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Journal articles on the topic 'P^2q'

Author: Grafiati

Published: 9 March 2023

Last updated: 19 July 2025

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1

Hernández Iglesias, Mauro Fernando. "Singularidad de la polar de una curva plana irreducible en K(2p,2q,2pq+d)." Pesquimat 22, no. 1 (2019): 1–8. http://dx.doi.org/10.15381/pes.v22i1.15758.

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Veremos que existe un abierto de Zariski en el conjunto de curvas planas irreducibles con exponentes característicos 2p; 2q y 2q+d, dado por K(2p; 2q; 2q+d) con mcd{p,q} = 1 y d impar, donde la polar es no degenerada, su topología es constante y determinada apenas por p y q.

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2

Tadee, Suton, and Apirat Siraworakun. "Nonexistence of Positive Integer Solutions of the Diophantine Equation p^x + (p + 2q)^ y = z^2 , where p, q and p + 2q are Prime Numbers." European Journal of Pure and Applied Mathematics 16, no. 2 (2023): 724–35. http://dx.doi.org/10.29020/nybg.ejpam.v16i2.4702.

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The Diophantine equation p^x + (p + 2q)^y = z^2 , where p, q and p + 2q are prime numbers, is studied widely. Many authors give q as an explicit prime number and investigate the positive integer solutions and some conditions for non-existence of positive integer solutions. In this work, we gather some conditions for odd prime numbers p and q for showing that the Diophantine equation p^x + (p + 2q)^y = z^2 has no positive integer solution. Moreover, many examples of Diophantine equations with no positive integer solution are illustrated.

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3

Yaying, Taja, Bipan Hazarika, and S. A. Mohiuddine. "Domain of Padovan q-difference matrix in sequence spaces lp and l∞." Filomat 36, no. 3 (2022): 905–19. http://dx.doi.org/10.2298/fil2203905y.

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In this study, we construct the difference sequence spaces lp (P?2q) = (lp)P?2q, 1 ? p ? ?, where P = (?rs) is an infinite matrix of Padovan numbers %s defined by ?rs = {?s/?r+5-2 0 ? s ? r, 0 s > r. and ?2q is a q-difference operator of second order. We obtain some inclusion relations, topological properties, Schauder basis and ?-, ?- and ?-duals of the newly defined space. We characterize certain matrix classes from the space lp (P?2q) to any one of the space l1, c0, c or l?. We examine some geometric properties and give certain estimation for von-Neumann Jordan constant and James constant of the space lp(P). Finally, we estimate upper bound for Hausdorff matrix as a mapping from lp to lp(P).

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Asbullah, Muhammad Asyraf, Normahirah Nek Abd Rahman, uhammad Rezal Kamel Ariffin, Siti Hasana Sapar, and Faridah Yunos. "CRYPTANALYSIS OF RSA KEY EQUATION OF N=p^2q FOR SMALL |2q – p| USING CONTINUED FRACTION." Malaysian Journal of Science 39, no. 1 (2020): 72–80. http://dx.doi.org/10.22452/mjs.vol39no1.6.

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Arifin, Muchammad Choerul, and Iwan Ernanto. "IDEMPOTENT ELEMENTS IN MATRIX RING OF ORDER 2 OVER POLYNOMIAL RING $\mathbb{Z}_{p^2q}[x]$." Journal of Fundamental Mathematics and Applications (JFMA) 6, no. 2 (2023): 136–47. http://dx.doi.org/10.14710/jfma.v6i2.19307.

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An idempotent element in the algebraic structure of a ring is an element that, when multiplied by itself, yields an outcome that remains unchanged and identical to the original element. Any ring with a unity element generally has two idempotent elements, 0 and 1, these particular idempotent elements are commonly referred to as the trivial idempotent elements However, in the case of rings $\mathbb{Z}_n$ and $\mathbb{Z}_n[x]$ it is possible to have non-trivial idempotent elements. In this paper, we will investigate the idempotent elements in the polynomial ring $\mathbb{Z}_{p^2q}[x]$ with $p,q$ different primes. Furthermore, the form and characteristics of non-trivial idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$ will be investigated. The results showed that there are 4 idempotent elements in $\mathbb{Z}_{p^2q}[x]$ and 7 idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$.

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6

Basher, Mohamed. "Even vertex odd mean labeling of uniform theta graphs." Proyecciones (Antofagasta) 43, no. 1 (2024): 153–62. http://dx.doi.org/10.22199/issn.0717-6279-4894.

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Let $G$ be a graph with $p$ vertices and $q$ edges. A total graph labeling $ f:V(G)\bigcup E(G)\rightarrow \{0,1,2,3,...,2q\}$ is called even vertex odd mean labeling of a graph $G$ if the vertices of the graph $G$ label by distinct even integers from the set $\{0,2,...,2q\}$ and the labels of the edges are defined as the mean of the labels of its end vertices and these labels are $2q-1$ distinct odd integers from the set $\{1,3,5,...,2q-1\}$. In this paper we investigate the even vertex odd mean labeling of uniform theta graphs.

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7

Zhao, Xin, та Wenming Zou. "On a class of critical elliptic systems in ℝ4". Advances in Nonlinear Analysis 10, № 1 (2020): 548–68. http://dx.doi.org/10.1515/anona-2020-0136.

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Abstract In the present paper, we consider the following classes of elliptic systems with Sobolev critical growth: $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u+\lambda_1u=\mu_1 u^3+\beta uv^2+\frac{2q}{p} y u^{\frac{2q}{p}-1}v^2\quad &\hbox{in}\;{\it\Omega}, \\ -{\it\Delta} v+\lambda_2v=\mu_2 v^3+\beta u^2v+2 y u^{\frac{2q}{p}}v\quad&\hbox{in}\;{\it\Omega}, \\ u,v \gt 0&\hbox{in}\;{\it\Omega}, \\ u,v=0&\hbox{on}\;\partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝ4 is a smooth bounded domain with smooth boundary ∂Ω; p, q are positive coprime integers with 1 < $\begin{array}{} \displaystyle \frac{2q}{p} \end{array}$ < 2; μi > 0 and λi ∈ ℝ are fixed constants, i = 1, 2; β > 0, y > 0 are two parameters. We prove a nonexistence result and the existence of the ground state solution to the above system under proper assumptions on the parameters. It seems that this system has not been explored directly before.

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8

ALZER, HORST. "ON AN INTEGRAL INEQUALITY OF R. BELLMAN." Tamkang Journal of Mathematics 22, no. 2 (1991): 187–91. http://dx.doi.org/10.5556/j.tkjm.22.1991.4597.

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   We prove: if $u$ and $v$ are non-negative, concave functions defined on $[0, 1]$ satisfying  \[\int_0^1 (u(x))^{2p} dx =\int_0^1 (v(x))^{2q} dx=1, \quad p>0, \quad q>0,\] then \[\int_0^1(u(x))^p (v(x))^q dx\ge\frac{2\sqrt{(2p+1)(2q+1)}}{(p+1)(q+1)}-1.\]  

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9

Kalita, N., and A. J. Dutta. "Spectral analysis of second order quantum difference operator over the sequence space lp (1 < p < ∞)." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 118, no. 2 (2025): 122–36. https://doi.org/10.31489/2025m2/122-136.

