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Example 0.1 Let \( {L}_{1} = \left\langle {a, b\left| {\;{a}^{5} = {b}^{5} = 1}\right. ,{ab} = {ba}}\right\rangle \) and \( {L}_{2} = \left\langle {{a}^{\prime },{b}^{\prime }}\right\rangle \) be a copy of \( {L}_{1} \) . Assume that \( \alpha \) is an automorphism of \( {L}_{1} \) of order 3 satisfying \( {a}^{\alpha ... | However, \( \left| {G : {N}_{G}\left( {H \cap {L}_{1}}\right) }\right| = \left| {G : {N}_{G}\left( {\langle a\rangle }\right) }\right| = 3 \) is not a 5-number, that is, \( H \) is not a generalized \( {CAP} \) -subgroup of \( G \) . | Yes |
Example 0.2 Let \( G = P \rtimes {A}_{5} \), where \( {A}_{5} \) is the alternating group of degree 5 and \( P \) is a simple \( {\mathbb{F}}_{3}{A}_{5} \) -modular which is faithful for \( {A}_{5} \) . Let \( H = {PB} \), where \( B \) is a Sylow 5-subgroup of \( {A}_{5} \) . Following [11], \( H \) is a generalized \... | By [11] again, \( H \) is neither a \( {CAP} \) -subgroup of \( G \) nor \( S \) -quasinormally embedded in \( G \) . Moreover, \( H \) is not \( {sn} \) -embedded in \( G \) . In fact, the normal subgroup \( T \) of \( G \) satisfying that \( {HT} \) is \( S \) -quasinormal in \( G \) must be \( G \) . In this case, \... | No |
Theorem 0.1 A subset \( E \) of \( {C}_{n}\left( \Omega \right) \) is \( a \) -minimally thin at infinity with respect to \( {C}_{n}\left( \Omega \right) \) if and only if | \[ \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{W\left( {2}^{k}\right) }{V\left( {2}^{k}\right) {C}_{\Omega }^{a}\left( {E}_{k}\right) } < \infty \] | No |
Theorem 0.2 A subset \( E \) of \( {C}_{n}\left( \Omega \right) \) is \( a \) -rarefied at infinity with respect to \( {C}_{n}\left( \Omega \right) \) if and only if | \[ \mathop{\sum }\limits_{{k = 0}}^{\infty }W\left( {2}^{k}\right) {C}_{\Omega }^{a}\left( {E}_{k}\right) < \infty \] | Yes |
Lemma 2.1 If the subset \( E \) of \( {C}_{n}\left( \Omega \right) \) is compact, then \( {\gamma }_{\Omega }^{a}\left( E\right) = + \infty \) if and only if \( {C}_{\Omega }^{a}\left( E\right) = 0. \) | Proof Assume that \( {C}_{\Omega }^{a}\left( E\right) > 0 \), and choose a non-negative Borel measure \( \mathrm{d}\mu \) supported in \( E \) via \( \mathrm{d}\mu \left( E\right) > 0 \) and\n\n\[ \n{G}_{\Omega }^{a}\left( {\mathrm{d}\mu }\right) \leq 1 \n\]\n\neverywhere.\n\nSet \( \mathrm{d}\nu = \frac{\mathrm{d}\mu ... | Yes |
Lemma 2.2 Suppose that \( {\widetilde{C}}_{\Omega }^{a}\left( A\right) = 0 \) and \( \mathrm{d}\mu \) is any non-negative Borel measure supported in a compact set of \( {C}_{n}\left( \Omega \right) \) with \( {G}_{\Omega }^{a}\left( {\mathrm{d}\mu }\right) \) being bounded from above in the set \( A \) . Then \( \mathr... | Proof Obviously, we see that the set \( A \) should be bounded. Suppose that \( \mathrm{d}\mu \left( E\right) > 0 \) for some compact set \( E \subseteq A \) and the restriction \( \mathrm{d}{\mu }_{E} = {\left. \mathrm{d}\mu \right| }_{E} \) . Then \( {G}_{\Omega }^{a}\left( {\mathrm{\;d}{\mu }_{E}}\right) \) is also ... | Yes |
Lemma 2.3 If \( {C}_{\Omega }^{a}\left( {A}_{k}\right) = 0 \) for all \( k \), then\n\n\[ \n{C}_{\Omega }^{a}\left( {\mathop{\bigcup }\limits_{{k = 1}}^{\infty }{A}_{k}}\right) = 0 \n\] | Proof Since \( {C}_{\Omega }^{a}\left( {A}_{k}\right) = 0 \) for all \( k \), we may choose open \( {U}_{k} \subset {\mathbb{R}}^{n} \) so that \( {A}_{k} \subset {U}_{k} \) and \( {C}_{\Omega }^{a}\left( {A}_{k}\right) \leq {C}_{\Omega }^{a}\left( {A}_{k}\right) + \frac{\varepsilon }{{2}^{k}} \) . For each \( k \), le... | Yes |
Theorem 3.1 Suppose that \( E \) is a compact set with \( {\gamma }_{\Omega }^{a}\left( E\right) > 0 \) . Consider two extremal problems as follows:\n\n(I) \( {\gamma }_{\Omega }^{a}\left( E\right) = \inf {I}_{\Omega }^{a}\left( {\mathrm{d}\mu }\right) \) over all probability Borel measures \( \mathrm{d}\mu \) supporte... | Proof At first we may assume that \( 0 < {\gamma }_{\Omega }^{a}\left( E\right) < + \infty \) . Otherwise, if \( {\gamma }_{\Omega }^{a}\left( E\right) = + \infty \), then \( {C}_{\Omega }^{a}\left( E\right) = 0 \) from Lemma 2.1. Since the \( {G}_{\Omega }^{a} \) -equilibrium measure \( \mathrm{d}{\mu }_{0} \) of \( E... | Yes |
Theorem 3.2 For a compact set \( E \) of \( {C}_{n}\left( \Omega \right) \), the extremal problem\n\n\[ \n{\gamma }_{\Omega }^{a}\left( E\right) = \inf {I}_{\Omega }^{a}\left( {\mathrm{\;d}\mu }\right) \n\]\n\nover all non-negative Borel measures with total mass equal to 1 supported in \( E \), i.e. probability Borel m... | Proof First, assume that \( {\gamma }_{\Omega }^{a}\left( E\right) < \infty \) . We consider a sequence of probability Borel measures \( \mathrm{d}{\mu }_{k} \) supported in \( E \) so that\n\n\[ \n{I}_{\Omega }^{a}\left( {\mathrm{\;d}{\mu }_{k}}\right) \rightarrow {\gamma }_{\Omega }^{a}\left( E\right) \;\text{ as }\;... | Yes |
Theorem 2.1 (Sphere-packing bound) Let \( t = \left\lfloor \frac{j - 1}{2}\right\rfloor \) . Then\n\n\[ \mathcal{A}\left( {{2\nu },{2j}, m}\right) \leq \frac{{\left\lbrack \begin{matrix} \nu \\ m \end{matrix}\right\rbrack }_{q}\mathop{\prod }\limits_{{i = \nu - m + 1}}^{\nu }\left( {{q}^{i} + 1}\right) }{\mathop{\sum }... | Proof Let \( \mathbb{C} \subseteq \mathcal{M}\left( {m,0;{2\nu }}\right) \) be a \( \left( {{2\nu }, M, d, m}\right) \) code. Then the spheres of radius \( {2t} \) about distinct codewords in \( \mathbb{C} \) are disjoint. By (2) each of these spheres contains\n\n\[ \mathop{\sum }\limits_{{i = 0}}^{t}N\left( {m - i;m,0... | Yes |
Theorem 2.2 (Wang-Xing-Safavi-Naini bound) Let \( j \leq m \) . Then \[ \mathcal{A}\left( {{2\nu },{2j}, m}\right) \leq \mathop{\prod }\limits_{{t = 0}}^{{m - j}}\frac{{q}^{2\left( {\nu - t}\right) } - 1}{{q}^{m - t} - 1} \mathrel{\text{:=}} {B}_{\mathrm{{WXS}}}. \] | Proof Let \( \mathbb{C} \subseteq \mathcal{M}\left( {m,0;{2\nu }}\right) \) be a \( \left( {{2\nu }, M,{2j}, m}\right) \) code. Then each codeword of \( \mathbb{C} \) contains exactly \( {\left\lbrack \begin{matrix} m \\ m - j + 1 \end{matrix}\right\rbrack }_{q} \) many \( \left( {m - j + 1}\right) \) -dimensional tota... | Yes |
