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Journal articles on the topic 'Math.CO'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Manturov, Vassily Olegovich. "Framed 4-valent graph minor theory II: Special minors and new examples." Journal of Knot Theory and Its Ramifications 24, no. 13 (November 2015): 1541004. http://dx.doi.org/10.1142/s0218216515410047.

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In the present paper, we proceed with the study of framed 4-graph minor theory initiated in [V. O. Manturov, Framed 4-valent graph minor theory I: Intoduction planarity criterion, arxiv: 1402.1564v1 [Math.Co]] and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in [Formula: see text].

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2

Manzoor, Saba, Nisar Fatima, Akhlaq Ahmad Bhatti, and Akbar Ali. "Zagreb Connection Indices of Some Nanostructures." Acta Chemica Iasi 26, no. 2 (December 1, 2018): 169–80. http://dx.doi.org/10.2478/achi-2018-0011.

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Abstract The first Zagreb index (occurred in an approximate formula of total π-electron energy, communicated in 1972) and the second Zagreb index (appeared in 1975, within the study of molecular branching) are among the most studied topological indices. Recently, three modified versions of the Zagreb indices were proposed independently in [A. Ali, N. Trinajstić, A novel/old modification of the first Zagreb index, arXiv:1705.10430 [math.CO], 2017] and [A. M. Naji, N. D. Soner, I. Gutman, On leap Zagreb indices of graphs, Commun. Comb. Optim., 2017, 2, 99–117], which were named as the Zagreb connection indices and the leap Zagreb indices, respectively. In this paper, we derive formulas for calculating these modified versions of the Zagreb indices of four well known nanostructures.

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3

Li, Zhenzhen, and Baoyindureng Wu. "Maximum value of conflict-free vertex-connection number of graphs." Discrete Mathematics, Algorithms and Applications 10, no. 05 (October 2018): 1850059. http://dx.doi.org/10.1142/s1793830918500593.

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A path in a vertex-colored graph is called conflict-free if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be conflict-free vertex-connected if any two vertices of the graph are connected by a conflict-free path. The conflict-free vertex-connection number, denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] conflict-free vertex-connected. Li et al. [Conflict-free vertex-connections of graphs, preprint (2017), arXiv:1705.07270v1[math.CO]] conjectured that for a connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. We confirm that the conjecture is true and poses two relevant conjectures.

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4

Fatima, Nisar, Akhlaq Ahmad Bhatti, Akbar Ali, and Wei Gao. "Zagreb Connection Indices of Two Dendrimer Nanostars." Acta Chemica Iasi 27, no. 1 (June 1, 2019): 1–14. http://dx.doi.org/10.2478/achi-2019-0001.

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Abstract It is well known fact that several physicochemical properties of chemical compounds are closely related to their molecular structure. Mathematical chemistry provides a method to predict the aforementioned properties of compounds using topological indices. The Zagreb indices are among the most studied topological indices. Recently, three modified versions of the Zagreb indices were proposed independently in [Ali, A.; Trinajstić, N. A novel/old modification of the first Zagreb index, arXiv:1705.10430 [math.CO] 2017; Mol. Inform. 2018, 37, 1800008] and [Naji, A. M.; Soner, N. D.; Gutman, I. On leap Zagreb indices of graphs, Commun. Comb. Optim. 2017, 2, 99–117], which were named as the Zagreb connection indices and the leap Zagreb indices, respectively. In this paper, we check the chemical applicability of the newly considered Zagreb connection indices on the set of octane isomers and establish general expressions for calculating these indices of two well-known dendrimer nanostars.

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5

Efimov, Alexander I. "Cohomological Hall algebra of a symmetric quiver." Compositio Mathematica 148, no. 4 (May 15, 2012): 1133–46. http://dx.doi.org/10.1112/s0010437x12000152.

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AbstractIn [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]], the authors, in particular, associate to each finite quiver Q with a set of vertices I the so-called cohomological Hall algebra ℋ, which is ℤI≥0-graded. Its graded component ℋγ is defined as cohomology of the Artin moduli stack of representations with dimension vector γ. The product comes from natural correspondences which parameterize extensions of representations. In the case of a symmetric quiver, one can refine the grading to ℤI≥0×ℤ, and modify the product by a sign to get a super-commutative algebra (ℋ,⋆) (with parity induced by the ℤ-grading). It is conjectured in [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]] that in this case the algebra (ℋ⊗ℚ,⋆) is free super-commutative generated by a ℤI≥0×ℤ-graded vector space of the form V =Vprim ⊗ℚ[x] , where x is a variable of bidegree (0,2)∈ℤI≥0×ℤ, and all the spaces ⨁ k∈ℤVprimγ,k, γ∈ℤI≥0. are finite-dimensional. In this paper we prove this conjecture (Theorem 1.1). We also prove some explicit bounds on pairs (γ,k) for which Vprimγ,k≠0 (Theorem 1.2). Passing to generating functions, we obtain the positivity result for quantum Donaldson–Thomas invariants, which was used by Mozgovoy to prove Kac’s conjecture for quivers with sufficiently many loops [S. Mozgovoy, Motivic Donaldson–Thomas invariants and Kac conjecture, Preprint (2011), arXiv:1103.2100v2[math.AG]]. Finally, we mention a connection with the paper of Reineke [M. Reineke, Degenerate cohomological Hall algebra and quantized Donaldson–Thomas invariants for m-loop quivers, Preprint (2011), arXiv:1102.3978v1[math.RT]].

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6

Kong, Liuzhen. "Strongly ($$\mathscr {X},\mathscr {Y},\mathscr {Z}$$)-Gorenstein Modules and Applications." Bulletin of the Iranian Mathematical Society 46, no. 2 (August 23, 2019): 503–17. http://dx.doi.org/10.1007/s41980-019-00272-w.

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7

Raza, M. A., and N. Rehman. "Полудифференцирования в первичных кольцах." Владикавказский математический журнал, no. 2 (June 24, 2021): 70–77. http://dx.doi.org/10.46698/d4945-5026-4001-v.