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In this article, we study the spectrum, fine spectrum and boundedness property of second order quantum difference operator ∆2q (0 < q < 1) over the class of sequence lp (1 < p < ∞), the pth summable sequence space. The second order quantum difference operator ∆2q is a lower triangular triple band matrix ∆2q(1,−(1+ q),q). We also determine the approximate point spectrum, defect spectrum, compression spectrum, and Goldberg classification of the operator on the class of sequence. We obtained the results by solving an infinite system of linear equations and computing the inverse of a lower triangular infinite matrix. We also provide appropriate examples along with graphical representations where necessary.

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10

Kamaraj, T., and J. Thangakani. "Edge even and edge odd graceful labelings of Paley Graphs." Journal of Physics: Conference Series 1770, no. 1 (2021): 012068. http://dx.doi.org/10.1088/1742-6596/1770/1/012068.

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Abstract Edge even graceful labeling is a novel graceful labelling, introduced in 2017 by Elsonbaty and Daoud. A graph G with p vertices and q edges is called an edge even graceful if there is a bijection f: E(G) → {2, 4,. . ., 2q} such that, when each vertex is assigned the sum of the labels of all edges incident to it mod 2k, where k = max (p, q), the resulting vertex labels are distinct. A labeling of G is called edge odd graceful labeling, if there exists a bijection f from the set of edges E(G) to the set {1,3,5,…,2q-1} such that the induced the map f* from the set of vertices V(G) to {0,1,2,.,.,2q-1} given by f*(u) = Σ uv∈E(G) f(uv) (mod 2q) is an injection. A graph which admits edge even (odd) graceful labeling is called an edge even (odd) graceful graph. Paley graphs are dense undirected graphs raised from the vertices as elements of an appropriate finite field by joining pairs of vertices that differ by a quadratic residue. In this paper, we study the construction of edge even (odd) graceful labeling for Paley graphs and prove that Paley graphs of prime order are edge even (odd) graceful.

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11

Selvaraj, R., S. Vidyanandini, and Soumya Ranjan Nayak. "Even vertex odd mean labeling of some graphs." Journal of Discrete Mathematical Sciences and Cryptography 27, no. 7 (2024): 2169–77. http://dx.doi.org/10.47974/jdmsc-2089.

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This work introduces the principle of “an even point (vertex) odd ratio (mean) labeling, which is specifically applied to a graph ‘G’ consisting of ‘p’ vertices and ‘q’ edges. Even point (vertex) odd ratio (mean) labeling is exhibited by a graph G in the presence of an injectionbased function f : V of G → {0, 2, 4, ... 2q – 2, 2q} ensuring that the function derived from it (induced map) g* : E of G→{1, 3, 5, ... 2q – 1} specified by g* (uv) = g(u)+g(v)/2 is a bijection. Graphs that meet these criteria are termed an even point (vertex) odd ratio (mean) graphs. This paper explores the properties of an even point (vertex) odd ratio (mean) labeling in various graph structures.

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12

Jeyanthi, P., S. Philo, and M. K. Siddiqui. "Odd harmonious labeling of super subdivisión graphs." Proyecciones (Antofagasta) 38, no. 1 (2019): 1–11. https://doi.org/10.22199/issn.0717-6279-3408.

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A graph G(p, q) is said to be odd harmonious if there exists an injection ?: V (G) → {0, 1, 2, · · · , 2q − 1} such that the induced function ?∗: E(G) → {1, 3, · · · , 2q − 1} defined by ?∗(uv) = ? (u) + ? (v) is a bijection. In this paper we prove that super subdivision of any cycle Cm with m ≥ 3 ,ladder, cycle Cn for n ≡ 0(mod 4) with K1,m and uniform fire cracker are odd harmonious graphs.

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13

Aguirre, J., and M. Escobedo. "On the blow-up of solutions of a convective reaction diffusion equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 123, no. 3 (1993): 433–60. http://dx.doi.org/10.1017/s0308210500025828.

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SynopsisWe study the blow-up of positive solutions of the Cauchy problem for the semilinear parabolic equationwhere u is a scalar function of the spatial variable x ∈ ℝN and time t > 0, a ∈ ℝV, a ≠ 0, 1 < p and 1 ≦ q. We show that: (a) if p > 1 and 1 ≦ q ≦ p, there always exist solutions which blow up in finite time; (b) if 1 < q ≦ p ≦ min {1 + 2/N, 1 + 2q/(N + 1)} or if q = 1 and 1 < p ≦ l + 2/N, then all positive solutions blow up in finite time; (c) if q > 1 and p > min {1 + 2/N, 1 + 2q/N + 1)}, then global solutions exist; (d) if q = 1 and p > 1 + 2/N, then global solutions exist.

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14

Virk, Abaid ur Rehman, and A. Riasat. "Odd Graceful Labeling of W -Tree W T ( n , k ) and its Disjoint Union." Utilitas Mathematica 118 (January 8, 2024): 51–62. http://dx.doi.org/10.61091/um118-05.

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Let G=(V(G),E(G)) be a graph with p vertices and q edges. A graph G of size q is said to be odd graceful if there exists an injection λ:V(G)→0,1,2,…,2q−1 such that assigning each edge xy the label or weight |λ(x)–λ(y)| results in the set of edge labels being 1,3,5,…,2q−1. This concept was introduced in 1991 by Gananajothi. In this paper, we examine the odd graceful labeling of the W-tree, denoted as WT(n,k).

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15

Liu, Hailin, Bengong Lou, and Bo Ling. "Tetravalent half-arc-transitive graphs of order $p^2q^2$." Czechoslovak Mathematical Journal 69, no. 2 (2019): 391–401. http://dx.doi.org/10.21136/cmj.2019.0335-17.

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16

Firmansah, Fery, and Muhammad Ridlo Yuwono. "Odd Harmonious Labeling on Pleated of the Dutch Windmill Graphs." CAUCHY 4, no. 4 (2017): 161. http://dx.doi.org/10.18860/ca.v4i4.4043.

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A graph G(p,q) with p=|V(G)| vertices and q=|E(G)| edges. The graph G(p,q) is said to be odd harmonious if there exist an injection f: V(G)-&gt;{0,1,2,...,2q-1} such that the induced function f*: E(G)-&gt;{1,2,3,...,2q-1} defined by f*(uv)=f(u)+f(v) which is a bijection and f is said to be odd harmonious labeling of G(p,q). In this paper we prove that pleated of the Dutch windmill graphs C_4^(k)(r) with k&gt;=1 and r&gt;=1 are odd harmonious graph. Moreover, we also give odd harmonious labeling construction for the union pleated of the Dutch windmill graph C_4^(k)(r) union C_4^(k)(r) with k&gt;=1 and r&gt;=1.