Lemma 3. \( {2}^{\left\lbrack 1,\text{Lemma 1}\right\rbrack } \) Let \( \Gamma = \left( {V, E}\right) \) be a graph that admits a transitive group of auto-morphisms Aut \( \left( \Gamma \right) \) and let \( A, B \) be arbitrary subsets of the vertex set \( V \) . Then there exists some \( g \in \operatorname{Aut}\left... | \[ \frac{\left| g\left( A\right) \cap B\right| }{\left| B\right| } \geq \frac{\left| A\right| }{\left| V\right| } \] | No |
Lemma 3.3 Let \( \mathbb{C} \subseteq \mathcal{M}\left( {m,0;{2\nu }}\right) \) be a \( \left( {{2\nu }, M,{2j}, m}\right) \) code. Then for an arbitrary subset \( \mathcal{B} \subseteq \mathcal{M}\left( {m,0;{2\nu }}\right) \), there exists a \( \left( {{2\nu },{M}^{ * },{2j}, m}\right) \) code \( {\mathbb{C}}^{ * } \... | Proof Define a graph \( \Gamma \) with the vertex set \( \mathcal{M}\left( {m,0;{2\nu }}\right) \), and two vertices \( P \) and \( Q \) are adjacent if \( \dim \left( {P \cap Q}\right) = m - 1 \) . Then \( \Gamma \) admits a transitive group of automorphisms \( S{p}_{2\nu }\left( {\mathbb{F}}_{q}\right) \) . By Lemma ... | Yes |
Theorem 4.1 (Johnson bound I) Let \( j \leq m - 1 \) . Then\n\n\[ \mathcal{A}\left( {{2\nu },{2j}, m}\right) \leq \frac{{q}^{2\nu } - 1}{{q}^{m} - 1}\mathcal{A}\left( {2\left( {\nu - 1}\right) ,{2j}, m - 1}\right) . \] | Proof Let \( \mathbb{C} \subseteq \mathcal{M}\left( {m,0;{2\nu }}\right) \) be a \( \left( {{2\nu }, M,{2j}, m}\right) \) code. Then each codeword of \( \mathbb{C} \) contains exactly \( {\left\lbrack \begin{array}{l} m \\ 1 \end{array}\right\rbrack }_{q} \) many one-dimensional subspaces. Since the total number of one... | Yes |
Theorem 4.2 (Johnson bound II) Let \( j \leq m \leq \nu - 1 \) . Then\n\n\[ \mathcal{A}\left( {{2\nu },{2j}, m}\right) \leq \frac{{q}^{2\nu } - 1}{{q}^{2\left( {\nu - m}\right) } - 1}\mathcal{A}\left( {2\left( {\nu - 1}\right) ,{2j}, m}\right) . \] | Proof Let \( \mathbb{C} \subseteq \mathcal{M}\left( {m,0;{2\nu }}\right) \) be a \( \left( {{2\nu }, M,{2j}, m}\right) \) code. For each subspace \( V \) of type \( (2(\nu - \) 1), \( \nu - 1 \) ) of \( {\mathbb{F}}_{q}^{2\nu } \), define\n\n\[ {\mathbb{C}}_{V} = \{ U \in \mathbb{C} \mid U \subseteq V\} \]\n\nThen \( {... | Yes |
Corollary 4.2 Let \( j \leq m \leq \nu - 1 \) . Then\n\n\[ \n\mathcal{A}\left( {{2\nu },{2j}, m}\right) \leq \mathop{\prod }\limits_{{t = 0}}^{{m - j}}\left( {{q}^{m - t} + 1}\right) \mathop{\prod }\limits_{{t = 0}}^{{\nu - m - 1}}\frac{{q}^{2\left( {\nu - t}\right) } - 1}{{q}^{2\left( {\nu - t - m}\right) } - 1}.\n\] | Proof From Theorems 4.2 and 2.2 we deduce that\n\n\[ \n\mathcal{A}\left( {{2\nu },{2j}, m}\right) \leq \mathcal{A}\left( {{2m},{2j}, m}\right) \mathop{\prod }\limits_{{t = 0}}^{{\nu - m - 1}}\frac{{q}^{2\left( {\nu - t}\right) } - 1}{{q}^{2\left( {\nu - t - m}\right) } - 1}\n\]\n\n\[ \n\leq \mathop{\prod }\limits_{{t =... | Yes |
Theorem 4.3 (Gilbert-Varshamov bound) Let \( j \leq m \) . Then\n\n\[ \mathcal{A}\left( {{2\nu },{2j}, m}\right) \geq \frac{{\left\lbrack \begin{matrix} \nu \\ m \end{matrix}\right\rbrack }_{q}\mathop{\prod }\limits_{{t = \nu - m + 1}}^{\nu }\left( {{q}^{t} + 1}\right) }{\mathop{\sum }\limits_{{i = 0}}^{{j - 1}}N\left(... | Proof Let \( \mathbb{C} \subseteq \mathcal{M}\left( {m,0;{2\nu }}\right) \) be a \( \left( {{2\nu }, M,{2j}, m}\right) \) code. Then there is no subspace \( U \) in \( \mathcal{M}\left( {m,0;{2\nu }}\right) \) such that \( d\left( {U, W}\right) \geq {2j} \) for all \( W \in \mathbb{C} \) . Therefore, for any subspace \... | Yes |
Lemma 5.1 The total number of blocks in an \( {S}_{q}\left\lbrack {\ell, m;{2\nu }}\right\rbrack \) is \( \mathop{\prod }\limits_{{t = 0}}^{{l - 1}}\frac{{q}^{2\left( {\nu - t}\right) } - 1}{{q}^{m - t} - 1} \) . | Proof Count pairs \( \left( {U, W}\right) \in \mathcal{M}\left( {\ell ,0;{2\nu }}\right) \times \mathcal{M}\left( {m,0;{2\nu }}\right) \) such that \( \dot{U} \subseteq W \) in two ways. Since each block of \( {S}_{q}\left\lbrack {\ell, m;{2\nu }}\right\rbrack \) contains \( {\left\lbrack \begin{matrix} m \\ \ell \end{... | No |
Lemma 5.2 Let \( \ell \geq 2 \) . If \( {S}_{q}\left\lbrack {\ell, m;{2\nu }}\right\rbrack \) exists, then \( {S}_{q}\left\lbrack {\ell - 1, m - 1;2\left( {\nu - 1}\right) }\right\rbrack \) exists. | Proof Let \( S \subseteq \mathcal{M}\left( {m,0;{2\nu }}\right) \) be an \( {S}_{q}\left\lbrack {\ell, m;{2\nu }}\right\rbrack \) . Pick a fixed one-dimensional subspace \( \left\langle {e}_{1}\right\rangle \) . Let\n\n\[
{S}^{\prime } = \left\{ {W \cap \left\langle {{e}_{2},\cdots ,{e}_{\nu },{e}_{\nu + 2},\cdots ,{e}... | Yes |
Theorem 5.1 A Steiner structure \( {S}_{q}\left\lbrack {\ell, m;{2\nu }}\right\rbrack \) is a \( \left( {{2\nu }, M,{2j}, m}\right) \) code with \( j = m - \ell + 1 \) and \( M = \mathop{\prod }\limits_{{t = 0}}^{{l - 1}}\frac{{q}^{2\left( {\nu - t}\right) } - 1}{{q}^{m - t} - 1} \) . | Proof For any two distinct blocks \( U, W \in {S}_{q}\left\lbrack {\ell, m;{2\nu }}\right\rbrack \), since every \( \ell \) -dimensional totally isotropic subspace is contained in exactly one block of \( {S}_{q}\left\lbrack {\ell, m;{2\nu }}\right\rbrack \), we have \( \dim \left( {U \cap W}\right) \leq \ell - 1 \) . B... | Yes |
Theorem 5.2 A \( \\left( {{2\\nu }, M,{2j}, m}\\right) \) code \( \\mathbb{C} \) achieves the Wang-Xing-Safavi-Naini bound, i.e., \( M = \\mathop{\\prod }\\limits_{{t = 0}}^{{m - j}}\\frac{{q}^{2\\left( {\\nu - t}\\right) } - 1}{{q}^{m - t} - 1} \), if and only if \( \\mathbb{C} \) is a Steiner structure \( {S}_{q}\\le... | Proof \\( \\overset{\\hat{} }{B}y \\) Theorems 2.2 and 5.1, a Steiner structure \( {S}_{q}\\left\\lbrack {m - j + 1, m;{2\\nu }}\\right\\rbrack \) is a \( \\left( {{2\\nu }, M,{2j}, m}\\right) \) code achieving the Wang-Xing-Safavi-Naini bound. On the other hand, suppose that \( \\mathbb{C} \) is a \( \\left( {{2\\nu }... | Yes |