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Let $\mathscr{R}$ be a prime ring with the extended centroid $\mathscr{C}$ and the Matrindale quotient ring $\mathscr{Q}$. An additive mapping $\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}$ is called a semiderivation associated with a mapping $\mathscr{G}: \mathscr{R}\rightarrow \mathscr{R}$, whenever $ \mathscr{F}(xy)=\mathscr{F}(x)\mathscr{G}(y)+x\mathscr{F}(y)= \mathscr{F}(x)y+ \mathscr{G}(x)\mathscr{F}(y) $ and $ \mathscr{F}(\mathscr{G}(x))= \mathscr{G}(\mathscr{F}(x))$ holds for all $x, y \in \mathscr{R}$. In this manuscript, we investigate and describe the structure of a prime ring $\mathscr{R}$ which satisfies $\mathscr{F}(x^m\circ y^n)\in \mathscr{Z(R)}$ for all $x, y \in \mathscr{R}$, where $m,n \in \mathbb{Z}^+$ and $\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}$ is a semiderivation with an~automorphism $\xi$ of $\mathscr{R}$. Further, as an application of our ring theoretic results, we discussed the nature of $\mathscr{C}^*$-algebras. To be more specific, we obtain for any primitive $\mathscr{C}^*$-algebra $\mathscr{A}$. If an anti-automorphism $ \zeta: \mathscr{A} \to \mathscr{A}$ satisfies the relation $(x^n)^\zeta+x^{n*}\in \mathscr{Z}(\mathscr{A})$ for every ${x,y}\in \mathscr{A},$ then $\mathscr{A}$ is $\mathscr{C}^{*}-\mathscr{W}_{4}$-algebra, i.\,e., $\mathscr{A}$ satisfies the standard identity $\mathscr{W}_4(a_1,a_2,a_3,a_4)=0$ for all $a_1,a_2,a_3,a_4\in \mathscr{A}$.

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8

Yang, Xiaoyan. "Gorenstein categories $\mathcal G(\mathscr X,\mathscr Y,\mathscr Z)$ and dimensions." Rocky Mountain Journal of Mathematics 45, no. 6 (December 2015): 2043–64. http://dx.doi.org/10.1216/rmj-2015-45-6-2043.

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9

Zrotowski, R. "Normality and $\mathscr{P}(\kappa)/\mathscr{J}$." Journal of Symbolic Logic 56, no. 3 (September 1991): 1064–67. http://dx.doi.org/10.2178/jsl/1183743751.

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10

Абрашкин, Виктор Александрович, and Viktor Aleksandrovich Abrashkin. "Фильтрация ветвления и деформации." Математический сборник 212, no. 2 (2021): 3–37. http://dx.doi.org/10.4213/sm9322.

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Пусть $\mathscr K$ - поле формальных рядов Лорана с коэффициентами в конечном поле характеристики $p$, $\mathscr G_{<p}$ - максимальный фактор группы Галуа поля $\mathscr K$ периода $p$ и класса нильпотентности $<p$ и $\{\mathscr G_{<p}^{(v)}\}_{v\geqslant 1}$ - фильтрация подгрупп ветвления в верхней нумерации. Пусть $\mathscr G_{<p}=G(\mathscr L)$ - отождествление нильпотентной теории Артина-Шрайера: здесь $G(\mathscr L)$ - группа, полученная из проконечной $\mathbb{F}_p$-алгебры Ли $\mathscr L$ с помощью группового закона Кемпбелла-Хаусдорфа. В работе изложен новый подход к описанию идеалов $\mathscr L^{(v)}$ таких, что $G(\mathscr L^{(v)})=\mathscr G_{<p}^{(v)}$, и построению их явных образующих. Для заданного $v_0\geqslant 1$ строится эпиморфизм алгебр Ли $\overline\eta^{\dagger }\colon \mathscr L\to \overline{\mathscr L}^{\dagger }$ и действие $\Omega_U$ формальной группы порядка $p$, $\alpha_p=\operatorname{Spec}\mathbb{F}_p[U]$, $U^p=0$, на $\overline{\mathscr L}^{\dagger }$. Пусть $d\Omega_U=B^{\dagger }U$, где $B^{\dagger }\in\operatorname{Diff}\overline{\mathscr L}^{\dagger }$, и $\overline{\mathscr L}^{\dagger }[v_0]$ - идеал в $\overline{\mathscr L}^{\dagger }$, порожденный элементами $B^{\dagger }(\overline{\mathscr L}^{\dagger })$. Основной результат работы утверждает, что $\mathscr L^{(v_0)}=(\overline\eta^{\dagger })^{-1}\overline{\mathscr L}^{\dagger }[v_0]$. В заключительных параграфах этот результат связывается с явным описанием образующих идеала $\mathscr L^{(v_0)}$, полученным ранее автором, и формулируется его более эффективная версия, позволяющая восстанавливать всю фильтрацию ветвления группы $\mathscr G_{<p}$ по множеству ее скачков. Библиография: 13 названий.

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11

Moslehian, Mohammad Sal, and Ali Zamani. "Mappings preserving approximate orthogonality in Hilbert $C^*$-modules." MATHEMATICA SCANDINAVICA 122, no. 2 (April 8, 2018): 257. http://dx.doi.org/10.7146/math.scand.a-102945.

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We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(\delta , \varepsilon )$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(\delta , \varepsilon )$-orthogonality preserving. In particular, if $\mathscr {E}$ is a full Hilbert $\mathscr {A}$-module with $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T, S\colon \mathscr {E}\to \mathscr {E}$ are two linear mappings satisfying $|\langle Sx, Sy\rangle | = \|S\|^2|\langle x, y\rangle |$ for all $x, y\in \mathscr {E}$ and $\|T - S\| \leq \theta \|S\|$, then we show that $T$ is a $(\delta , \varepsilon )$-orthogonality preserving mapping. We also prove whenever $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T\colon \mathscr {E} \to \mathscr {F}$ is a nonzero $\mathscr {A}$-linear $(\delta , \varepsilon )$-orthogonality preserving mapping between $\mathscr {A}$-modules, then \[ \bigl \|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle \bigr \|\leq \frac {4(\varepsilon - \delta )}{(1 - \delta )(1 + \varepsilon )} \|Tx\|\|Ty\|\qquad (x, y\in \mathscr {E}). \] As a result, we present some characterizations of the orthogonality preserving mappings.

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12

Rieffel, Marc A. "Concrete Realizations of Quotients of Operator Spaces." MATHEMATICA SCANDINAVICA 114, no. 2 (May 6, 2014): 205. http://dx.doi.org/10.7146/math.scand.a-17107.

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Let $\mathscr{B}$ be a unital C*-subalgebra of a unital C*-algebra $\mathscr{A}$, so that $\mathscr{A}/\mathscr{B}$ is an abstract operator space. We show how to realize $\mathscr{A}/\mathscr{B}$ as a concrete operator space by means of a completely contractive map from $\mathscr{A}$ into the algebra of operators on a Hilbert space, of the form $A \mapsto [Z, A]$ where $Z$ is a Hermitian unitary operator. We do not use Ruan's theorem concerning concrete realization of abstract operator spaces. Along the way we obtain corresponding results for abstract operator spaces of the form $\mathscr{A}/\mathscr{V}$ where $\mathscr{V}$ is a closed subspace of $\mathscr{A}$, and then for the more special cases in which $\mathscr{V}$ is a $*$-subspace or an operator system.

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13

HILBERDINK, HENDRIK. "Inclusions for partiality." Mathematical Structures in Computer Science 25, no. 1 (December 2, 2014): 46–82. http://dx.doi.org/10.1017/s0960129514000036.