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17

MORIWAKI, MASAYASU. "MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP." International Journal of Mathematics 19, no. 10 (2008): 1187–201. http://dx.doi.org/10.1142/s0129167x08005084.

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Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G′ of G, the restriction π|G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p1) × U(p2) × U(q) for p1, p2 ≥ 1 with p1 + p2 = p is multiplicity-free, whereas the restriction to U(p1) × U(p2) × U(q1) × U(q2) for q1, q2 ≥ 1 with q1 + q2 = q has infinite multiplicities.

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Li, Qiang, and Baoquan Yuan. "A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field." AIMS Mathematics 7, no. 3 (2022): 4168–75. http://dx.doi.org/10.3934/math.2022231.

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<abstract><p>In this paper, we establish a regularity criterion for the 3D nematic liquid crystal flows. More precisely, we prove that the local smooth solution $ (u, d) $ is regular provided that velocity component $ u_{3} $, vorticity component $ \omega_{3} $ and the horizontal derivative components of the orientation field $ \nabla_{h}d $ satisfy</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \int_{0}^{T}|| u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}}+||\nabla_{h} d||_{L^{a}}^{\frac{2a}{a-3}} \mbox{d} t&lt;\infty,\nonumber \\ with\ \ 3&lt; p\leq\infty,\ \frac{3}{2}&lt; q\leq\infty,\ 3&lt; a\leq\infty. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> </abstract>

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Damelin, S. B., and D. S. Lubinsky. "Necessary and Sufficient Conditions for Mean Convergence of Lagrange Interpolation for Erdős Weights." Canadian Journal of Mathematics 48, no. 4 (1996): 710–36. http://dx.doi.org/10.4153/cjm-1996-037-1.

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AbstractWe investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn(W2, x) for Erdös weights W2 = e-2Q. The archetypal example is Wk,α = exp(—Qk,α), whereα > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < ∞, Δ ∊ ℝ, k > 0. Let Ln[f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn(W2, x) = pn(e-2Q, x). Then forto hold for every continuous function ƒ: ℝ —> ℝ satisfyingit is necessary and sufficient that

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20

YALÇIN, NAZMİYE FEYZA. "SEIDEL LAPLACIAN ENERGY of ZERO-DIVISOR GRAPH Г[Z_n]". Journal of Science and Arts 24, № 4 (2024): 875–80. https://doi.org/10.46939/j.sci.arts-24.4-a10.

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The Seidel Laplacian energy of graphs has recently been defined. In the present work, we compute the Seidel Laplacian energy of the zero-divisor graph Г[Z_n] for n=p^2,n=pq, n=2q, and n=p^3, where p,q are distinct prime numbers.

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21

Subramanian, Nagarajan, M. R. Bivin, and Nallaswamy Saivaraju. "The Generalized Non-absolute type of sequence spaces." Boletim da Sociedade Paranaense de Matemática 34, no. 2 (2015): 263–74. http://dx.doi.org/10.5269/bspm.v34i1.25674.

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In this paper we introduce the notion of $\lambda_{mn}-\chi^{2}$ and $\Lambda^{2}$ sequences. Further, we introduce the spaces $\left[\chi^{2q\lambda}_{f\mu },\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ and $\left[\Lambda^{2q\lambda}_{f\mu },\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)},$ which are of non-absolute type and we prove that these spaces are linearly isomorphic to the spaces $\chi^{2}$ and $\Lambda^{2},$ respectively. Moreover, we establish some inclusion relations between these spaces.

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Damelin, S. B., and D. S. Lubinsky. "Necessary and Sufficient Conditions for Mean Convergence of Lagrange Interpolation for Erdős Weights II." Canadian Journal of Mathematics 48, no. 4 (1996): 737–57. http://dx.doi.org/10.4153/cjm-1996-038-9.

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AbstractWe complete our investigations of mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn(W2, x) for Erdős weights W2 = e-2Q. The archetypal example is Wk,α = exp(—Qk,α), whereα > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < 4 and α ∊ ℝ Let Ln[f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn(W2, x) = pn(e-2Q, x). Then forto hold for every continuous function ƒ:ℝ. —> ℝ satisfyingit is necessary and sufficient that α > 1/p. This is, essentially, an extension of the Erdös-Turan theorem on L2 convergence. In an earlier paper, we analyzed convergence for all p > 1, showing the necessity and sufficiency of using the weighting factor 1 + Q for all p > 4. Our proofs of convergence are based on converse quadrature sum estimates, that are established using methods of H. König.

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23

Kim, Youngjoon, Youngho Kim, and Jeong Seop Sim. "An Improved Order-Preserving Pattern Matching Algorithm Using Fingerprints." Mathematics 10, no. 12 (2022): 1954. http://dx.doi.org/10.3390/math10121954.

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Two strings of the same length are order isomorphic if their relative orders are the same. The order-preserving pattern matching problem is to find all substrings of text T that are order isomorphic to pattern P when T(|T|=n) and P(|P|=m) are given. An O(mn+nqlogq+q!)-time algorithm using the O(m+q!) space for the order-preserving pattern matching problem has been proposed utilizing fingerprints of q-grams based on the factorial number system and the bad character heuristic. In this paper, we propose an O(mn+2q)-time algorithm using the O(m+2q) space for the order-preserving pattern matching problem, but utilizing fingerprints of q-grams converted to binary numbers. A comparative experiment using three types of time series data demonstrates that the proposed algorithm is faster than existing algorithms because it reduces the number of order isomorphism tests.

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Abd Ghafar, Amir Hamzah, and Muhammad Rezal Kamel Ariffin. "SPA on Rabin variant with public key $$N=p^2q$$ N = p 2 q." Journal of Cryptographic Engineering 6, no. 4 (2016): 339–46. http://dx.doi.org/10.1007/s13389-016-0118-5.

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Bisgaard, Søren. "The Design and Analysis of 2k–p× 2q–rSplit Plot Experiments." Journal of Quality Technology 32, no. 1 (2000): 39–56. http://dx.doi.org/10.1080/00224065.2000.11979970.

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26

李, 世荣. "关于方程|Aut(G)|=p~2q~2的解". Chinese Science Bulletin 40, № 23 (1995): 2124–27. http://dx.doi.org/10.1360/csb1995-40-23-2124.

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Agdgomelashvili, Zurab. "Some interesting tasks from the classical number theory." Works of Georgian Technical University, no. 4(518) (December 15, 2020): 150–88. http://dx.doi.org/10.36073/1512-0996-2020-4-150-188.