Lemma 6.1 Let \( U \in \mathcal{M}\left( {\ell ,0;{2\nu }}\right) \) . Then \[ \left| {{S}_{r}\left( U\right) }\right| = \mathop{\sum }\limits_{{k = 0}}^{\nu }\mathop{\sum }\limits_{{t = \lceil \left( {\ell + k - r}\right) /2\rceil }}^{{\min \{ \ell, k\} }}N\left( {t;\ell ,0;k,0;{2\nu }}\right) . \] | Proof \( \begin{aligned} \text{ For any }W \in {S}_{r}\left( U\right) \cap \mathcal{M}\left( {k,0;{2\nu }}\right) , & \text{ by }\left( 1\right) d\left( {U, W}\right) \leq r \\ & \text{ if and only if }\ell + k - 2\dim (U \cap \end{aligned} \) \( \left. W\right) \leq r \), which implies that \( \dim \left( {U \cap W}\r... | Yes |
Proposition 0.1 Let \( {\left\{ {P}_{t}\right\} }_{t \geq 0} \) be a strong Feller semigroup. Assume that \( {\mu }_{1} \) and \( {\mu }_{2} \) are two ergodic invariant probability measures of \( {\left\{ {P}_{t}\right\} }_{t \geq 0} \), then \( \operatorname{Supp}{\mu }_{1} \cap \operatorname{Supp}{\mu }_{2} = \varno... | Proof Since \( {\mu }_{1} \) and \( {\mu }_{2} \) are ergodic invariant probability measures of \( {\left\{ {P}_{t}\right\} }_{t \geq 0} \), then by \( \lbrack 3 \) , Proposition 3.2.5], there exist two measurable sets \( {A}_{1} \) and \( {A}_{2} \) such that\n\n\[ \n{\mu }_{1}\left( {A}_{1}\right) = 1,\;{\mu }_{2}\le... | Yes |
Lemma 1.1 Let \( u \) be a locally Lipschitz function on \( {\mathbb{R}}^{m} \). (1) If \( K \) is a compact subset of \( {\mathbb{R}}^{m} \), then \[ \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}{u}_{\varepsilon }\left( x\right) = u\left( x\right) ,\;\text{ uniformly on }K. \] | Proof (1) For any \( \varepsilon > 0 \) , \[ \left| {{J}_{\varepsilon } * u\left( x\right) - u\left( x\right) }\right| = \left| {{\int }_{{\mathbb{R}}^{m}}{J}_{\varepsilon }\left( {x - y}\right) \left\lbrack {u\left( y\right) - u\left( x\right) }\right\rbrack \mathrm{d}y}\right| \] \[ \leq \mathop{\sup }\limits_{{\left... | Yes |
Lemma 2.3 For any \( x, y \in {\mathbb{R}}^{m},\phi \in {\mathcal{B}}_{b}\left( {\mathbb{R}}^{m}\right) \) and \( \varepsilon > 0 \) ,\n\n\[ \left| {\mathbb{E}\phi \left( {X}_{t}^{x,\varepsilon }\right) - \mathbb{E}\phi \left( {X}_{t}^{y,\varepsilon }\right) }\right| \leq M\parallel \phi {\parallel }_{\infty }\left| {x... | Proof Let us prove the following so-called Bismut’s formula: for any \( \phi \in {C}^{2}\left( {\mathbb{R}}^{m}\right) \) and \( h \in {\mathbb{R}}^{m}, \n\n\[ {\nabla }_{h}\mathbb{E}\phi \left( {X}_{t \land {\tau }_{n}^{\varepsilon }}^{x,\varepsilon }\right) = \frac{1}{t}\mathbb{E}\left\lbrack {\phi \left( {X}_{t \lan... | No |
Lemma 2.5 Assume \( \left( {\mathrm{A}{1}^{\prime }}\right) \) ,(A2) and (A3) hold, then \( {\left\{ {P}_{t}\right\} }_{t > 0} \) is irreducible. | Proof Step 1: First, we assume \( \nu \left( {B}_{0}\right) < \infty \) . Let \( {\left\{ {\tau }_{i}\right\} }_{i \geq 1} \) be the interactive times of the Poisson random measure \( N \) . Then \( {\left\{ {Z}_{{\tau }_{i}}\right\} }_{i \geq 1} \) is a point process associated with the Poisson random measure \( N \) ... | Yes |
Theorem 2.1 Suppose the assumptions (1.2)-(1.5), and the equation (1.1) has a global solution. If there exists a positive function \( V \in {\mathcal{C}}^{2}\left( {\mathbb{R}}^{m}\right) \) such that\n\n\[ \mathop{\sup }\limits_{{\left| x\right| \geq R}}\mathcal{L}V\left( x\right) \rightarrow - \infty ,\;\text{ as }R ... | We omit the proof, since it is similar to the proof of [7, Theorem 3.7]. | No |
Corollary 2.1 If there are some \( c, D > 0 \) such that\n\n\[ \mathcal{L}V\left( x\right) \leq - {cV}\left( x\right) + D,\;V\left( x\right) \rightarrow \infty ,\]\n\nthen the semigroup \( {P}_{t} \) has an invariant probability measure \( \mu \) satisfying \( \mu \left( V\right) \leq \frac{D}{c}. \) | Proof By Itô formula, one has\n\n\[ \mathbb{E}V\left( {X}_{t \land {\tau }_{n}}^{x}\right) {\mathrm{e}}^{{ct} \land {\tau }_{n}} = V\left( x\right) + \mathbb{E}{\int }_{0}^{t \land {\tau }_{n}}\left\lbrack {\mathcal{L}V\left( {X}_{s}^{x}\right) {\mathrm{e}}^{cs} + {cV}\left( {X}_{s}^{x}\right) {\mathrm{e}}^{cs}}\right\... | Yes |
Example 4.1 Let \( b\left( x\right) ,\sigma \left( x\right) \) and \( c\left( {x, z}\right) \) in (0.1) satisfy (A1). Assume that there are positive constants \( {c}_{1},{c}_{2} \) and \( {c}_{3} \) satisfying\n\n\[ \langle b\left( x\right), x\rangle \leq {c}_{1}{\left| x\right| }^{2},\;\parallel \sigma \left( x\right)... | \[ \leq \left( {{c}_{1} + {c}_{2} + {c}_{3}}\right) {\left| x\right| }^{2} \]\n\n\[ = \left( {{c}_{1} + {c}_{2} + {c}_{3}}\right) V\left( x\right) \]\n\nIt is easy to see that (0.7) holds. Thus (0.1) has a unique global solution. Furthermore, if the non-degenerate condition (A3) holds, then the semigroup \( {P}_{t} \) ... | Yes |
Example 4.2 Let \( b\left( x\right) ,\sigma \left( x\right) \) and \( c\left( {x, z}\right) \) in (0.1) satisfy (A1). Assume\n\n\[ \left| {b\left( x\right) }\right| \leq \left( {1 + \left| x\right| \ln \left| x\right| }\right) ,\;\parallel \sigma \left( x\right) {\parallel }_{2}^{2} \leq \left( {1 + {\left| x\right| }^... | Note that\n\n\[ \ln \left( {1 + x}\right) - x \leq {x}^{2},\;x \in \left( {-1,\infty }\right) .\n\nThen we have\n\n\[ {\int }_{{B}_{0}}\left( {V\left( {x + c\left( {x, z}\right) }\right) - V\left( x\right) -\langle \nabla V\left( x\right), c\left( {x, z}\right) \rangle }\right) \nu \left( {\mathrm{d}z}\right) \n\n\[ = ... | Yes |
Example 4.3 Consider the system\n\n\[ \left\{ \begin{array}{l} \mathrm{d}{x}_{t} = \left\lbrack {{x}_{t} - {y}_{t} - {x}_{t}\left( {{x}_{t}^{2} + {y}_{t}^{2}}\right) }\right\rbrack \mathrm{d}t + \left( {a + {x}_{t}^{2} + {y}_{t}^{2}}\right) \mathrm{d}{W}_{1}\left( t\right) + {\int }_{{B}_{0}}{c}_{1}\left( {{x}_{t - }, ... | Set\n\n\[ V\left( {x, y}\right) = {x}^{2} + {y}^{2} \]\n\nThen\n\n\[ \mathcal{L}V\left( {x, y}\right) = 2\left( {{x}^{2} + {y}^{2}}\right) \left( {1 - \left( {{x}^{2} + {y}^{2}}\right) }\right) + 2{\left( a + {x}^{2} + {y}^{2}\right) }^{2} + {\int }_{{B}_{0}}{\left| {c}_{1}\left( x, z\right) + {c}_{2}\left( y, z\right)... | Yes |