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In this paper we introduce stable systems of inclusions, which feature chosen arrows A ↪ B to capture the notion that A is a subobject of B, and proposes them as an alternative context to stable systems of monics to discuss partiality. A category C equipped with such a system $\mathscr{I}$, called an i-category, is shown to give rise to an associated category ∂(C,$\mathscr{I}$) of partial maps, which is a split restriction category whose restriction monics are inclusions. This association is the object part of a 2-equivalence between such inclusively split restriction categories and i-categories. $\mathscr{I}$ determines a stable system of monics $\mathscr{I}$+ on C, and, conversely, a stable system of monics $\mathscr{M}$ on C yields an i-category (C[$\mathscr{M}$],$\mathscr{M}$+), giving a 2-adjunction between i-categories and m-categories. The category of partial maps Par(C,$\mathscr{M}$) is isomorphic to the full subcategory of ∂(C[$\mathscr{M}$],$\mathscr{M}$+) comprising the objects of C, and ∂(C,$\mathscr{I}$) ≅ Par(C,$\mathscr{I}$+).

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14

Morales, M., A. A. Yazdan Pour, and R. Zaare-Nahandi. "Regularity and Free Resolution of Ideals Which Are Minimal To $d$-Linearity." MATHEMATICA SCANDINAVICA 118, no. 2 (June 9, 2016): 161. http://dx.doi.org/10.7146/math.scand.a-23685.

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For given positive integers $n\geq d$, a $d$-uniform clutter on a vertex set $[n]=\{1,\dots,n\}$ is a collection of distinct $d$-subsets of $[n]$. Let $\mathscr{C}$ be a $d$-uniform clutter on $[n]$. We may naturally associate an ideal $I(\mathscr{C})$ in the polynomial ring $S=k[x_1,\dots,x_n]$ generated by all square-free monomials \smash{$x_{i_1}\cdots x_{i_d}$} for $\{i_1,\dots,i_d\}\in\mathscr{C}$. We say a clutter $\mathscr{C}$ has a $d$-linear resolution if the ideal \smash{$I(\overline{\mathscr{\mathscr{C}}})$} has a $d$-linear resolution, where \smash{$\overline{\mathscr{C}}$} is the complement of $\mathscr{C}$ (the set of $d$-subsets of $[n]$ which are not in $\mathscr C$). In this paper, we introduce some classes of $d$-uniform clutters which do not have a linear resolution, but every proper subclutter of them has a $d$-linear resolution. It is proved that for any two $d$-uniform clutters $\mathscr{C}_1$, $\mathscr{C}_2$ the regularity of the ideal $I(\overline{\mathscr{C}_1 \cup \mathscr{C}_2})$, under some restrictions on their intersection, is equal to the maximum of the regularities of $I(\overline{\mathscr{C}}_1)$ and $I(\overline{\mathscr{C}}_2)$. As applications, alternative proofs are given for Fröberg's Theorem on linearity of edge ideals of graphs with chordal complement as well as for linearity of generalized chordal hypergraphs defined by Emtander. Finally, we find minimal free resolutions of the ideal of a triangulation of a pseudo-manifold and a homology manifold explicitly.

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15

Fošner, Ajda, and Maja Fošner. "Equations Related to Superderivations on Prime Superalgebras." MATHEMATICA SCANDINAVICA 115, no. 2 (December 3, 2014): 303. http://dx.doi.org/10.7146/math.scand.a-19227.

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In this paper we investigate equations related to superderivations on prime superalgebras. We prove the following result. Let $D=D_0+D_1$ be a nonzero superderivation on a prime associative superalgebra $\mathscr{A}$ satisfying the relations $D_i(x)[D_i(x),x]_s=0$, $[D_i(x),x]_sD_i(x)=0$ for all $x\in \mathscr{A}$, $i=0,1$. Then one of the following is true: (a) $\mathscr{A}_1=0$ and $D(\mathscr{A}_0) \subseteq Z(\mathscr{A})$ or (b) $ D(\mathscr{A}_0)=0$ and $\mathscr{A}$ is commutative or (c) $D^2=0$. The research is a generalization of the results in [10] and [4] by using the theory of superalgebras.

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16

Matsumoto, Kengo. "Cuntz-Krieger Algebras Associated with Hilbert $C^*$-Quad Modules of Commuting Matrices." MATHEMATICA SCANDINAVICA 117, no. 1 (September 28, 2015): 126. http://dx.doi.org/10.7146/math.scand.a-22239.

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Let $\mathscr{O}_{\mathscr{H}^{A,B}_{\kappa}}$ be the $C^*$-algebra associated with the Hilbert $C^*$-quad module arising from commuting matrices $A,B$ with entries in $\{0,1\}$. We will show that if the associated tiling space $X_{A,B}^\kappa$ is transitive, the $C^*$-algebra $\mathscr{O}_{\mathscr{H}^{A,B}_{\kappa}}$ is simple and purely infinite. In particular, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^*$-algebra $\mathscr{O}_{\mathscr{H}^{[N],[M]}_{\kappa}}$ are computed by using the Euclidean algorithm.

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17

KOUSHESH, M. R. "REPRESENTATION THEOREMS FOR NORMED ALGEBRAS." Journal of the Australian Mathematical Society 95, no. 2 (June 17, 2013): 201–22. http://dx.doi.org/10.1017/s1446788713000207.

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AbstractWe show that for a normal locally-$\mathscr{P}$ space $X$ (where $\mathscr{P}$ is a topological property subject to some mild requirements) the subset ${C}_{\mathscr{P}} (X)$ of ${C}_{b} (X)$ consisting of those elements whose support has a neighborhood with $\mathscr{P}$, is a subalgebra of ${C}_{b} (X)$ isometrically isomorphic to ${C}_{c} (Y)$ for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone–Čech compactification $\beta X$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, ${C}_{\mathscr{P}} (X)$ coincides with the set of those elements of ${C}_{b} (X)$ whose support has $\mathscr{P}$, it moreover becomes a Banach algebra, and simultaneously, $Y$ satisfies ${C}_{c} (Y)= {C}_{0} (Y)$. This includes the cases when $\mathscr{P}$ is the Lindelöf property and $X$ is either a locally compact paracompact space or a locally-$\mathscr{P}$ metrizable space. In either of the latter cases, if $X$ is non-$\mathscr{P}$, then $Y$ is nonnormal and ${C}_{\mathscr{P}} (X)$ fits properly between ${C}_{0} (X)$ and ${C}_{b} (X)$; even more, we can fit a chain of ideals of certain length between ${C}_{0} (X)$ and ${C}_{b} (X)$. The known construction of $Y$ enables us to derive a few further properties of either ${C}_{\mathscr{P}} (X)$ or $Y$. Specifically, when $\mathscr{P}$ is the Lindelöf property and $X$ is a locally-$\mathscr{P}$ metrizable space, we show that $$\begin{eqnarray*}\dim C_{\mathscr{P}}(X)= \ell \mathop{(X)}\nolimits ^{{\aleph }_{0} } ,\end{eqnarray*}$$ where $\ell (X)$ is the Lindelöf number of $X$, and when $\mathscr{P}$ is countable compactness and $X$ is a normal space, we show that $$\begin{eqnarray*}Y= {\mathrm{int} }_{\beta X} \upsilon X\end{eqnarray*}$$ where $\upsilon X$ is the Hewitt realcompactification of $X$.