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The article considers the following issues: – It’s of great interest for p and q primes to determine the number of those prime number divisors of a number 1 1 pq A p    that are less than p. With this purpose we have considered: Theorem 1. Let’s p and q are odd prime numbers and p  2q 1. Then from various individual divisors of the 1 1 pq A p    number, only one of them is less than p. A has at least two different simple divisors; Theorem 2. Let’s p and q are odd prime numbers and p  2q 1. Then all prime divisors of the number 1 1 pq A p    are greater than p; Theorem 3. Let’s q is an odd prime number, and p N \{1}, p]1;q] [q  2; 2q] , then each of the different prime divisors of the number 1 1 pq A p    taken separately is greater than p; Theorem 4. Let’s q is an odd prime number, and p{q1; 2q1}, then from different prime divisors of the number 1 1 pq A p    taken separately, only one of them is less than p. A has at least two different simple divisors. Task 1. Solve the equation 1 2 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 2. Solve the equation 1 3 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 3. Solve the equation 1 1 z x y p y    where p{6; 7; 11; 13;} are the prime numbers, x, y  N and y is a prime number. There is a lema with which the problem class can be easily solved: Lemma ●. Let’s a, b, nN and (a,b) 1. Let’s prove that if an  0 (mod| ab|) , or bn  0 (mod| ab|) , then | ab|1. Let’s solve the equations ( – ) in natural x , y numbers: I. 2 z x y z z x y          ; VI. (x  y)xy  x y ; II. (x  y)z  (2x)z  yz ; VII. (x  y)xy  yx ; III. (x  y)z  (3x)z  yz ; VIII. (x  y) y  (x  y)x , (x  y) ; IV. ( y  x)x y  x y , (y  x) ; IX. (x  y)x y  xxy ; V. ( y  x)x y  yx , (y  x) ; X. (x  y)xy  (x  y)x , (y  x) . Theorem . If a, bN (a,b) 1, then each of the divisors (a2  ab  b2 ) will be similar. The concept of pseudofibonacci numbers is introduced and some of their properties are found.

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Rottey, Sara, and Geertrui Van de Voorde. "Unitals with many Baer secants through a fixed point." Advances in Geometry 19, no. 1 (2019): 21–30. http://dx.doi.org/10.1515/advgeom-2017-0010.

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Abstract We show that a unital U in PG(2, q2) containing a point P, such that at least q2 − ϵ of the secant lines through P intersect U in a Baer subline, is an ovoidal Buekenhout–Metz unital (where ϵ ≈ 2q for q even and ϵ ≈ q3/2/2 for q odd).

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MADETI, PRABHAKAR, and RAMA MISHRA. "MINIMAL DEGREE SEQUENCE FOR TORUS KNOTS OF TYPE (p, q)." Journal of Knot Theory and Its Ramifications 18, no. 04 (2009): 485–91. http://dx.doi.org/10.1142/s021821650900704x.

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In this paper we prove the following result: for coprime positive integers p and q with p < q, if r is the least positive integer such that 2p-1 and q + r are coprime, then the minimal degree sequence for a torus knot of type (p, q) is the triple (2p - 1, q + r, d) or (q + r, 2p - 1, d) where q + r + 1 ≤ d ≤ 2q - 1.

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30

Sharutin, V. "Synthesis and structure of derivatives tetra(para-tolyl)antimony." Bulletin of the South Ural State University series "Chemistry" 15, no. 1 (2023): 50–57. http://dx.doi.org/10.14529/chem230105.

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Tetra(p-tolyl)antimony oximates and carboxylates p-Tol4SbX (X = ONCHR, R = CHCHPh (1), C6H4(Br-3 ) (2), X = OC(O)R', R' = CH2OC6H3Cl2-2.4 (3), CF2CF2C(O)OH (4). X-ray diffraction analysis, antimony atoms in complexes 1–3 have a distorted trigonal bipyramid coordination with three aryl ligands in the equatorial plane, while the CSbO axial angles are 178.94(5), 174.4(2), and 176.95(5). Crystal 4 consists of distorted tetrahedral tetra(p-tolyl)stibonium cations (CSbC angles 106.6(2)112.46(19)) and singly charged tetrafluoroethanedioic acid anions. X-ray diffraction data: (1) [C37H36NOSb, M = 632.42; triclinic syngony, sp. gr. P-1; cell parameters: a = 10.789(4) Å, b = 10.811(5) Å, c = 14.558(5) Å;  = 73.389(18), β = 75.201(15),  = 87.55(2), V = 1572.3(11) Å3, Z = 2; (calc.) = 1.336 g/cm3;  = 0.906 mm–1; F(000) = 648.0; region 2q collection: 6.04–75.9; –18 ≤ h ≤ 18, –18 ≤ k ≤ 18, –25 ≤ l ≤ 25; total reflections 115476; independent reflections 16980 (Rint = 0,0449); GOOF = 1.003; R-factor 4,58 %]; (2) [C35H33NOSbBr, M = 685.28; triclinic syngony, sp. gr. P-1; cell parameters: a = 10.719(18) Å, b = 10.731(13) Å, c = 15.85(2) Å;  = 101.53(4), β = 92.31(8),  = 119.11(5), V = 1541(4) Å3, Z = 2; (calc.) = 1.477 g/cm3;  = 2.219 mm–1; F(000) = 688.0; region 2q collection: 5.5–77.08; –16 ≤ h ≤ 16, –17 ≤ k ≤ 17, –25 ≤ l ≤ 25; total reflections 60962; independent reflections 12480 (Rint = 0.0604); GOOF = 1.429; R-factor 10.99 %]; (3) [C36H33O3Cl2Sb, M = 706.27; triclinic syngony, sp. gr. P–1; cell parameters: a = 10.621(5) Å, b = 11.016(5) Å, c = 15.809(9) Å;  = 103.55(2), β = 108.00(2),  = 98.34(2), V = 1662.1(14) Å3, Z = 2; (calc.) = 1.411 g/cm3;  = 1.024 mm–1; F(000) = 716.0; region 2q collection: 5.68–60.22; –14 ≤ h ≤ 14, –15 ≤ k ≤ 15, –22 ≤ l ≤ 22; total reflections 110814; independent reflections 9738 (Rint = 0.0348); GOOF = 1.041; R-factor 2.74 %]; (4) [C32H29F4O4Sb, M = 675,30; triclinic syngony, sp. gr. P–1; cell parameters: a = 10.223(15) Å, b = 12.011(14) Å, c = 12.949(14) Å;  = 74.32(3), β = 89.65(7),  = 86.99(5), V = 1529(3) Å3, Z = 2; (calc.) = 1.467 g/cm3;  = 0.961 mm–1; F(000) = 680.0; region collection for 2q: 6.536–56.708; –13 ≤ h ≤ 13, –16 ≤ k ≤ 16, –17 ≤ l ≤ 17; total reflec-tions 44836; independent reflections 7568 (Rint = 0.0449); GOOF = 1.052; R-factor 6.02 %]. Complete tables of atomic coordinates, bond lengths, and bond angles for compounds 1, 2, 3 and 4 have been deposited at the Cambridge Crystallographic Data Center (CCDC 2130472, 2131085, 2131084, 2126158; deposit@ccdc.cam.ac.uk; http:// www.ccdc.cam.ac.uk).

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31

Khmaladze, Estáte V. "Convergence Properties in Certain Occupancy Problems Including the Karlin-Rouault Law." Journal of Applied Probability 48, no. 4 (2011): 1095–113. http://dx.doi.org/10.1239/jap/1324046021.