Example 4.4 Consider the following system\n\n\[ \left\{ \begin{array}{l} \mathrm{d}{x}_{t} = {y}_{t}\mathrm{\;d}t + {\sigma }_{1}\mathrm{\;d}{W}_{1}\left( t\right) , \\ \mathrm{d}{y}_{t} = \left( {-{x}_{t} + \left( {1 - {x}_{t}^{2}}\right) {y}_{t}}\right) \mathrm{d}t + {\sigma }_{2}\mathrm{\;d}{W}_{1}\left( t\right) + ... | Let\n\n\[ V\left( {x, y}\right) = \frac{1}{2}{\left( y + \frac{{x}^{2}}{3} - x\right) }^{2} + \frac{{x}^{2}}{2} + \frac{{\sigma }_{2}}{4}. \]\n\nThen\n\n\[ \frac{\partial V\left( {x, y}\right) }{\partial x} = \left( {y + \frac{{x}^{2}}{3} - x}\right) \left( {\frac{2}{3}x - 1}\right) + x,\;\frac{\partial V\left( {x, y}\... | Yes |
Example 4.5 Consider the following stochastic Lorenz equation\n\n\[ \left\{ \begin{array}{l} \mathrm{d}{X}_{t} = \left( {-\sigma {X}_{t} - \sigma {Y}_{t}}\right) \mathrm{d}t + {\sigma }_{1}\mathrm{\;d}{W}_{1}\left( t\right) + {\int }_{{B}_{0}}{c}_{1}\left( {{X}_{t - },{Y}_{t - },{Z}_{t - };v}\right) {\widetilde{N}}_{1}... | where \( \sigma ,\beta \) and \( \mu \) are positive constants. Let\n\n\[ V\left( {x, y, z}\right) = {x}^{2} + {y}^{2} + {\left( z - \sigma - \mu \right) }^{2}, \]\n\nthen\n\n\[ \frac{\partial V}{\partial x} = {2x},\;\frac{\partial V}{\partial y} = {2y},\;\frac{\partial V}{\partial z} = 2\left( {z - \sigma - \mu }\righ... | Yes |
Theorem 1.1 \( {}^{\left\lbrack {10},{17}\right\rbrack } \) Let \( G = G\left\lbrack {X, Y}\right\rbrack \) be a bipartite graph with minimum degree \( \delta \geq t \) \( \left( {t \geq 1}\right) \) and \( m \) edges, where \( \left| X\right| = \left| Y\right| = n \geq {2t} \) . If\n\n\[ m \geq {n}^{2} - {tn} + {t}^{2... | Proof Suppose that \( G \) is a non-Hamiltonian bipartite graph with minimum degree \( \delta \) and degree sequence \( \left( {{d}_{1},{d}_{2},\cdots ,{d}_{2n}}\right) \), where \( {d}_{1} \leq {d}_{2} \leq \cdots \leq {d}_{2n} \) . By Lemma 1.2, there exists an integer \( k \leq \frac{n}{2} \) such that \( {d}_{k} \l... | Yes |
Theorem 1.2 Let \( G = G\left\lbrack {X, Y}\right\rbrack \) be a bipartite graph with minimum degree \( \delta \geq t\left( {t \geq 1}\right) \) , where \( \left| X\right| = \left| Y\right| = n \geq {2t} \) . If\n\n\[ \rho \left( G\right) \geq \sqrt{{n}^{2} - {tn} + {t}^{2}} \]\n\nthen \( G \) is Hamiltonian unless \( ... | Proof By Lemma 1.1, we have \( \sqrt{{n}^{2} - {tn} + {t}^{2}} \leq \rho \left( G\right) \leq \sqrt{m} \), where \( m \) is the number of edges in \( G \) . Then \( m \geq {n}^{2} - {tn} + {t}^{2} \) . By Theorem 1.1, the result follows. | Yes |
Theorem 2.2 Let \( G \) be a connected graph of order \( n \geq 4 \) . If\n\n\[ m \geq \left( \begin{matrix} n - 2 \\ 2 \end{matrix}\right) + 2 \]\n\nthen \( G \) is traceable unless \( G \cong \{ {K}_{1} \vee \left( {{K}_{n - 3} + 2{K}_{1}}\right) ,{K}_{2} \vee \left( {3{K}_{1} + {K}_{2}}\right) ,{K}_{4} \vee 6{K}_{1}... | Proof Suppose that \( G \) is a non-traceable graph and its degree sequence is \( \left( {{d}_{1},{d}_{2},\cdots ,{d}_{n}}\right) , \) where \( {d}_{1} \leq {d}_{2} \leq \cdots \leq {d}_{n} \) . By Lemma 2.4, there is an integer \( k < \frac{\left( n + 1\right) }{2} \) such that \( {d}_{k} \leq k - 1 \) and \( {d}_{n -... | Yes |
Lemma 1.5 For a graph \( G \), let \( u \) and \( v \) be two non-adjacent vertices of \( G \), and \( {G}^{\prime } = G + {uv}. \) Let \( \rho \left( G\right) \) and \( \rho \left( {G}^{\prime }\right) \) be the spectral radii of \( G \) and \( {G}^{\prime } \), respectively. Then \( \rho \left( {G}^{\prime }\right) \... | Proof Let \( \mathbf{x} \) and \( \mathbf{y} \) be the principal eigenvector of \( G \) and \( {G}^{\prime } \), respectively. By the wellknown Rayleigh-Ritz Theorem we have\n\n\[ \rho \left( G\right) = \mathop{\max }\limits_{{\parallel \mathbf{x}\parallel = 1}}{\mathbf{x}}^{\mathrm{T}}A\left( G\right) \mathbf{x} = 2\m... | Yes |
Theorem 1.2 For connected graphs \( G \) and \( H \), let \( \rho \left( G\right) ,\rho \left( H\right) \) and \( \rho \left( {GuvH}\right) \) be the spectral radii of graphs \( G, H \) and \( {GuvH} \), respectively. Then\n\n\[ \max \{ \rho \left( G\right) ,\rho \left( H\right) \} < \rho \left( {\operatorname{Guv}H}\r... | Proof Clearly, \( G \cup H \) is a proper subgraph of \( G{uvH} \), by Lemma 1.2 we have \( \rho \left( {G{uvH}}\right) > \) \( \rho \left( {G \cup H}\right) = \max \{ \rho \left( G\right) ,\rho \left( H\right) \} \) . On the other hand, since \( {GuvH} = G \cup H + {uv} \), it follows from Lemma 1.5 that \( \rho \left... | Yes |
For \( G = {K}_{6} \) and \( H = {K}_{4} \), let \( {\rho }_{n} \) be the spectral radius of \( {B}_{n}\left( {{K}_{6},{K}_{4}}\right) \) and \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\rho }_{n} = \rho \) . According to Theorem 2.2, \( \rho \) is the largest positive root of \( {\left( x + \sqrt{{x}^{2} - 4}\r... | Using MATLAB software we obtain that \( \rho \approx {5.0355} \) | Yes |
Theorem 2.3 Let \( {\rho }_{n} \) be the spectral radius of \( {B}_{n}\left( {G, H}\right) \) . Suppose that \( {\rho }_{n} \rightarrow \rho > 2 \) as \( n \rightarrow \infty \) . Then \( \rho \) is the largest positive solution of the equation\n\n\[ \n\frac{1}{2}{\left( x + \sqrt{{x}^{2} - 4}\right) }^{2} - \left( {x ... | Proof According to Lemma 2.2 and Theorem 2.2, we have\n\n\[ \n\frac{1}{2}{\left( x + \sqrt{{x}^{2} - 4}\right) }^{2}{\phi }_{G}\left( x\right) {\phi }_{H}\left( x\right) - \left( {x + \sqrt{{x}^{2} - 4}}\right) \left( {\mathop{\sum }\limits_{{k = 1}}^{{m}_{1}^{\prime }}\frac{{\alpha }_{k, u}^{2}}{x - {\mu }_{k}}{\phi }... | Yes |
Let \( {\rho }_{n} \) be the spectral radius of \( {B}_{n}\left( {{K}_{3},{K}_{5}}\right) \) and \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\rho }_{n} = \rho \) . It is easy to obtain that \( {K}_{3} \) has two distinct eigenvalues \( {\mu }_{1} = 2 \) and \( {\mu }_{2} = - 1 \) ; and for any vertex \( u \) of ... | \[ {15}{\left( x + \sqrt{{x}^{2} - 4}\right) }^{2} - 2\left( {x + \sqrt{{x}^{2} - 4}}\right) \left( {\frac{5}{x - 2} + \frac{10}{x + 1} + \frac{3}{x - 4} + \frac{12}{x + 1}}\right) \] \[ + 4\left( {\frac{1}{\left( {x - 2}\right) \left( {x - 4}\right) } + \frac{4}{\left( {x - 2}\right) \left( {x + 1}\right) } + \frac{2}... | Yes |