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18

Gutik, O. V., and A. S. Savchuk. "On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers." Carpathian Mathematical Publications 11, no. 2 (December 31, 2019): 296–310. http://dx.doi.org/10.15330/cmp.11.2.296-310.

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In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

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19

Cox, Sonja G., and Kristin Kirchner. "Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle–Matérn fields." Numerische Mathematik 146, no. 4 (November 16, 2020): 819–73. http://dx.doi.org/10.1007/s00211-020-01151-x.

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AbstractWe analyze several types of Galerkin approximations of a Gaussian random field $$\mathscr {Z}:\mathscr {D}\times \varOmega \rightarrow \mathbb {R}$$ Z : D × Ω → R indexed by a Euclidean domain $$\mathscr {D}\subset \mathbb {R}^d$$ D ⊂ R d whose covariance structure is determined by a negative fractional power $$L^{-2\beta }$$ L - 2 β of a second-order elliptic differential operator $$L:= -\nabla \cdot (A\nabla ) + \kappa ^2$$ L : = - ∇ · ( A ∇ ) + κ 2 . Under minimal assumptions on the domain $$\mathscr {D}$$ D , the coefficients $$A:\mathscr {D}\rightarrow \mathbb {R}^{d\times d}$$ A : D → R d × d , $$\kappa :\mathscr {D}\rightarrow \mathbb {R}$$ κ : D → R , and the fractional exponent $$\beta >0$$ β > 0 , we prove convergence in $$L_q(\varOmega ; H^\sigma (\mathscr {D}))$$ L q ( Ω ; H σ ( D ) ) and in $$L_q(\varOmega ; C^\delta (\overline{\mathscr {D}}))$$ L q ( Ω ; C δ ( D ¯ ) ) at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on $$H^{1+\alpha }(\mathscr {D})$$ H 1 + α ( D ) -regularity of the differential operator L, where $$0<\alpha \le 1$$ 0 < α ≤ 1 . For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in $$L_{\infty }(\mathscr {D}\times \mathscr {D})$$ L ∞ ( D × D ) and in the mixed Sobolev space $$H^{\sigma ,\sigma }(\mathscr {D}\times \mathscr {D})$$ H σ , σ ( D × D ) , showing convergence which is more than twice as fast compared to the corresponding $$L_q(\varOmega ; H^\sigma (\mathscr {D}))$$ L q ( Ω ; H σ ( D ) ) -rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle–Matérn class, where $$A\equiv \mathrm {Id}_{\mathbb {R}^d}$$ A ≡ Id R d and $$\kappa \equiv {\text {const.}}$$ κ ≡ const. , and (b) an example of anisotropic, non-stationary Gaussian random fields in $$d=2$$ d = 2 dimensions, where $$A:\mathscr {D}\rightarrow \mathbb {R}^{2\times 2}$$ A : D → R 2 × 2 and $$\kappa :\mathscr {D}\rightarrow \mathbb {R}$$ κ : D → R are spatially varying.

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20

Kawaguchi, Shu, and Kazuhiko Yamaki. "Effective Faithful Tropicalizations Associated to Adjoint Linear Systems." International Mathematics Research Notices 2019, no. 19 (January 16, 2018): 6089–112. http://dx.doi.org/10.1093/imrn/rnx302.

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Abstract Let R be a complete discrete valuation ring of equi-characteristic zero with fraction field K. Let X be a connected smooth projective variety of dimension d over K, and let L be an ample line bundle over X. We assume that there exist a regular strictly semistable model ${\mathscr {X}}$ of X over R and a relatively ample line bundle ${\mathscr {L}}$ over ${\mathscr {X}}$ with $\left .{{\mathscr {L}}}\right \vert_{{X}} \cong L$. Let $S({\mathscr {X}})$ be the skeleton associated to ${\mathscr {X}}$ in the Berkovich analytification Xan of X. In this article, we study when $S({\mathscr {X}})$ is faithfully tropicalized into tropical projective space by the adjoint linear system |L⊗m ⊗ ωX|. Roughly speaking, our results show that if m is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of $S({\mathscr {X}})$.

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Robert, C. P., and James P. Hobert. "$\mathscr{P}$-admissibility." Annals of Statistics 27, no. 1 (March 1999): 361–73. http://dx.doi.org/10.1214/aos/1018031115.

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Иванов, Дмитрий Николаевич, and Dmitry Nikolaevich Ivanov. "$\mathscr H$-биекции групп и $\mathscr H_R$-изоморфизмы групповых колец." Математический сборник 188, no. 6 (1997): 27–46. http://dx.doi.org/10.4213/sm227.

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Bender, Carl M., C. Ford, Nima Hassanpour, and B. Xia. "Series solutions of ${\mathscr{P}}{\mathscr{T}}$-symmetric Schrödinger equations." Journal of Physics Communications 2, no. 2 (February 7, 2018): 025012. http://dx.doi.org/10.1088/2399-6528/aaa953.

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Balingit, Cherry Mae Rivas, and Julius Benitez. "Functions on $n$-generalized Topological Spaces." European Journal of Pure and Applied Mathematics 12, no. 4 (October 31, 2019): 1553–66. http://dx.doi.org/10.29020/nybg.ejpam.v12i4.3502.

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An $n$-generalized topological ($n$-GT) space is a pair $(X,\mathscr{G})$ of a nonempty set $X$ and a collection $\mathscr{G}$ of $n$ $(n\in\mathbb{N})$ distinct generalized topologies (in the sense of A. Cs\'{a}sz\'{a}r [1]) on the set $X$. In this paper, we look into $\mathscr{G}$-continuous maps, $\mathscr{G}$-open and $\mathscr{G}$-closed maps, as well as $\mathscr{G}$-homoemorphisms in terms of $n$-GT spaces and establish some of their basic properties and relationships. Moreover, these notions are also examined with respect to the component generalized topologies of the underlying spaces by defining and characterizing pairwise versions of the said types of mappings.

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Hassanzadeh, Mohammad, Masoud Khalkhali, and Ilya Shapiro. "Monoidal Categories, 2-Traces, and Cyclic Cohomology." Canadian Mathematical Bulletin 62, no. 02 (January 7, 2019): 293–312. http://dx.doi.org/10.4153/cmb-2018-016-4.