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Let x denote a vector of length q consisting of 0s and 1s. It can be interpreted as an ‘opinion’ comprised of a particular set of responses to a questionnaire consisting of q questions, each having {0, 1}-valued answers. Suppose that the questionnaire is answered by n individuals, thus providing n ‘opinions’. Probabilities of the answer 1 to each question can be, basically, arbitrary and different for different questions. Out of the 2q different opinions, what number, μn, would one expect to see in the sample? How many of these opinions, μn(k), will occur exactly k times? In this paper we give an asymptotic expression for μn / 2q and the limit for the ratios μn(k)/μn, when the number of questions q increases along with the sample size n so that n = λ2q, where λ is a constant. Let p(x) denote the probability of opinion x. The key step in proving the asymptotic results as indicated is the asymptotic analysis of the joint behaviour of the intensities np(x). For example, one of our results states that, under certain natural conditions, for any z > 0, ∑1{np(x) > z} = dnz−u, dn = o(2q).

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32

Cahen, M., A. Franc, and S. Gutt. "Spectrum of the Dirac operator on complex projective space P 2q-1(?)." Letters in Mathematical Physics 18, no. 2 (1989): 165–76. http://dx.doi.org/10.1007/bf00401871.

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33

Anderson, G. D., S. Banerjee, and C. S. David. "MHC class II A alpha and E alpha molecules determine the clonal deletion of V beta 6+ T cells. Studies with recombinant and transgenic mice." Journal of Immunology 143, no. 11 (1989): 3757–61. http://dx.doi.org/10.4049/jimmunol.143.11.3757.

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Abstract Interactions between MHC class II genes and minor lymphocyte stimulating (Mls) associated products are responsible for clonally deleting self-reactive T cells in mice. Here we demonstrate the role of the intact I-A and I-E molecules as well as the individual A alpha and E alpha chains in the deletion of cells bearing the V beta 6 TCR. DBA/1 (H-2q, Mls-1a) mice were crossed with various inbred congenic, recombinant, and transgenic strains and the F1's were screened for V beta 6 expression. All I-E+ strains were fully permissive in deleting V beta 6+ T cells. I-E- strains expressing I-A b,f,s,k,p permitted only partial deletion, while I-Aq strains showed no deletion. Recombinant I-Aq and I-Af strains which expressed E kappa alpha chain in the absence of E beta chain showed a decrease in V beta 6+ T cells as compared to their H-2q and H-2f counterparts. Furthermore, transgenic mice expressing E kappa alpha Aq beta gene in an H-2q haplotype (E kappa alpha Aq beta?) gave similar results to that of the recombinants in deleting V beta 6 T-cells. The role of the 1-A molecule was also shown by the partial deletion of V beta 6+ T cells in H-2q mice expressing transgenic I-Ak molecules. These results demonstrate that the E alpha chain is important in the deletion of V beta 6 T-cells in Mls-1a mice. The role of A alpha chain is also implied by the permissiveness of E kappa alpha Aq beta but not Aq alpha Aq beta molecules in the deletion of V beta 6+ T cells.

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34

Geng, Fengjie, and Hairong Lian. "Bifurcation of Limit Cycles from a Quasi-Homogeneous Degenerate Center." International Journal of Bifurcation and Chaos 25, no. 01 (2015): 1550007. http://dx.doi.org/10.1142/s0218127415500078.

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In this paper, we deal with the following differential system [Formula: see text] where p, q are positive integers, and P(x, y), Q(x, y) are real polynomials of degree n, we obtain an upper bound for the maximum number of limit cycles bifurcating from the period annulus of a quasi-homogeneous center, that is (n - 1)p1 + (t + 1)q - 1 + 2rp1q1(q + 3) + 2tqrp1q1, where t = [n/2q] + 2, (p, q) = r(p1, q1), p1 and q1 are coprime.

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35

Numasawa, H., N. Yamamoto, and T. Shibahara. "P.42 Loss of heterozygosity on chromosome 2q inhuman oral squamous cell carcinoma." Oral Oncology Supplement 1, no. 1 (2005): 158. http://dx.doi.org/10.1016/s1744-7895(05)80406-7.

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36

Sharutin, V. "Structural features of binuclear aryl compounds of antimony." Bulletin of the South Ural State University series "Chemistry" 16, no. 4 (2024): 86–94. https://doi.org/10.14529/chem240407.

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The structures of three binuclear aryl compounds of antimony (Ar3SbX)2O (Ar = Ph, X = Cl (1), Ar = Ph, X = OC6H3(Cl-2)(F-4) (2), Ar = 3-FC6H4, X = OSO2CH2CF3 (3) were determined by X-ray diffrac-tion analysis. Crystal (1): C36H30OCl2Sb2, М 793.00; monoclinic system, symmetry group P21/n; cell parameters: a = 9.158(6), b = 19.911(14), c = 18.426(16) Å; β = 98.60(3)°, V = 3322(4) Å3; Z = 4, rcal = 1.585 g/cm3; 2q 6.06-52 deg; total reflections 56802; independent reflections 6507; number of re-fined parameters 371; Rint = 0.0331; R1 = 0.0247, wR2 = 0.0546; residual electron density (max/min): 0.49/-1.46 e/Å3; (2): C48H36O3F2Cl2Sb2 M 1013.17; symmetry group P21; cell parameters: a = 11.694(10), b = 12.754(8), c = 14.487(11) Å; V = 2161(3) Å3, Z = 2; rcal = 1.557 g/cm3; 2q 5.62-71,84 deg; total reflections 91787; independent reflections 11155; number of specified parameters 514; Rint = 0.0401; GOOF 1.033; R1 = 0.0307, wR2 = 0.0757; residual electron density (max/min): 0.49/-1.46 e/Å3]; (3) C40H28O7F12S2Sb2, M 1156.31; triclinic system, symmetry group P-1; cell parameters: a = 10.946(5), b = 20.130(10), c = 20.282(12) Å; a = 76.57(3)°, β = 78.284(18)°, g = 89.672(17)°; V = 4252(4) Å3, Z = 2; rcal = 1.689 g/cm3; 2q 5.762-52.138 deg; total reflections 93721; independent re-flections 16665; number of specified parameters 1113; Rint = 0.0548; GOOF 1.048; R1 = 0.0426, wR2 = 0.1088; residual electron density (max/min): 1.01/-0.60 e/Å3].

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37

Bemm, Laerte, and Priscila Costa Ferreira de Jesus Bemm. "Potências de p em bases da forma 2p." REMAT: Revista Eletrônica da Matemática 9, no. 2 (2023): e3003. http://dx.doi.org/10.35819/remat2023v9i2id6625.

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Nesse trabalho, vamos explorar o cálculo de potências de p em bases da forma 2p. Veremos que se p é par e n>=2, então a potência p^n (na base 2p), tem o algarismo da unidade igual a 0. Para o caso em que p é ímpar da forma p=2q+1, com q par, a potência p^n (na base 2p), tem o algarismo da unidade igual a p e o algarismo de segunda ordem igual a q. Mais ainda, veremos que tais potências podem ser obtidas recursivamente por meio de uma divisão por 2 e acréscimo de algarismos específicos a direita desse quociente.