Theorem 1.2 \( {}^{\left\lbrack 1\right\rbrack } \) Let \( G \) be a graph of order \( n \) with vertex set \( V\left( G\right) = \left\{ {{v}_{1},{v}_{2},\cdots ,{v}_{n}}\right\} \) . Then for any \( N \subseteq V\left( G\right) \) with \( \left| N\right| = n - k + 1 \), we have\n\n\[ \n{\lambda }_{k}\left( G\right) \... | Proof Let \( N \subseteq V\left( G\right) \) with \( \left| N\right| = n - k + 1 \) and \( W = V\left( G\right) \smallsetminus N \) . Let \( B \) be the principal submatrix of \( {D}^{-\frac{1}{2}}\left( G\right) \mathcal{L}\left( G\right) {D}^{\frac{1}{2}}\left( G\right) = {D}^{-1}\left( G\right) L\left( G\right) \) f... | Yes |
Example 1.6 If \( R \) is a semiregular ring, \( S \) is local and let \( {}_{R}{M}_{S} \) be bimodule, then \( \left( \begin{matrix} R & M \\ 0 & S \end{matrix}\right) \) is a feckly semiregular ring. | Proof Let \( T = \left( \begin{matrix} R & M \\ 0 & S \end{matrix}\right) \) . For any \( \alpha = \left( \begin{matrix} a & m \\ b & b \end{matrix}\right) \in T \) . Since \( S \) is local, \( b \) or \( 1 - b \) is invertible. Assume that \( b \) is invertible. Note that \( a \) is a semiregular element in \( R \), s... | Yes |
Proposition 1.7 Let \( R \) be a feckly semiregular ring. Then the following results hold.\n\n(1) Every homomorphic image of \( R \) is feckly semiregular. | Proof (1) It is trivial by Lemma 1.1 (3). | No |
Proposition 1.9 Let \( R = \mathop{\prod }\limits_{{\alpha \in I}}{R}_{\alpha } \) . Then \( R \) is a feckly semiregular ring if and only if there exists \( {\alpha }_{0} \in I \), such that \( {R}_{{\alpha }_{0}} \) is a feckly semiregular ring and for each \( \alpha \in I - {\alpha }_{0},{R}_{\alpha } \) is semiregu... | Proof ( \( \Leftarrow \) ) Let \( x = \left( {x}_{\alpha }\right) \in R,\alpha \in I \) . By hypothesis, \( {x}_{{\alpha }_{0}} \) or \( {1}_{{R}_{{\alpha }_{0}}} - {x}_{{\alpha }_{0}} \) is semiregular in \( {R}_{{\alpha }_{0}} \) . If \( {x}_{{\alpha }_{0}} \) is semiregular in \( {R}_{{\alpha }_{0}} \), then \( x \)... | Yes |
Theorem 1.11 The following are equivalent for a ring \( R \) .\n\n(1) \( R \) is a feckly semiregular ring;\n\n(2) \( R/J\left( R\right) \) is a VNL-ring and \( R \) is an exchange ring;\n\n(3) \( R/J\left( R\right) \) is a VNL-ring and idempotents can be lifted modulo \( J\left( R\right) \) . | Proof \( \;\left( 2\right) \Leftrightarrow \left( 3\right) \) follows from \( \left\lbrack {{14},\text{Proposition 1.5}}\right\rbrack \) and the fact that VNL-rings are exchange rings.\n\n\( \left( 1\right) \Rightarrow \left( 3\right) \) . Clearly, \( R/J\left( R\right) \) is a VNL-ring. For any \( a \in R,{a}^{2} - a ... | Yes |
Lemma 1.15 If \( R \) is a feckly semiregular ring, then either \( {eRe} \) or \( \left( {1 - e}\right) R\left( {1 - e}\right) \) is a semiregular ring for any \( {e}^{2} = e \in R \) . | Proof If \( R \) is feckly semiregular, \( \bar{R} = R/J\left( R\right) \) is a VNL-ring by Theorem 1.11. Then either \( \bar{e}\bar{R}\bar{e} \cong \overline{eRe} \) or \( \overline{\left( 1 - e\right) }\bar{R}\overline{\left( 1 - e\right) } \cong \overline{\left( {1 - e}\right) \bar{R}\left( {1 - e}\right) } \) is a ... | Yes |
Proposition 1.17 The following are equivalent for an abelian ring \( R \) .\n\n(1) \( R \) is a feckly semiregular ring;\n\n(2) \( R \) is an exchange ring such that for every \( {e}^{2} = e \in R \), either \( {eRe} \) or \( \left( {1 - e}\right) R\left( {1 - e}\right) \) is semiregular. | Proof (1) \( \Rightarrow \) (2). It follows by Theorem 1.11 and Lemma 1.15.\n\n\( \left( 2\right) \Rightarrow \left( 1\right) \) . For any \( a \in R \), as \( R \) is an exchange ring, there exists \( {e}^{2} = e \in R \) such that \( e \in {aR} \) and \( 1 - e \in \left( {1 - a}\right) R \) . Now if \( {eRe} = {eR} \... | Yes |
Lemma 1.18 Let \( R \) be a ring in which idempotents can be lifted modulo \( J\left( R\right) \) . Then \( M\left( {R/{M}^{\prime }\left( R\right) }\right) = 0. \) | Proof For convenience, we write \( \bar{R} = R/J\left( R\right) \) . Then\n\n\[ \n{M}^{\prime }\left( R\right) /J\left( R\right) = \{ \overline{a} \in \overline{R}\;|\;\left( a\right) \;\text{is a semiregular ideal in}\;R\} \n\]\n\n\[ \n= \{ \overline{a} \in \overline{R}\;|\;\left( \overline{a}\right) \text{ is a regul... | Yes |
Theorem 1.19 The following hold for an abelian ring \( R \) .\n\n(1) If \( R \) is a feckly semiregular ring, then \( R/{M}^{\prime }\left( R\right) \) is a local ring. | Proof (1) Assume that \( R \) is an abelian, feckly semiregular ring. Then \( R/J\left( R\right) \) is an abelian VNL-ring, and hence \( \frac{R/J\left( R\right) }{M\left( {R/J\left( R\right) }\right) } \) is a local ring by [6, Lemma 2.7]. Note that\n\n\[ R/{M}^{\prime }\left( R\right) \cong \frac{R/J\left( R\right) }... | Yes |
Proposition 1.20 Let \( R \) be an abelian ring. Then the following are equivalent:\n\n(1) \( R \) is a feckly semiregular ring;\n\n(2) whenever \( {\left( S\right) }_{r} = R \) for some nonempty subset \( S \) of \( R \), at least one element in \( S \) is semiregular, where \( {\left( S\right) }_{r} \) is a right ide... | Proof (2) \( \Rightarrow \) (1). For any \( a \in R \), let \( S = \{ a,1 - a\} \) . Then \( {\left( S\right) }_{r} = R \), and hence either \( a \) or \( 1 - a \) is semiregular.\n\n\( \left( 1\right) \Rightarrow \left( 2\right) \) . If \( R \) is semiregular, we are done. Then one suppose that \( R \) is a feckly sem... | Yes |
Theorem 2.1 Let \( T = \left( \begin{matrix} R & M \\ S & S \end{matrix}\right) \) be a Morita context such that \( {MN} \subseteq J\left( R\right) \) and \( {NM} \subseteq \) \( J\left( S\right) \) . Then \( T \) is feckly semiregular if and only if one of \( R \) and \( S \) is semiregular and the other is feckly sem... | Proof By [17, Lemma 3.1], \( T/J\left( T\right) \cong R/J\left( R\right) \times S/J\left( S\right) \) . If \( T \) is feckly semiregular, then \( T/J\left( T\right) \) is a VNL-ring, and hence one of \( R/J\left( R\right) \) and \( S/J\left( S\right) \) is regular and the other is a VNL-ring by [16, Theorem 3.1]. Note ... | Yes |