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AbstractIn this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category ( $\mathscr{C},\otimes$ ) endowed with a symmetric 2-trace, i.e., an $F\in \text{Fun}(\mathscr{C},\text{Vec})$ satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in $F$ ”. Furthermore, we observe that if $\mathscr{M}$ is a $\mathscr{C}$ -bimodule category and $(F,M)$ is a stable central pair, i.e., $F\in \text{Fun}(\mathscr{M},\text{Vec})$ and $M\in \mathscr{M}$ satisfy certain conditions, then $\mathscr{C}$ acquires a symmetric 2-trace. The dual notions of symmetric 2-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories.

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Вильданова, Венера Фидарисовна, and Venera Fidarisovna Vil'danova. "Существование и единственность слабого решения интегро-дифференциального уравнения агрегации на римановом многообразии." Математический сборник 211, no. 2 (2020): 74–105. http://dx.doi.org/10.4213/sm9216.

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На компактном римановом многообразии $\mathscr{M}$ рассматривается класс интегро-дифференциальных уравнений агрегации с нелинейным параболическим членом $b(x,u)_t$. Дивергентный член в уравнениях может вырождаться с потерей коэрцитивности и содержит нелинейности с переменными показателями. Краевое условие "непротекания" на границе $\partial\mathscr{M}\times[0,T]$ цилиндра $Q^T=\mathscr{M}\times[0,T]$ обеспечивает при отсутствии внешних источников сохранение "массы" $\displaystyle\int_\mathscr{M}b(x,u(x,t)) d\nu=\mathrm{const}$. В цилиндре $Q^T$ с достаточно малым $T$ доказано существование ограниченного решения смешанной задачи для уравнения агрегации. При дополнительных условиях доказано существование ограниченного решения задачи в цилиндре $Q^{\infty}=\mathscr{M}\times[0,\infty)$. Для уравнений вида $b(x,u)_t=\Delta A(x,u)-\operatorname{div}(b(x,u)\mathscr{G}(u))+f(x,u)$ с оператором Лапласа-Бельтрами $\Delta$ и интегральным оператором $\mathscr{G}(u)$ доказана единственность ограниченного решения смешанной задачи. Библиография: 26 названий.

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Amend, Nils, Pierre Deligne, and Gerhard Röhrle. "On the -problem for restrictions of complex reflection arrangements." Compositio Mathematica 156, no. 3 (January 20, 2020): 526–32. http://dx.doi.org/10.1112/s0010437x19007796.

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Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X(\mathscr{A}(W))$ of the reflection arrangement $\mathscr{A}(W)$ is a $K(\unicode[STIX]{x1D70B},1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr{A}(W)$, let $X(\mathscr{A}(W)^{Y})$ be the complement in $Y$ of the hyperplanes in $\mathscr{A}(W)$ not containing $Y$. We hope that $X(\mathscr{A}(W)^{Y})$ is always a $K(\unicode[STIX]{x1D70B},1)$. We prove it in case of the monomial groups $W=G(r,p,\ell )$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.

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Ciaglia, F. M., A. Ibort, J. Jost, and G. Marmo. "Manifolds of classical probability distributions and quantum density operators in infinite dimensions." Information Geometry 2, no. 2 (October 24, 2019): 231–71. http://dx.doi.org/10.1007/s41884-019-00022-1.

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Abstract The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of $$C^{*}$$C∗-algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given $$C^{*}$$C∗-algebra $$\mathscr {A}$$A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $$\mathscr {S}$$S of a possibly infinite-dimensional, unital $$C^{*}$$C∗-algebra $$\mathscr {A}$$A is partitioned into the disjoint union of the orbits of an action of the group $$\mathscr {G}$$G of invertible elements of $$\mathscr {A}$$A. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space $$\mathcal {H}$$H are smooth, homogeneous Banach manifolds of $$\mathscr {G}=\mathcal {GL}(\mathcal {H})$$G=GL(H), and, when $$\mathscr {A}$$A admits a faithful tracial state $$\tau $$τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through $$\tau $$τ is a smooth, homogeneous Banach manifold for $$\mathscr {G}$$G.

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Сырцева, Ксения Александровна, and Ksenia Alexandrovna Syrtseva. "Алгебры свободных голоморфных функций и локализации." Математический сборник 210, no. 9 (2019): 89–106. http://dx.doi.org/10.4213/sm9130.

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Рассматриваются алгебры голоморфных функций на свободном полидиске $\mathscr{F}^T(\mathbb{D}_R^n)$, $\mathscr{F}(\mathbb{D}_R^n)$ и алгебра голоморфных функций на свободном шаре $\mathscr{F}(\mathbb{B}_r^n)$. Показано, что алгебра $\mathscr{F}(\mathbb{D}_R^n)$ является локализацией свободной алгебры и, более того, свободной аналитической алгеброй с $n$ образующими (в смысле Дж. Л. Тейлора), а алгебра $\mathscr{F}(\mathbb{B}_r^n)$ не является локализацией свободной алгебры. Кроме того, доказано, что класс локализаций свободных алгебр и класс свободных аналитических алгебр замкнуты относительно операции свободного произведения Аренса-Майкла. Библиография: 21 название.

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30

Qian, Zicheng. "Dilogarithm and higher ℒ-invariants for 𝒢ℒ₃(𝐐_{𝐩})." Representation Theory of the American Mathematical Society 25, no. 12 (May 3, 2021): 344–411. http://dx.doi.org/10.1090/ert/567.