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38

Kajiura, Hiroshige. "Cyclicity in homotopy algebras and rational homotopy theory." Georgian Mathematical Journal 25, no. 4 (2018): 545–70. http://dx.doi.org/10.1515/gmj-2018-0058.

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AbstractKadeishvili proposes a minimal{C_{\infty}}-algebra as a rational homotopy model of a space. We discuss a cyclic version of this Kadeishvili{C_{\infty}}-model and apply it to classifying rational Poincaré duality spaces. We classify 1-connected minimal cyclic{C_{\infty}}-algebras whose cohomology algebras are those of{(S^{p}\times S^{p+2q-1})\sharp(S^{q}\times S^{2p+q-1})}, where{2\leq p\leq q}. We also include a proof of the decomposition theorem for cyclic{A_{\infty}}and{C_{\infty}}-algebras.

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39

V. Pandichelvi and B. Umamaheswari. "ERCEIVING SOLUTIONS FOR AN EXPONENTIAL DIOPHANTINE EQUATION LINKING SAFE AND SOPHIE GERMAIN PRIMES qx + py= z2." jnanabha 52, no. 02 (2022): 165–67. http://dx.doi.org/10.58250/jnanabha.2022.52219.

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In this article, an exponential Diophantine equation qx + py= z2 where p , q are Safe primes and q Sophie Germain primes respectively and x, y, z are positive integers is measured for all the opportunities of x+y = 0, 1, 2, 3 and showed that all conceivable integer solutions are (p, q, x, y, z) = (7, 3, 1, 0, 2), (11, 5, 1, 1, 4), (5, 2, 3, 0, 3), (2q + 1, q, 2, 1, q + 1) by retaining basic rules of Mathematics.

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40

Chen, Beibei, Jialin Wang, and Dongni Liao. "Gradient regularity for nonlinear sub-elliptic systems with the drift term: sub-quadratic growth case." AIMS Mathematics 10, no. 1 (2025): 1407–37. https://doi.org/10.3934/math.2025065.

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<p>This paper focuses on nonlinear sub-elliptic systems with drift terms in divergence form, under Dini continuity conditions, where the growth rate satisfies $ \frac{2Q}{Q+2} &lt; m &lt; 2 $, and $ Q $ represents the homogeneous dimension in the Heisenberg group. By generalizing the $ \mathcal{A} $-harmonic approximation technique to accommodate sub-quadratic growth, we establish the $ C^1 $ regularity associated with the horizontal gradient of weak solutions away from a negligible set.</p>

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41

Sharutin, V., and O. Sharutina. "Structural features of tetraphenylantimony carboxylates Ph4SbOC(O)R (R = CH2Cl, CH2Br, CH2l, C6H3F2-2,3) and tetraphenylantimony nitrate hydrate Ph4SbONO2 ∙ H2O." Bulletin of the South Ural State University series "Chemistry" 16, no. 2 (2024): 37–45. http://dx.doi.org/10.14529/chem240203.

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The structure of tetraphenylantimony carboxylates Ph4SbOC(O)R [R = CH2Cl (1), CH2Br (2), CH2l (3), C6H3F2-2,3) (4)] and nitrate Ph4SbONO2 ∙ H2O (5) was established by X-ray diffraction analysis (XRD). According to the XRD data, the antimony atoms in complexes 1–5 have the coordination of a distorted trigonal bipyramid with an electronegative ligand in the axial position. The CSbO axial angles are 174.05(7); 171.6(2), 170.3(2); 173.10(12); 177.93(5); 178.02(9), 168.11(9), and 169.33(9) respectively. The X-ray diffraction data: (1) [C26H22O2ClSb, M = 523.64; rhombic system, sp. gr. Pbca; cell parameters: a = 14.382(8) Å, b = 16.681(10) Å, c = 19.270(11) Å; β = 90.00, V = 4623(5) Å3, Z = 8; calc = 1.505 g/cm3;  = 1.328 mm–1; F(000) = 2096.0; region 2q collection: 5.64–56.6; –19 ≤ h ≤ 19, –22 ≤ k ≤ 21, –23 ≤ l ≤ 25; total reflections 69348; independent reflections 5710 (Rint = 0.0398); GOOF = 1.067; R-factor 0.0261]; (2) [C52H46O4Br2Sb2, M = 1138.19; triclinic syngony, sp. gr. P–1; cell parameters: a = 11.096(13) Å, b = 12.510(13) Å, c = 17.62(2) Å;  = 78.01(6), β = 89.35(7),  = 89.71(5), V = 2393(5) Å3, Z = 2; calc = 1.577 g/cm3;  = 2.841 mm–1; F(000) = 1120.0; region 2q collection: 5.16–69.06; –16 ≤ h ≤ 16, –14 ≤ k ≤ 14, –23 ≤ l ≤ 23; total reflections 89320; independent reflections 11788 (Rint = 0.0568); GOOF = 1.034; R-factor 0.0519]; (3) [C26H22O2SbI, M = 615.09; monoclinic syngony, sp. gr. P21/c; cell parameters: a = 12.779(6) Å, b = 10.864(4) Å, c = 17.542(9) Å; β = 100.18(3), V = 2397(2) Å3, Z = 4; calc = 1.704 g/cm3;  = 2.458 mm–1; F(000) = 1192.0; region 2q collection: 6.02–71.46; –20 ≤ h ≤ 20, –17 ≤ k ≤ 17, –28 ≤ l ≤ 28; total reflections 70960; independent reflections 11043 (Rint = 0.0510); GOOF = 1.018; R-factor 0,0537]; (4) [C31H23O2F2Sb, M = 587.24; triclinic syngony, sp. gr. P–1; cell parameters: a = 9.862(13) Å, b = 10.154(13) Å, c = 14.298(2) Å;  = 84.03(6), β = 82.76(7),  = 68.41(5), V = 1318.2(5) Å3, Z = 2; calc = 1.479 g/cm3;  = 1.086 mm–1; F(000) = 588.0; region 2q collection: 6.08–74.28; –16 ≤ h ≤ 16, –17 ≤ k ≤ 17, –24 ≤ l ≤ 24; total reflections 88852; independent reflections 13477 (Rint = 0.0353); GOOF = 1.026; R-factor 0.0359]; (5) [C72H62N3O10Sb3, M = 1494.50; monoclinic syngony, sp. gr. P21/n; cell parameters: a = 23.072(7) Å, b = 10.427(3) Å, c = 27.040(10) Å; β = 95.860(13), V = 6472(4) Å3, Z = 4; calc = 1.534 g/cm3;  = 1.305 mm–1; F(000) = 2992.0; region 2q collection: 5.6–62.16; –33 ≤ h ≤ 29, –15 ≤ k ≤ 15, –39 ≤ l ≤ 39; total reflections 228547; independent reflections 20667 (Rint = 0.0432); GOOF = 1.041; R-factor 0.0303]. Complete tables of atomic coordinates, bond lengths, and bond angles for compounds 1–5 are deposited at the Cambridge Crystallographic Data Center (CCDC 2169943, 2170138, 2213768, 2170205, 2147525; deposit@ccdc.cam.ac.uk; http://www.ccdc.cam.ac.uk).

