”
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Even Order Half-Linear Differential Equations with Regularly Varying Coefficients by Vojtěch Růžička

“…We establish nonoscillation criterion for the even order half-linear differential equation <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi></msup><msup><mfenced separators="" open="(" close=")"><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>Φ</mo><mfenced separators="" open="(" close=")"><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></mfenced></mfenced><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msup><msub><mi>β</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><msup><mfenced separators="" open="(" close=")"><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>Φ</mo><mfenced separators="" open="(" close=")"><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>l</mi><mo>)</mo></mrow></msup></mfenced></mfenced><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>l</mi><mo>)</mo></mrow></msup><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math display="inline"><semantics><mrow><msub><mi>β</mi><mn>0</mn></msub><mo>,</mo><msub><mi>β</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>β</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> are real numbers, <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mo>Φ</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><msup><mfenced open="|" close="|"><mi>s</mi></mfenced><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>sgn</mi><mi>s</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub></semantics></math></inline-formula> is a regularly varying (at infinity) function of the index <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>−</mo><mi>l</mi><mi>p</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. …”
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Additive energies on discrete cubes by Jaume de Dios Pont, Rachel Greenfeld, Paata Ivanisvili, Jos\'e Madrid

“…If one devises a suitable inductive hypothesis involving functions taking values in $[0,1]$ rather than simply in $\{0,1\}$, most of the proof ends up being straightforward, but one is left needing an inequality for the real numbers, which often turns out to be surprisingly delicate and surprisingly hard to prove. …”
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On the Domain of the Four-Dimensional Sequential Band Matrix in Some Double Sequence Spaces by Orhan Tuğ, Vladimir Rakočević, Eberhard Malkowsky

“…Let <i>E</i> represent any of the spaces <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">M</mi> <mi>u</mi> </msub> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">C</mi> <mi>ϑ</mi> </msub> </semantics> </math> </inline-formula><inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>ϑ</mi> <mo>=</mo> <mo>{</mo> <mi>b</mi> <mo>,</mo> <mi>b</mi> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>}</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">L</mi> <mi>q</mi> </msub> </semantics> </math> </inline-formula><inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mo>∞</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> of bounded, <inline-formula> <math display="inline"> <semantics> <mi>ϑ</mi> </semantics> </math> </inline-formula>-convergent, and <i>q</i>-absolutely summable double sequences, respectively, and <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>E</mi> <mo stretchy="false">˜</mo> </mover> </semantics> </math> </inline-formula> be the domain of the four-dimensional (4D) infinite sequential band matrix <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>(</mo> <mover accent="true"> <mi>r</mi> <mo stretchy="false">˜</mo> </mover> <mo>,</mo> <mover accent="true"> <mi>s</mi> <mo stretchy="false">˜</mo> </mover> <mo>,</mo> <mover accent="true"> <mi>t</mi> <mo stretchy="false">˜</mo> </mover> <mo>,</mo> <mover accent="true"> <mi>u</mi> <mo stretchy="false">˜</mo> </mover> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> in the double sequence space <i>E</i>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>r</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi>m</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mo>∞</mo> </msubsup> <mo>,</mo> <mspace width="3.33333pt"></mspace> <mover accent="true"> <mi>s</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>m</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mo>∞</mo> </msubsup> <mo>,</mo> <mspace width="3.33333pt"></mspace> <mover accent="true"> <mi>t</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mo>∞</mo> </msubsup> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>u</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mo>∞</mo> </msubsup> </mrow> </semantics> </math> </inline-formula> are given sequences of real numbers in the set <inline-formula> <math display="inline"> <semantics> <mrow> <mi>c</mi> <mo>∖</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula>. …”
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Algebraic subjects / theory of matrices : and applications by Afriat, S

“…The <em>upper modulus</em> |<em>a</em>|* and the <em>lower modulus</em> |<em>a</em>|_* of any matrix <em>a</em> with complex elements are defined as the non- negative real numbers whose squares are the maximum and the minimum characteristic values respectively of the non-negative definite Hermitian matrix <em>&amp;amacr;</em>'<em>a</em>, and the <em>absolute trace</em> |<em>a</em>|₍₊₎ and the <em>absolute determinant</em> |<em>a</em>|_(x) are defined at the non-negative real numbers whose square are the trace and the determinant respectively of <em>&amp;amacr;</em>'<em>a</em>.…”

New Results on the SSIE with an Operator of the form <i>F</i>_Δ ⊂ <i>Ɛ</i> + <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>F</mi><mi>x</mi><mo>′</m... by Bruno de Malafosse

“…Given any sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mi>a</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> of positive real numbers and any set <i>E</i> of complex sequences, we can use <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>a</mi></msub></semantics></math></inline-formula> to represent the set of all sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mi>y</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>/</mo><mi>a</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mi>y</mi><mi>n</mi></msub><mo>/</mo><msub><mi>a</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub><mo>∈</mo><mi>E</mi></mrow></semantics></math></inline-formula>. …”
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