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The primary purpose of this paper is to clarify the relation between previous results in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145], [Amer. J. Math. 141 (2019), pp. 661–703], and [Camb. J. Math. 8 (2020), p. 775–951] via the construction of some interesting locally analytic representations. Let E E be a sufficiently large finite extension of Q p \mathbf {Q}_p and ρ p \rho _p be a p p -adic semi-stable representation G a l ( Q p ¯ / Q p ) → G L 3 ( E ) \mathrm {Gal}(\overline {\mathbf {Q}_p}/\mathbf {Q}_p)\rightarrow \mathrm {GL}_3(E) such that the associated Weil–Deligne representation W D ( ρ p ) \mathrm {WD}(\rho _p) has rank two monodromy and the associated Hodge filtration is non-critical. A computation of extensions of rank one ( φ , Γ ) (\varphi , \Gamma ) -modules shows that the Hodge filtration of ρ p \rho _p depends on three invariants in E E . We construct a family of locally analytic representations Σ m i n ( λ , L 1 , L 2 , L 3 ) \Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3) of G L 3 ( Q p ) \mathrm {GL}_3(\mathbf {Q}_p) depending on three invariants L 1 , L 2 , L 3 ∈ E \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3 \in E , such that each representation in the family contains the locally algebraic representation A l g ⊗ S t e i n b e r g \mathrm {Alg}\otimes \mathrm {Steinberg} determined by W D ( ρ p ) \mathrm {WD}(\rho _p) (via classical local Langlands correspondence for G L 3 ( Q p ) \mathrm {GL}_3(\mathbf {Q}_p) ) and the Hodge–Tate weights of ρ p \rho _p . When ρ p \rho _p comes from an automorphic representation π \pi of a unitary group over Q \mathbf {Q} which is compact at infinity, we show (under some technical assumption) that there is a unique locally analytic representation in the above family that occurs as a subrepresentation of the Hecke eigenspace (associated with π \pi ) in the completed cohomology. We note that [Amer. J. Math. 141 (2019), pp. 611–703] constructs a family of locally analytic representations depending on four invariants ( cf. (4) in that publication ) and proves that there is a unique representation in this family that embeds into the Hecke eigenspace above. We prove that if a representation Π \Pi in Breuil’s family embeds into the Hecke eigenspace above, the embedding of Π \Pi extends uniquely to an embedding of a Σ m i n ( λ , L 1 , L 2 , L 3 ) \Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3) into the Hecke eigenspace, for certain L 1 , L 2 , L 3 ∈ E \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3\in E uniquely determined by Π \Pi . This gives a purely representation theoretical necessary condition for Π \Pi to embed into completed cohomology. Moreover, certain natural subquotients of Σ m i n ( λ , L 1 , L 2 , L 3 ) \Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3) give an explicit complex of locally analytic representations that realizes the derived object Σ ( λ , L _ ) \Sigma (\lambda , \underline {\mathscr {L}}) in (1.14) of [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145]. Consequently, the locally analytic representation Σ m i n ( λ , L 1 , L 2 , L 3 ) \Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3) gives a relation between the higher L \mathscr {L} -invariants studied in [Amer. J. Math. 141 (2019), pp. 611–703] as well as the work of Breuil and Ding and the p p -adic dilogarithm function which appears in the construction of Σ ( λ , L _ ) \Sigma (\lambda , \underline {\mathscr {L}}) in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145].

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31

KOUSHESH, M. R. "ONE-POINT EXTENSIONS AND LOCAL TOPOLOGICAL PROPERTIES." Bulletin of the Australian Mathematical Society 88, no. 1 (August 2, 2012): 12–16. http://dx.doi.org/10.1017/s0004972712000524.

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AbstractA space $Y$ is called an extension of a space $X$ if $Y$ contains $X$ as a dense subspace. An extension $Y$ of $X$ is called a one-point extension of $X$ if $Y\setminus X$ is a singleton. P. Alexandroff proved that any locally compact non-compact Hausdorff space $X$ has a one-point compact Hausdorff extension, called the one-point compactification of $X$. Motivated by this, Mrówka and Tsai [‘On local topological properties. II’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.19 (1971), 1035–1040] posed the following more general question: For what pairs of topological properties ${\mathscr P}$ and ${\mathscr Q}$ does a locally-${\mathscr P}$ space $X$ having ${\mathscr Q}$ possess a one-point extension having both ${\mathscr P}$ and ${\mathscr Q}$? Here, we provide an answer to this old question.

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32

Мощевитин, Николай Германович, and Nikolai Germanovich Moshchevitin. "О множествах вида $\mathscr A+\mathscr B$ и конечных цепных дробях." Математический сборник 198, no. 4 (2007): 95–116. http://dx.doi.org/10.4213/sm3772.

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33

Znojil, Miloslav, and František Růžička. "Nonlinearity of perturbations in ${\mathscr{P}}{\mathscr{T}}$-symmetric quantum mechanics." Journal of Physics: Conference Series 1194 (April 2019): 012120. http://dx.doi.org/10.1088/1742-6596/1194/1/012120.

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34

Walasik, Wiktor, Chicheng Ma, and Natalia M. Litchinitser. "Dissimilar directional couplers showing ${\mathscr{P}}{\mathscr{T}}$-symmetric-like behavior." New Journal of Physics 19, no. 7 (July 12, 2017): 075002. http://dx.doi.org/10.1088/1367-2630/aa7092.

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35

Alexeeva, N. V., I. V. Barashenkov, and Y. S. Kivshar. "Solitons in ${\mathscr{P}}{\mathscr{T}}$-symmetric ladders of optical waveguides." New Journal of Physics 19, no. 11 (November 22, 2017): 113032. http://dx.doi.org/10.1088/1367-2630/aa8fdd.

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36

Dong, Yuanhao, Wen-Jing Zhang, Jing Liu, and Xiao-Tao Xie. "Analytic solutions for generalized ${\mathscr{P}}{\mathscr{T}}$-symmetric Rabi models." Chinese Physics B 28, no. 11 (October 2019): 114202. http://dx.doi.org/10.1088/1674-1056/ab457b.

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37

Kondyrev, Grigory, and Artem Prikhodko. "CATEGORICAL PROOF OF HOLOMORPHIC ATIYAH–BOTT FORMULA." Journal of the Institute of Mathematics of Jussieu 19, no. 5 (December 20, 2018): 1739–63. http://dx.doi.org/10.1017/s1474748018000543.

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Given a $2$-commutative diagramin a symmetric monoidal $(\infty ,2)$-category $\mathscr{E}$ where $X,Y\in \mathscr{E}$ are dualizable objects and $\unicode[STIX]{x1D711}$ admits a right adjoint we construct a natural morphism $\mathsf{Tr}_{\mathscr{E}}(F_{X})\xrightarrow[{}]{~~~~~}\mathsf{Tr}_{\mathscr{E}}(F_{Y})$ between the traces of $F_{X}$ and $F_{Y}$, respectively. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty ,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah–Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

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Кабелло Санчес, Феликс, Félix Cabello Sanchez, Хесус М. Ф. Кастильо, Jesus María Fernandez Castillo, Рикардо Гарсия, and Ricardo García. "Гомологические размерности банаховых пространств." Математический сборник 212, no. 4 (2021): 91–112. http://dx.doi.org/10.4213/sm9425.

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Цель этой статьи - заложить основы исследования вопроса, когда $\operatorname{Ext}^n(X,Y)=0$ для банаховых пространств. Мы приводим несколько примеров пар $X$, $Y$, для которых $\operatorname{Ext}^n(X,Y)$ равно (или не равно) $0$. Мы покажем, что $\operatorname{Ext}^n(\mathscr K,\mathscr K)\neq0$ для всех $n\in\mathbb{N}$, если $\mathscr K$ - пространство Кадеца. В частности, как проективная, так и инъективная размерности $\mathscr K$ бесконечны. Библиография: 48 названий.

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Eleftherakis, G. K. "Morita Equivalence of Nest Algebras." MATHEMATICA SCANDINAVICA 113, no. 1 (September 1, 2013): 83. http://dx.doi.org/10.7146/math.scand.a-15483.

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Let $\mathscr{N}_1$ (resp. $\mathscr{N}_2$) be a nest, $A$ (resp. $B$) be the corresponding nest algebra, $A_0$ (resp. $B_0$) be the subalgebra of compact operators. We prove that the nests $\mathscr{N}_1, \mathscr{N}_2$ are isomorphic if and only if $A$ and $B$ are weakly-$*$ Morita equivalent if and only if $A_0$ and $ B_0$ are strongly Morita equivalent. We characterize the nest isomorphisms which implement stable isomorphism between the corresponding nest algebras.