Theorem 2.3 Let \( R \) be a ring. Then the following are equivalent:\n\n(1) \( R \) is a semiregular ring;\n\n(2) \( {M}_{n}\left( R\right) \) is a semiregular ring for every \( n > 1 \) ;\n\n(3) \( {M}_{n}\left( R\right) \) is a feckly semiregular ring for every \( n > 1 \) . | Proof (1) \( \Leftrightarrow \) (2) follows from [15, Corollary B.55]. (2) \( \Rightarrow \) (3) is trivial.\n\n\( \left( 3\right) \Rightarrow \left( 1\right) \) . Because \( {M}_{n}\left( R\right) \) is feckly semiregular, \( {M}_{n}\left( R\right) \) is an exchange ring, and then so is \( R \) . Thus idempotents can ... | Yes |
Proposition 2.5 The following are equivalent for a ring \( R \) .\n\n(1) \( R \) is a feckly semiregular ring;\n\n(2) \( R\left\lbrack \left\lbrack x\right\rbrack \right\rbrack \) is a feckly semiregular ring. | Proof (1) \( \Rightarrow \) (2). As \( R\left\lbrack \left\lbrack x\right\rbrack \right\rbrack /J\left( {R\left\lbrack \left\lbrack x\right\rbrack \right\rbrack }\right) = R\left\lbrack \left\lbrack x\right\rbrack \right\rbrack /J\left( R\right) \left\lbrack \left\lbrack x\right\rbrack \right\rbrack \cong R/J\left( R\r... | Yes |
Proposition 2.6 Let \( C \) be a subring of a ring \( D \) . Then the following are equivalent:\n\n(1) \( R\left\lbrack {D, C}\right\rbrack \) is a VNL-ring;\n\n(2) \( D \) is regular and \( C \) is a VNL-ring. | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) . For convenience, let \( S = R\left\lbrack {D, C}\right\rbrack \) . We construct a homomorphism \( f \) : \( S \rightarrow C \) given by \( f\left( {{a}_{1},\cdots ,{a}_{n}, b, b,\cdots }\right) = b \) . Thus \( C \) is a VNL-ring because it is a homomorphism ima... | Yes |
Corollary 2.7 Let \( C \) be a subring of a ring \( D \) . Then the following are equivalent:\n\n(1) \( R\left\lbrack {D, C}\right\rbrack \) is a feckly semiregular ring;\n\n(2) \( D \) is semiregular, \( \frac{C}{J\left( D\right) \cap J\left( C\right) } \) is a VNL-ring and idempotents in \( C \) can be lifted modulo ... | Proof By [3, Theorem 2.2], we have\n\n\[ \frac{R\left\lbrack {D, C}\right\rbrack }{J\left( {R\left\lbrack {D, C}\right\rbrack }\right) } = \frac{R\left\lbrack {D, C}\right\rbrack }{R\left\lbrack {J\left( D\right), J\left( D\right) \cap J\left( C\right) }\right\rbrack } \cong R\left\lbrack {\frac{D}{J\left( D\right) },\... | Yes |
Lemma 2.10 The following are equivalent for a ring \( R \) and a bimodule \( M \) :\n\n(1) Idempotents of \( R \) can be lifted modulo \( J\left( R\right) \) ;\n\n(2) Idempotents of \( R \propto M \) can be lifted modulo \( J\left( {R \propto M}\right) \) . | Proof \( \;\left( 1\right) \Rightarrow \left( 2\right) . \) For any \( {\left( x, m\right) }^{2} - \left( {x, m}\right) \in J\left( {R \propto M}\right) ,{x}^{2} - x \in J\left( R\right) \) . By hypothesis, there exists \( {e}^{2} = e \in R \) such that \( e - x \in J\left( R\right) \) . Thus, \( {\left( e,0\right) }^{... | Yes |
Proposition 2.11 Let \( R \) be a ring and \( M \) a bimodule over \( R \) . Then the following are equivalent:\n\n(1) \( R \) is feckly semiregular;\n\n(2) \( R \propto M \) is feckly semiregular. | Proof Note that \( \frac{R \propto M}{J\left( {R \propto M}\right) } \cong R/J\left( R\right) \), then the results follow from Theorem 1.11 and Lemma 2.10. | No |
Theorem 1.4 \( {}^{\left\lbrack 7,\text{ Theorem 3.1 }\right\rbrack } \) Let \( p \) be the smallest prime in \( \pi \left( G\right) \) and \( P \) a Sylow \( p \) -subgroup of \( G \) . Then \( G \) is \( p \) -nilpotent if every maximal subgroup of \( P \) is a partial CAP-subgroup of \( {N}_{G}\left( P\right) \) and... | Let \( p \) be the smallest prime in \( \pi \left( G\right) \) . Bearing Theorem 1.3 in mind, we can see that \( {N}_{G}\left( P\right) \) is \( p \) -nilpotent under the hypotheses of Theorem 1.4. Hence, Theorem 1.4 can be restated as follows: | Yes |
Theorem 0.1 Suppose that \( {T}_{a, b} \) is defined as in (0.1), and the kernel \( K \) satisfies (0.2). Then we have\n\n(i) If \( a > 1, b > 1 \), then \( {T}_{a, b} \) is bounded from \( {L}^{\frac{a + b}{a}} \) to itself;\n\n(ii) \( {T}_{a, b} \) is bounded from \( {L}^{2} \) to itself if \( a = b = 1 \) ;\n\n(iii)... | Part (i) of Theorem 0.1 was established in [7], while Part (ii) can be found in [6] (see also [8]). Pan provided a proof of Part (iii) in 1998 (see [2]). | Yes |
Lemma 1.1 Let \( \left( {X, S,\mu }\right) \) be defined as in the above definitions. If \( {p}_{n} \geq {p}_{n - 1} \geq \cdots \geq \) \( {p}_{1} \geq 1 \) and \( f \in {L}^{{p}_{n},{p}_{n - 1},\cdots ,{p}_{1}}\left( X\right) \), then \( f \in {L}^{{p}_{1},{p}_{2},\cdots ,{p}_{n}}\left( X\right) \) and there holds\n\... | For the proof of the lemma, readers can refer to [4]. | No |
Lemma 1.2 (Riesz-Thorin) Let \( \\left( {X,\\mu }\\right) \) and \( \\left( {Y,\\nu }\\right) \) be two measure spaces. Let \( T \) be a linear operator defined on the set of all simple functions on \( X \) and taking values in the set of measurable functions on \( Y \) . Let \( 1 \\leq {p}_{0},{p}_{1},{q}_{0},{q}_{1} ... | By density, \( T \) has a unique extension as a bounded operator from \( {L}^{p}\\left( {X,\\mu }\\right) \) into \( {L}^{q}\\left( {Y,\\nu }\\right) \) for all \( p \) and \( q \) as in (1.3). | Yes |
Theorem 2.3 The operator \( {T}_{a, b} \) defined as in (0.1) is bounded from \( {L}^{p}\left( \mathbb{R}\right) \) to \( {L}^{q}\left( \mathbb{R}\right) \) , where \( \frac{1}{p} = \theta + \frac{a\left( {1 - \theta }\right) }{a + b} \) and \( \frac{1}{q} = \frac{a\left( {1 - \theta }\right) }{a + b} \) hold for all \... | Proof From Theorem \( {0.1},{T}_{a, b} \) is bounded from \( {L}^{\frac{a + b}{a}}\left( \mathbb{R}\right) \) to itself. Noting that the kernel \( K \) satisfies \( \left| {K\left( {x, y}\right) }\right| \leq {A}_{00} \) for all \( x \neq y \), the operator is bounded from \( {L}^{1}\left( \mathbb{R}\right) \) to \( {L... | Yes |
Lemma 1.4 \( {}^{\left\lbrack 1\right\rbrack } \) Let \( p \in \lbrack 1,\infty ), f \in {L}^{p}\left( \mu \right) \) and \( \zeta \in \left( {1,\infty }\right) (\zeta > \left\lbrack \frac{\gamma \parallel f{\parallel }_{{L}^{p}\left( \mu \right) }}{\mu {\left( \mathcal{X}\right) }^{1/p}}\right\rbrack \) when \( \mu \l... | \[ \n{\int }_{\mathcal{X}}{\varphi }_{j}\mathrm{\;d}\mu \left( x\right) = {\int }_{6{B}_{j}}f\left( x\right) {\omega }_{j}\left( x\right) \mathrm{d}\mu \left( x\right) ,\;\mathop{\sum }\limits_{j}\left| {{\varphi }_{j}\left( x\right) }\right| \leq \zeta {\gamma }_{0}\n\]\n\nfor \( \mu \) -almost every \( x \in \mathcal... | Yes |