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KLYUEV, ALEXEY V., EVGENY I. SHMELEV, and ARKADY V. YAKIMOV. "MODIFICATION OF VAN DER ZIEL RELATION FOR SPECTRUM OF NOISE IN p–n JUNCTION." Fluctuation and Noise Letters 11, no. 02 (2012): 1250015. http://dx.doi.org/10.1142/s0219477512500150.

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Spectrum SiD of the white current noise iD(t) in p–n junction with the ideality factor η of the current–voltage characteristic, which is greater than one, is investigated here. It is shown, that the Van der Ziel relation, SiD = 2q(ID + 2Is), intended for η = 1, is inapplicable if η > 1; here q is the elementary charge, ID is the current through the junction, and Is is the saturation current. As the first step, the simplest case is considered, η = 2. That is the main recombination of injected electrons and holes takes place in the middle of the junction depleted region. Such junction may be modeled by two identical junctions with η = 1, which are connected in series. The current noise iD(t) is determined by noise sources of both junctions. Obtained result is generalized by the use of the Gupta theorem for the thermal noise spectrum in nonlinear resistive systems. The current noise spectrum is found, SiD = (2q/η) · (ID + 2Is). This result is the modification of the Van der Ziel relation extended to η ≥ 1. Experimental proof of the modified relation is made by analysis of the white noise spectrum in Schottky diode with δ-doping (having η = 2.2) in the vicinity of thermodynamical equilibrium.

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43

Park, Young-Taek, JeongYun Park, Ji Soo Jeon, Young Jae Kim, and Kwang Gi Kim. "Changes in Nurse Staffing Grades of Korean Hospitals during COVID-19 Pandemic." International Journal of Environmental Research and Public Health 18, no. 11 (2021): 5900. http://dx.doi.org/10.3390/ijerph18115900.

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The global COVID-19 pandemic is creating challenges to manage staff ratios in clinical units. Nurse staffing level is an important indicator of the quality of care. This study aimed to identify any changes in the nurse staffing levels in the general wards of hospitals in Korea during the COVID-19 pandemic. The unit of analysis was the hospitals. This longitudinal study observed the quarterly change of the nurse staffing grades in 969 hospitals in 2020. The nurse staffing grades ranged from 1 to 7 according to the nurse–patient ratio measured by the number of patients (or beds) per nurse. The major dependent and independent variables were the change of nurse staffing grades and three quarterly observation points being compared with those during the 1st quarter (1Q) of 2020, respectively. A generalized linear model was used. Unexpectedly, the nurse staffing grades significantly improved (2Q: RR, 27.2%; 95% confidence interval (CI), 15.1–27.6; p < 0.001; 3Q: RR, 95% CI, 20.2%; 16.9–21.6; p < 0.001; 4Q: RR, 26.6%; 95% CI, 17.8–39.6; p < 0.001) quarterly, indicating that the nurse staffing levels increased. In the comparison of grades at 2Q, 3Q, and 4Q with those at 1Q, most figures improved in tertiary, general, and small hospitals (p < 0.05), except at 3Q and 4Q of general hospitals. In conclusion, the nurse staffing levels did not decrease, but nursing shortage might occur.

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44

Shirai, M., M. S. Vacchio, R. J. Hodes, and J. A. Berzofsky. "Preferential V beta usage by cytotoxic T cells cross-reactive between two epitopes of HIV-1 gp160 and degenerate in class I MHC restriction." Journal of Immunology 151, no. 4 (1993): 2283–95. http://dx.doi.org/10.4049/jimmunol.151.4.2283.

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Abstract The T cell response to HIV-1 gp160 is among the most thoroughly studied immune responses to HIV-1 products. In our previous work, the MHC class I molecule Dd as well as H-2u, p, and q, were found to present P18 and HP53, two determinants of HIV-1 gp160, to CD8+ CTL in mice. We have studied the TCR V beta chain expression in CTL lines, either cross-reactive for these two peptides or specific for P18 alone, in these four different MHC haplotypes. The usage of V beta in T cells showing cross-reaction between these two peptides was remarkably conserved (primarily V beta 8 family, with some use of V beta 14) despite the extensive TCR V beta diversity of the non-cross-reactive CTL, which did not use V beta 8 or 14. This correlation of V beta usage with fine specificity was consistent in H-2d, u, and p (p &lt; 0.01), but not in H-2q. The correlation of V beta use with peptide fine specificity independent of MHC restriction was unexpected. The strong predominance of V beta 8 family TCR was all the more surprising in view of the finding that mice bearing a genomic deletion of V beta 8 can still produce T cells with the cross-reactive phenotype, implying that other V beta chains can produce this specificity. We therefore asked whether the complexes of P18 with H-2d, p, and u are recognized as identical, and observed the surprising result that H-2d, p, and u cells mutually cross-present the peptides P18 and HP53 to allogeneic CTL lines and individual clones of each of the other haplotypes, whereas none of these cross-present to H-2q CTL, nor do H-2q targets present to CTL of the other haplotypes. This degeneracy of MHC restriction is novel for class I molecules. Moreover, the observed restriction in V beta usage occurs only in the unique set of CTL that exhibit both peptide-cross-reactive fine specificity and MHC allogeneic cross-presentation. The observation that a strain of mice in which the V beta 8 family is genomically deleted can still make CTL of this phenotype using another V beta demonstrates the plasticity of the class I MHC-restricted repertoire when the dominating receptor is not available.

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IKEDA, TORU. "BOUNDARIES OF INCOMPRESSIBLE SURFACES IN GRAPH KNOT EXTERIORS." Journal of Knot Theory and Its Ramifications 19, no. 01 (2010): 71–79. http://dx.doi.org/10.1142/s0218216510007747.

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We will study boundaries of incompressible surfaces properly embedded in graph knot exteriors. We will first show that any bounded two-sided surface is meridional or longitudinal. In particular, iterated torus knot exteriors contain no bounded two-sided essential surface which is meridional or preferred longitudinal but Seifert surfaces. Even if the surface is possibly one-sided, the boundary is not of type (2p, 2q + 1) for any integers p and q.

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46

Zhelezov, Ognyan Ivanov, and Valentina Markova Petrova. "Application of the Transpositions matrix for obtaining n-dimensional rotation matrices." International Journal of Chemistry, Mathematics and Physics 6, no. 5 (2022): 12–18. http://dx.doi.org/10.22161/ijcmp.6.5.3.

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This article proposes an algorithm for generation of N-dimensional rotation matrix R, N=m+n, m=2p, n=2q, p,q  [2, 4, 8] which rotates given N-dimensional vector X in the direction of coordinate axis x1 Algorithm uses block diagonal matrix, composed by Transpositions matrices. As practical realization article gives Matlab code of functions, which creates Householder and Transpositions matrices and V matrix for given n-dimensional vector X.