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CALIMERI, FRANCESCO, CARMINE DODARO, DAVIDE FUSCÀ, SIMONA PERRI, and JESSICA ZANGARI. "Efficiently Coupling the I-DLV Grounder with ASP Solvers." Theory and Practice of Logic Programming 20, no. 2 (December 4, 2018): 205–24. http://dx.doi.org/10.1017/s1471068418000546.

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We present ${{{{$\mathscr{I}$}-}\textsc{dlv}}+{{$\mathscr{MS}$}}}$, a new answer set programming (ASP) system that integrates an efficient grounder, namely ${{{$\mathscr{I}$}-}\textsc{dlv}}$, with an automatic selector that inductively chooses a solver: depending on some inherent features of the instantiation produced by ${{{$\mathscr{I}$}-}\textsc{dlv}}$, machine learning techniques guide the selection of the most appropriate solver. The system participated in the latest (7th) ASP competition, winning the regular track, category SP (i.e., one processor allowed).

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41

Любимцев, Олег Владимирович, and Oleg Vladimirovich Ljubimtsev. "Вполне разложимые факторно делимые абелевы группы с изоморфными полугруппами эндоморфизмов." Matematicheskie Zametki 108, no. 2 (2020): 224–35. http://dx.doi.org/10.4213/mzm12536.

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Пусть $\Lambda$ - некоторый класс абелевых групп. Говорят, что группа $A\in\Lambda$ определяется своей полугруппой $E^\star(A)$ эндоморфизмов в классе $\Lambda$, если всякий раз из изоморфизма $E^\star(A)\cong E^\star(B)$, где $B\in\Lambda$, следует изоморфизм $A\cong B$. В статье описаны абелевы группы из класcа $\mathscr Q\mathscr D_{\mathrm{cd}}$ вполне разложимых факторно делимых абелевых групп, которые определяются своими полугруппами эндоморфизмов в классе $\mathscr Q\mathscr D_{\mathrm{cd}}$. Библиография: 14 названий.

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42

Lee, Wan, Wan Lee, Soogil Seo, and Soogil Seo. "Об арифметике модифицированных групп классов иделей." Известия Российской академии наук. Серия математическая 84, no. 3 (2020): 119–67. http://dx.doi.org/10.4213/im8849.

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Пусть $k$ - числовое поле и $S$, $T$ - множества точек поля $k$. Для любого простого $p$ мы определяем инвариант $\mathscr{G}=\mathscr{G}_p(k_\infty/k,S,T)$, связанный с группой Галуа максимального абелева расширения поля $k$, которое не разветвлено вне $S$ и вполне распадается в $T$. В основной теореме мы интерпретируем $\mathscr{G}$ в терминах другого арифметического объекта $\mathscr{U}$, затрагивающего различные группы единиц и использующего теорию родов, примененную к некоторым модулям, которые получены некоторыми техническими модификациями из групп иделей. Мы показываем, что эта интерпретация функториальна относительно $S$ и $T$ и, вследствие этого, приводит к интересным взаимосвязям арифметических объектов $\mathscr{G}$ и $\mathscr{U}$ при меняющихся $S$ и $T$. Наш подход и методы новы и отличны от классических методов теории родов для групп иделей. Преимущество новых методов на конечном уровне не только обобщает, но также усиливает некоторые известные результаты, затрагивающие максимальную $p$-абелеву проконечную группу Галуа поля $k$, не разветвленную вне $S$ и распадающуюся в $T$, в терминах арифметики некоторых единиц поля $k$. На бесконечном уровне наши методы связывают глубокую арифметику специальных единиц с арифметикой проконечных групп Галуа. Например, для специального выбора $S$ и $T$ инварианты $\mathscr{G}$ связаны с гипотезами Гросса (или Кузьмина-Гросса) и Леопольдта. Соответственно, функториальная интерпретация $\mathscr{G}$ при вариации $S$ и $T$ в специальных случаях включает интересные связи между гипотезами Гросса и Леопольдта, полученные более простым и конкретным образом. Как результат, мы высказываем предположение, что $\mathscr{G}$ конечен для всех конечных непересекающихся множеств $S$, $T$ над круговой $\mathbb{Z}_p$-башней поля $k$, что включает гипотезы Гросса и Леопольдта как специальные случаи. Библиография: 23 наименования.

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Brown, Morgan V., and Enrica Mazzon. "The essential skeleton of a product of degenerations." Compositio Mathematica 155, no. 7 (June 13, 2019): 1259–300. http://dx.doi.org/10.1112/s0010437x19007346.

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We study the problem of how the dual complex of the special fiber of a strict normal crossings degeneration$\mathscr{X}_{R}$changes under products. We view the dual complex as a skeleton inside the Berkovich space associated to$X_{K}$. Using the Kato fan, we define a skeleton$\text{Sk}(\mathscr{X}_{R})$when the model$\mathscr{X}_{R}$is log-regular. We show that if$\mathscr{X}_{R}$and$\mathscr{Y}_{R}$are log-smooth, and at least one is semistable, then$\text{Sk}(\mathscr{X}_{R}\times _{R}\mathscr{Y}_{R})\simeq \text{Sk}(\mathscr{X}_{R})\times \text{Sk}(\mathscr{Y}_{R})$. The essential skeleton$\text{Sk}(X_{K})$, defined by Mustaţă and Nicaise, is a birational invariant of$X_{K}$and is independent of the choice of$R$-model. We extend their definition to pairs, and show that if both$X_{K}$and$Y_{K}$admit semistable models,$\text{Sk}(X_{K}\times _{K}Y_{K})\simeq \text{Sk}(X_{K})\times \text{Sk}(Y_{K})$. As an application, we compute the homeomorphism type of the dual complex of some degenerations of hyper-Kähler varieties. We consider both the case of the Hilbert scheme of a semistable degeneration of K3 surfaces, and the generalized Kummer construction applied to a semistable degeneration of abelian surfaces. In both cases we find that the dual complex of the$2n$-dimensional degeneration is homeomorphic to a point,$n$-simplex, or$\mathbb{C}\mathbb{P}^{n}$, depending on the type of the degeneration.

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44

Wei, Tie. "Integral properties of turbulent-kinetic-energy production and dissipation in turbulent wall-bounded flows." Journal of Fluid Mechanics 854 (September 10, 2018): 449–73. http://dx.doi.org/10.1017/jfm.2018.578.