Theorem 0.3 For \( x \) near \( \partial \Omega \), choose \( y\left( x\right) \) so that \( \operatorname{dist}\left( {x, y}\right) = \operatorname{dist}\left( {x,\partial \Omega }\right) \) . Then if \( \frac{n + 1}{s} \) is an integer and \( \frac{n + 1}{s} + \frac{n - 1 - {2p}}{2} \geq 0 \), or if \( \frac{n + 1}{s... | \[ {\chi }_{p}\left( x\right) = {\left( 2\pi \right) }^{\frac{n - 1}{2}}\Gamma \left( \frac{{2p} - n + 1}{2}\right) \cdot G{\left( y\right) }^{\frac{n - p}{n + 1}}{\left( -v\left( x\right) \right) }^{\frac{n - 1 - {2p}}{2}} \] \[ + \mathop{\sum }\limits_{{i = 1}}^{\left\lbrack p - \frac{n}{2}\right\rbrack }{\varepsilon... | Yes |
Lemma 2.3 The derivatives of \( \varphi \) up to third order have finite continuous values on \( \partial \Omega \) except that \( {\varphi }_{nnn} \) is of order \( {d}^{-1} \) at most. | By (2.10), we have\n\n\[ \n{g}_{ij} = \frac{1}{2}\left( {-\frac{{\varphi }_{ij}}{\varphi } + \frac{{\varphi }_{i}{\varphi }_{j}}{2{\varphi }^{2}}}\right) = \frac{1}{{2p} + 1 - n}\frac{{\chi }_{ij}}{\chi } - \frac{{2p} + 2 - n}{{\left( 2p + 1 - n\right) }^{2}}\frac{{\chi }_{i}}{\chi }\frac{{\chi }_{j}}{\chi }. \n\]\n\n\... | No |
Lemma 3.1 The derivatives of \( \chi \) satisfy\n\n\[ \frac{1}{{\chi }^{2}}\sum {g}^{{i}_{1}{j}_{1}}{g}^{{i}_{2}{j}_{2}}\cdots {g}^{{i}_{k}{j}_{k}}{\chi }_{{i}_{1}{i}_{2}\cdots {i}_{k}}{\chi }_{{j}_{1}{j}_{2}\cdots {j}_{k}} = O\left( 1\right) ,\;k = 1,2,\cdots . \] | Proof There exist multi-indices \( \alpha = \left( {{\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{n}}\right) \) and \( \beta = \left( {{\beta }_{1},{\beta }_{2},\cdots ,{\beta }_{n}}\right) \) such that\n\n\[ {\chi }_{{i}_{1}{i}_{2}\cdots {i}_{k}} = {D}^{\alpha }\chi ,\;{\chi }_{{j}_{1}{j}_{2}\cdots {j}_{k}} = {D}^{\b... | Yes |
Lemma 3.4 For any multi-index \( \alpha = \left( {{\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{n}}\right) \) , \[ \frac{{D}^{\alpha }\left( {R}_{{i}_{1}{i}_{2}\cdots {i}_{s}}\right) }{\sqrt{{\lambda }_{1}^{{\alpha }_{1}}{\lambda }_{2}^{{\alpha }_{2}}\cdots {\lambda }_{n}^{{\alpha }_{n}} \cdot {\lambda }_{{i}_{1}}{\la... | Proof We proceed by introduction on \( s \) . For \( s = 4 \), it is obtained in Lemma 3.3. Now suppose that for any multi-index \( \alpha \) the estimates (3.8) hold for \( s \), we need to prove (3.8) for \( s + 1 \) . \[ {D}^{\alpha }\left( {R}_{{i}_{1}{i}_{2}\cdots {i}_{s}{i}_{0}}\right) = {D}^{\alpha }\left( {\fra... | Yes |
Lemma 1.1 Let \( \phi : M \rightarrow \left( {\widetilde{M},\widetilde{F}}\right) \) be a submanifold isometrically immersed in a Randers manifold. Suppose that the navigation data of \( \widetilde{F} \) is \( \left( {\widetilde{h},\widetilde{W}}\right) \). Then the navigation data of the induced metric \( F \) satisfi... | Proof The navigation data \( \left( {h, W}\right) \) of the induced Randers metric \( F = \alpha + \beta \) is determined by\n\n\[ {h}_{ij} = \left( {1 - \parallel \beta {\parallel }_{\alpha }^{2}}\right) \left( {{a}_{ij} - {b}_{i}{b}_{j}}\right) ,\;{W}^{i} = - \frac{{a}^{ij}{b}_{j}}{1 - \parallel \beta {\parallel }_{\... | Yes |
Lemma 1.2 Let \( \phi : M \rightarrow \left( {\widetilde{M},\widetilde{F}}\right) \) be a submanifold. Suppose that the navigation data of \( \widetilde{F} \) is \( \left( {\widetilde{h},\widetilde{W}}\right) \) . If \( {\widetilde{W}}^{ * } \) is Killing, then the induced 1-form \( {\widetilde{W}}^{ * } = {\phi }^{ * ... | Proof We take the local coordinate system at \( x \), then\n\n\[ {\bar{W}}_{i \mid j} = {\left( {\widetilde{W}}_{\alpha }{\phi }_{i}^{\alpha }\right) }_{\mid j} \]\n\n\[ = {\widetilde{W}}_{\alpha |\beta }{\phi }_{i}^{\alpha }{\phi }_{j}^{\beta } + {\widetilde{W}}_{\alpha }{\phi }_{i|j}^{\alpha } \]\n\n\[ = - {\widetild... | Yes |
Lemma 1.3 Let \( \phi : M \rightarrow \left( {\widetilde{M},\widetilde{F}}\right) \) be a submanifold. The navigation data of \( \widetilde{F} \) and \( F \) are \( \left( {\widetilde{h},\widetilde{W}}\right) \) and \( \left( {h, W}\right) \), respectively. Suppose that \( \mathbf{n} \) is a unit normal vector field of... | Proof Set \( \rho : T\widetilde{M} \rightarrow \mathrm{d}{\phi TM} \) to be an orthogonal projection with respect to \( \widetilde{h} \) . It follows from \( \langle \widetilde{W},\mathbf{n}{\rangle }_{\widetilde{h}} = 0 \) that \( \widetilde{W} = {\widetilde{W}}^{\mathrm{T}} \mathrel{\text{:=}} \rho \widetilde{W} \) .... | Yes |
For a Randers metric \( F = \frac{\sqrt{\lambda {h}^{2} + {W}_{0}^{2}}}{\lambda } - \frac{{W}_{0}}{\lambda } \), where \( {W}_{0} \mathrel{\text{:=}} {W}_{i}{y}^{i} \), it has constant flag curvature \( K = k \) if and only if \( h \) has constant sectional curvature \( \bar{K} = k + {c}^{2} \) and \( W \) satisfies\n\... | In our case, \( {W}_{0 \mid 0} = 0 \), so \( F \) has constant flag curvature \( K = k \) if and only if \( h \) has constant sectional curvature \( \bar{K} = k \) . It follows from (5)-(7) that\n\n\[ \n{h}_{ab} = \mathop{\sum }\limits_{i}{p}^{2}{f}_{a}^{i}{f}_{b}^{i},\;{h}_{an} = 0,\;{h}_{nn} = {\left( {p}^{\prime }\r... | Yes |
Theorem 0.1 On \( {\mathbb{B}}^{n}\left( \mu \right) \), a spherically symmetric Finsler metric \( F\left( {x, y}\right) = {u\phi }\left( {r, s}\right) \) has relatively isotropic Landsberg curvature if and only if | \[ \left\{ \begin{array}{l} 3{\phi }_{s}{P}_{ss} + \phi {P}_{sss} + \left( {{s\phi } + \left( {{r}^{2} - {s}^{2}}\right) {\phi }_{s}}\right) {Q}_{sss} = c\left( x\right) \left( {3{\phi }_{s}{\phi }_{ss} + \phi {\phi }_{sss}}\right) , \\ - {s\phi }{P}_{ss} + {\phi }_{s}\left( {P - s{P}_{s}}\right) + \left( {{s\phi } + \... | Yes |