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47

Boyde, Guy. "Bounding size of homotopy groups of Spheres." Proceedings of the Edinburgh Mathematical Society 63, no. 4 (2020): 1100–1105. http://dx.doi.org/10.1017/s001309152000036x.

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AbstractLet p be prime. We prove that, for n odd, the p-torsion part of πq(Sn) has cardinality at most $p^{2^{{1}/({p-1})(q-n+3-2p)}}$ and hence has rank at most 21/(p−1)(q−n+3−2p). for p = 2, these results also hold for n even. The best bounds proven in the existing literature are $p^{2^{q-n+1}}$ and 2q−n+1, respectively, both due to Hans–Werner Henn. The main point of our result is therefore that the bound grows more slowly for larger primes. As a corollary of work of Henn, we obtain a similar result for the homotopy groups of a broader class of spaces.

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Dholey, S. "Effect of Magnetic Field on the Unsteady Boundary Layer Flows Induced by an Impulsive Motion of a Plane Surface." Zeitschrift für Naturforschung A 75, no. 4 (2020): 343–55. http://dx.doi.org/10.1515/zna-2019-0334.

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AbstractThe unsteady laminar boundary layer flow of an electrically conducting viscous fluid near an impulsively started flat plate of infinite extent is considered, with a view to examine the influence of transverse magnetic field fixed to the fluid. A new type of similarity transformation is proposed, which renews the governing partial differential equation into a linear ordinary differential equation with four physical parameters, viz. unsteadiness parameter β, magnetic parameter M, and the velocity indices (p, q). The analytic solution of this equation has been found in terms of a first kind confluent hypergeometric function for some specific parameter regimes. This solution shows the structure of a new type of boundary layer flow that includes the solution of the first Stokes problem as a special case. For non-zero values of (p, q), there is a definite range of p (either −∞ < p < 2q or 2q < p < ∞ according to β < or > 0) for which this flow problem will be valid. This analysis reveals an important relation $(p\beta+{M^{2}}=q\beta)$ at which separation appears inside the layer and has been detected as the separation threshold of the problem. Indeed, this relation gives us the critical value of one when the others are known. Flow separation inside the layer is delayed with an increasing value of q but cannot be completely removed whatever is the value of q (>0). The present analysis ensures that the reverse flow can be suppressed by the use of a proper amount of magnetic field M depending upon the values of p, q, and β. The obtained result provides insight into the stability of the boundary layer flows.

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Sharutin, V. "Features of the structure of tetra(para- tolyl)antimony derivatives p-Tol4SbX (X = Br, OC(O)Ph∙PhH, OSO2C6H2Me3-2,4,6)." Bulletin of the South Ural State University series "Chemistry" 15, no. 4 (2023): 109–16. http://dx.doi.org/10.14529/chem230402.

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The structure of tetra(para-tolyl)antimony compounds p-Tol4SbX [X = Br (1), OC(O)Ph∙PhH (2), OSO2C6Me3-2,4,6 (3)] was established by X-ray diffraction analysis (XRD). According to the X-ray diffraction data, the antimony atoms in complexes 1–3 have a distorted trigonal bipyramidal coordination with three aryl ligands in the equatorial plane, while the axial angles CSbX are 174.75(8), 175.13(9), and 174.51(6). The X-ray diffraction data: (1) [C28H28BrSb, M = 566.16; monoclinic syngony, sp. gr. P21/n; cell parameters: a = 9.868(6) Å, b = 23.312(11) Å, c = 12.106(6) Å; β = 113.15(2), V = 2561(2) Å3, Z = 4; calc = 1.469 g/cm3;  = 2.649 mm–1; F(000) = 1128.0; region 2q collection: 6.4–56.76; –13 ≤ h ≤ 13, –31 ≤ k ≤ 31, –16 ≤ l ≤ 16; total reflections 42998; independent reflections 6359 (Rint = 0.0346); GOOF = 1.080; R-factor 0.0325]; (2) [C41H39O2Sb, M = 685.47; monoclinic syngony, sp. gr. C2/c; cell parameters: a = 28.186(13) Å, b = 15.116(6) Å, c = 17.629(8) Å; β = 91.73(2), V = 7507(6) Å3, Z = 8; calc = 1.213 g/cm3;  = 0.765 mm–1; F(000) = 2816.0; region 2q collection: 6.572–56.996; –37 ≤ h ≤ 37, –20 ≤ k ≤ 20, –23 ≤ l ≤ 23; total reflections 116806; independent reflections 9489 (Rint = 0.0492); GOOF = 1.102; R-factor 0.0363]; (3) [C37H39O3SSb, M = 685.49; monoclinic syngony, sp. gr. P21/n; cell options: a = 12.172(4) Å, b = 18.802(5) Å, c = 15.433(6) Å; β = 108.744(12), V = 3345(2) Å3, Z = 4; calc = 1.361 g/cm3;  = 0.921 mm–1; F(000) = 1408.0; region 2q collection: 5.96–63.02; –16 ≤ h ≤ 17, –27 ≤ k ≤ 27, –22 ≤ l ≤ 21; total reflections 138835; independent reflections 11081 (Rint = 0.0373); GOOF = 1.045; R-factor 0.0304]. Complete tables of atomic coordinates, bond lengths, and bond angles for compounds 1–3 have been deposited at the Cambridge Crystallographic Data Center (CCDC 2182608, 2149953, 2171918; deposit@ccdc.cam.ac.uk; http://www.ccdc.cam. ac.uk).

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Shirai, M., S. Kozlowski, D. H. Margulies, and J. A. Berzofsky. "Degenerate MHC restriction reveals the contribution of class I MHC molecules in determining the fine specificity of CTL recognition of an immunodominant determinant of HIV-1 gp160 V3 loop." Journal of Immunology 158, no. 7 (1997): 3181–88. http://dx.doi.org/10.4049/jimmunol.158.7.3181.

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Abstract The novel allogeneic presentation of an immunodominant determinant within the HIV-1 gp160 V3 loop by three different class I MHC molecules to the same CD8+ CTL is used to study the influence of the MHC molecule on the fine specificity of CTL recognition. We previously reported that four distinct class I molecules of H-2d,u,p,q presented the V3 decapeptide P18-I10 (RGPGRAFVTI) to CTL. Surprisingly, we found that H-2d,u,p cells mutually cross-present the P18-I10 peptide to allogeneic CTL clones of each of the other haplotypes, whereas none of these cross-presents to H-2q CTL, nor do H-2q targets present to CTL of the other haplotypes. Here, we explore the critical amino acid residues for the cross-presentation using 10 variant peptides with single amino acid substitutions. The fine specificity examined using these mutant peptides presented by the same MHC class I molecule showed striking similarity among the CTL of each haplotype, expressing either V beta 8.1 or V beta 14. In contrast, the fine specificity is different between the distinct MHC class I molecules even for the lysis by the same CTL, as shown by reciprocal effects of the same substitutions. Thus, peptide fine specificity of a single TCR is influenced by changes in the class I MHC molecules presenting the Ag.

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