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Turbulent-kinetic-energy (TKE) production $\mathscr{P}_{k}=R_{12}(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y)$ and TKE dissipation $\mathscr{E}_{k}=\unicode[STIX]{x1D708}\langle (\unicode[STIX]{x2202}u_{i}/x_{k})(\unicode[STIX]{x2202}u_{i}/x_{k})\rangle$ are important quantities in the understanding and modelling of turbulent wall-bounded flows. Here $U$ is the mean velocity in the streamwise direction, $u_{i}$ or $u,v,w$ are the velocity fluctuation in the streamwise $x$- direction, wall-normal $y$- direction, and spanwise $z$-direction, respectively; $\unicode[STIX]{x1D708}$ is the kinematic viscosity; $R_{12}=-\langle uv\rangle$ is the kinematic Reynolds shear stress. Angle brackets denote Reynolds averaging. This paper investigates the integral properties of TKE production and dissipation in turbulent wall-bounded flows, including turbulent channel flows, turbulent pipe flows and zero-pressure-gradient turbulent boundary layer flows (ZPG TBL). The main findings of this work are as follows. (i) The global integral of TKE production is predicted by the RD identity derived by Renard & Deck (J. Fluid Mech., vol. 790, 2016, pp. 339–367) as $\int _{0}^{\unicode[STIX]{x1D6FF}}\mathscr{P}_{k}\,\text{d}y=U_{b}u_{\unicode[STIX]{x1D70F}}^{2}-\int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y)^{2}\,\text{d}y$ for channel flows, where $U_{b}$ is the bulk mean velocity, $u_{\unicode[STIX]{x1D70F}}$ is the friction velocity and $\unicode[STIX]{x1D6FF}$ is the channel half-height. Using inner scaling, the identity for the global integral of the TKE production in channel flows is $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\text{d}y^{+}=U_{b}^{+}-\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$. In the present work, superscript $+$ denotes inner scaling. At sufficiently high Reynolds number, the global integral of the TKE production in turbulent channel flows can be approximated as $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}\approx U_{b}^{+}-9.13$. (ii) At sufficiently high Reynolds number, the integrals of TKE production and dissipation are equally partitioned around the peak Reynolds shear stress location $y_{m}:\,\int _{0}^{y_{m}}\mathscr{P}_{k}\,\text{d}y\approx \int _{y_{m}}^{\unicode[STIX]{x1D6FF}}\mathscr{P}_{k}\,\text{d}y$ and $\int _{0}^{y_{m}}\mathscr{E}_{k}\,\text{d}y\approx \int _{y_{m}}^{\unicode[STIX]{x1D6FF}}\mathscr{E}_{k}\,\text{d}y$. (iii) The integral of the TKE production ${\mathcal{I}}_{\mathscr{P}_{k}}(y)=\int _{0}^{y}\mathscr{P}_{k}\,\text{d}y$ and the integral of the TKE dissipation ${\mathcal{I}}_{\mathscr{E}_{k}}(y)=\int _{0}^{y}\mathscr{E}_{k}\,\text{d}y$ exhibit a logarithmic-like layer similar to that of the mean streamwise velocity as, for example, ${\mathcal{I}}_{\mathscr{P}_{k}}^{+}(y^{+})\approx (1/\unicode[STIX]{x1D705})\ln (y^{+})+C_{\mathscr{P}}$ and ${\mathcal{I}}_{\mathscr{E}_{k}}^{+}(y^{+})\approx (1/\unicode[STIX]{x1D705})\ln (y^{+})+C_{\mathscr{E}}$, where $\unicode[STIX]{x1D705}$ is the von Kármán constant, $C_{\mathscr{P}}$ and $C_{\mathscr{E}}$ are addititve constants. The logarithmic-like scaling of the global integral of TKE production and dissipation, the equal partition of the integrals of TKE production and dissipation around the peak Reynolds shear stress location $y_{m}$ and the logarithmic-like layer in the integral of TKE production and dissipation are intimately related. It is known that the peak Reynolds shear stress location $y_{m}$ scales with a meso-length scale $l_{m}=\sqrt{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}}$. The equal partition of the integral of the TKE production and dissipation around $y_{m}$ underlines the important role of the meso-length scale $l_{m}$ in the dynamics of turbulent wall-bounded flows.

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45

Lu, Qian, and Qilong Liao. "Normal criterion and shared values by derivatives of meromorphic functions." Tamkang Journal of Mathematics 45, no. 2 (June 30, 2014): 109–17. http://dx.doi.org/10.5556/j.tkjm.45.2014.1014.

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Let $\mathscr{F}$ be a family of meromorphic functions in a plane domain $D$. If for every function $f\in\mathscr{F}$, all of whose zeros have,at least,multiplicity $l$ and poles have, at least,multiplicity $p$, and for each pair functions $f$ and $g$ in $\mathscr{F}$, $f^{(k)}$ and $g^{(k)}$ share 1 in $D$, where $k,l,$ and $p$ are three positive integer satisfying $\frac{k+1}{l}+\frac{1}{p}\leq 1$, then $\mathscr{F}$ is normal.

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46

S., Midhun, and Raji Pilakkat. "Connected $\mathscr{F}$ domination." Malaya Journal of Matematik 8, no. 4 (2020): 1894–97. http://dx.doi.org/10.26637/mjm0804/0093.

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47

Schmitt, Lothar M. "Characterization of $L^2(\mathscr{M})$ for injective W*-algebras $\mathscr{M}$." MATHEMATICA SCANDINAVICA 57 (December 1, 1985): 267. http://dx.doi.org/10.7146/math.scand.a-12117.

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48

Ivanov, D. N. "$ \mathscr H$-bijections of groups and $ \mathscr H_R$-isomorphisms of group rings." Sbornik: Mathematics 188, no. 6 (June 30, 1997): 823–41. http://dx.doi.org/10.1070/sm1997v188n06abeh000227.

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49

Moshchevitin, N. G. "Sets of the form $ \mathscr A+\mathscr B$ and finite continued fractions." Sbornik: Mathematics 198, no. 4 (April 30, 2007): 537–57. http://dx.doi.org/10.1070/sm2007v198n04abeh003848.

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50

STEINBERG, BENJAMIN. "MODULES OVER ÉTALE GROUPOID ALGEBRAS AS SHEAVES." Journal of the Australian Mathematical Society 97, no. 3 (September 9, 2014): 418–29. http://dx.doi.org/10.1017/s1446788714000342.

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AbstractThe author has previously associated to each commutative ring with unit $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Bbbk $ and étale groupoid $\mathscr{G}$ with locally compact, Hausdorff, totally disconnected unit space a $\Bbbk $-algebra $\Bbbk \, \mathscr{G}$. The algebra $\Bbbk \, \mathscr{G}$ need not be unital, but it always has local units. The class of groupoid algebras includes group algebras, inverse semigroup algebras and Leavitt path algebras. In this paper we show that the category of unitary$\Bbbk \, \mathscr{G}$-modules is equivalent to the category of sheaves of $\Bbbk $-modules over $\mathscr{G}$. As a consequence, we obtain a new proof of a recent result that Morita equivalent groupoids have Morita equivalent algebras.

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