Lemma 1.1 Let \( F\left( {x, y}\right) = \left| y\right| \phi \left( {\left| x\right| ,\frac{\langle x, y\rangle }{\left| y\right| }}\right) \) be a spherically symmetric Finsler metric on \( {\mathbb{B}}^{n}\left( \mu \right) \subseteq {\mathbb{R}}^{n} \) . Let \( u \mathrel{\text{:=}} \left| y\right|, r \mathrel{\tex... | A Finsler metric \( F = F\left( {x, y}\right) \) on an open domain \( U \subset {\mathbb{R}}^{n} \) is said to be projectively flat in \( U \) if all geodesics are straight lines. A simple computation shows that \( F \) is projectively flat if and only if \( Q = 0 \) . | Yes |
Proposition 2.1 For a spherically symmetric Finsler metric \( F = {u\phi }\left( {r, s}\right) \), the \( {C}_{ijk} \) of \( F \) is given by\n\n\[ \n{C}_{ijk} = \frac{1}{2u}\left\lbrack {{M}_{1}{x}^{i}{x}^{j}{x}^{k} + {M}_{2}\left( {{\delta }_{ij}{x}^{k} + {x}^{i}{\delta }_{jk} + {x}^{j}{\delta }_{ik}}\right) + {M}_{3... | Proof The metric tensor of a spherically symmetric Finsler metric \( F = {u\phi }\left( {r, s}\right) \) is given by (1.1). By a direct computation, we know\n\n\[ \n{\left\lbrack \phi \left( \phi - s{\phi }_{s}\right) \right\rbrack }_{{y}^{k}} = {s}_{{y}^{k}}\left( {\phi {\phi }_{s} - s{\phi }_{s}^{2} - {s\phi }{\phi }... | Yes |
Lemma 3.1 If a Finsler metric \( F = {u\phi }\left( {r, s}\right) \) has the relatively isotropic Landsberg curvature, then there exists functions \( {c}_{0}\left( r\right) ,{c}_{1}\left( r\right) ,{c}_{2}\left( r\right) ,{c}_{3}\left( r\right) \) so that its geodesic spray coefficients \( {G}^{i} \) satisfy\n\n\[ \n{G... | Proof Let \( \theta = P - s{P}_{s} - c\left( x\right) \left( {\phi - s{\phi }_{s}}\right) ,\eta = {s\phi } + \left( {{r}^{2} - {s}^{2}}\right) {\phi }_{s} \) . By a direct computation,\n\n\[ \n{\theta }_{s} = - s{P}_{ss} + c\left( x\right) s{\phi }_{ss} \n\]\n\n(3.1)\n\n\[ \n{\eta }_{s} = \phi - s{\phi }_{s} + \left( {... | Yes |
Lemma 4.1 If \( Q \) satisfies \( {Q}_{s} - s{Q}_{ss} = 0 \), the general solution of (4.2) is given by\n\n\[ \phi = \frac{{2c}\left( x\right) s + \sqrt{\left( {{4c}{\left( x\right) }^{2} + {\lambda A}}\right) {s}^{2} + \frac{1}{2}{\lambda C}}}{\lambda }, \]\n\nwhere \( A \) and \( C \) are functions of \( r \) on \( {... | Proof For a fixed \( r \) ,(4.2) is equivalent to the following equation\n\n\[ M\mathrm{\;d}\phi + N\mathrm{\;d}s = 0 \]\n\n(4.3)\n\nwhere \( M = A{s}^{2} + {2\phi c}\left( x\right) s + C \) and \( N = - {As\phi } - 2{\phi }^{2}c\left( x\right) \) . By a direct computation,\n\n\[ \frac{\partial M}{\partial s} = {2As} +... | Yes |
Theorem 4.1 Let \( F\left( {x, y}\right) = \left| y\right| \phi \left( {\left| x\right| ,\frac{\langle x, y\rangle }{\left| y\right| }}\right) \) be a spherically symmetric Finsler metric. Suppose that \( F \) is of relatively isotropic Landsberg on \( {\mathbb{B}}^{n}\left( \mu \right) \) and \( Q \) satisfies \( {Q}_... | Proof of Theorem 0.2 Suppose that \( F\left( {x, y}\right) = \left| y\right| \phi \left( {\left| x\right| ,\frac{\langle x, y\rangle }{\left| y\right| }}\right) \) is a projectively flat spherically symmetric Finsler metric. Then \( Q = 0 \) . In this case, we have \( {c}_{0} = {c}_{2} = {c}_{3} = 0 \) . Then\n\n\[ A =... | No |
Lemma 1.1 \( \; \) Let \( {A}^{\left( 1\right) }\left( {2n}\right) = \{ {2}^{{k}_{i}}i : i = 1,3,\cdots ,{2n} - 1\} \) where each \( {k}_{i} \) satisfies \( \frac{2n}{{3}^{{k}_{i} + 1}} < i \leq \) \( \frac{2n}{{3}^{{k}_{i}}} \), then \( {A}^{\left( 1\right) }\left( {2n}\right) \in \mathcal{A}\left( {2n}\right) \) . Mo... | Proof It is easy to show that \( {2}^{{k}_{i}}i \leq {2n} \) for each odd \( i \) . We now divide \( \{ 1,2,\cdots ,{2n}\} \) into the following \( k + 1 \) subsets:\n\n\[ \left( {\frac{2n}{{3}^{1}},\frac{2n}{{3}^{0}}}\right\rbrack ,\left( {\frac{2n}{{3}^{2}},\frac{2n}{{3}^{1}}}\right\rbrack ,\cdots ,\left( {\frac{2n}{... | Yes |
Theorem 1.2 For \( k \geq 2 \) and \( p = \frac{c}{n} \) with the constant \( c \) sufficiently large, w.h.p. \( u{m}_{k}\left( {G}_{n, p}\right) \) \( = \Omega \left( {\log n}\right) \) . | Proof By Claim 1.1, Claim 1.6 and Chebyshev’s inequality, Theorem 1.2 holds. | No |
Proposition 1.1 Let \( R \) be a ring and \( {e}^{2} = e \in R \) . If \( r \in {eRe} \) has the strong 2-sum property in \( {eRe} \), then \( r \) has the strong 2 -sum property in \( R \) . | Proof Since \( r \) has the strong 2-sum property in \( {eRe} \), there exist \( {u}_{1},{u}_{2} \in U\left( {eRe}\right) \) such that \( r = {u}_{1} + {u}_{2} \) with \( {u}_{1}{u}_{2} = {u}_{2}{u}_{1} \) . Let \( v = {u}_{1} - \left( {1 - e}\right) \) and \( w = {u}_{2} + \left( {1 - e}\right) \) . Then \( v \) and \... | Yes |
Theorem 1.1 Let \( e \) be the central idempotent of \( R \) . Then \( x \in R \) has the strong 2-sum property in \( R \) if and only if \( {ex} \) and \( \left( {1 - e}\right) x \) have the strong 2-sum property in \( {eR} \) and \( \left( {1 - e}\right) R \) , respectively. | Proof \( \;\left( \Rightarrow \right) \; \) Since \( \;x\; \) has the strong 2-sum property in \( R \), there exist \( {u}_{1},{u}_{2} \in U\left( R\right) \) such that \( x = {u}_{1} + {u}_{2} \) with \( {u}_{1}{u}_{2} = {u}_{2}{u}_{1} \) . Let \( v = e{u}_{1} \) and \( w = e{u}_{2} \) . Then \( v, w \in U\left( {eR}\... | Yes |
Theorem 1.2 Let \( R \) be a commutative local ring. Then \( e{\mathbb{T}}_{2}\left( R\right) e \) is a strong 2 -sum ring for any idempotent matrix \( e \) of \( {\mathbb{T}}_{2}\left( R\right) \) if and only if \( R \) is a strong 2 -sum ring. | Proof \( \left( \Rightarrow \right) \;{\mathbb{T}}_{2}\left( R\right) \) is a strong 2-sum ring if \( e = {I}_{2} \) . Let \( A = \left( {a}_{ij}\right) \in {\mathbb{T}}_{2}\left( R\right) \) . Note that \( \phi : {\mathbb{T}}_{2}\left( R\right) \rightarrow R \) with \( \phi \left( A\right) = {a}_{11} \) is an epimorph... | Yes |
Theorem 1.3 Let \( R \) be a commutative local ring. Then \( e{\mathbb{T}}_{3}\left( R\right) e \) is a strong 2 -sum ring for any idempotent matrix \( e \) of \( {\mathbb{T}}_{3}\left( R\right) \) if and only if \( R \) is a strong 2-sum ring. | Proof \( \;\left( \Rightarrow \right) \; \) The proof is similar to that of Theorem 1.2.\n\n\( \left( \Leftarrow \right) \) Let \( e \) be an idempotent of \( {\mathbb{T}}_{3}\left( R\right) \) . Then\n\n\[ e = \left( \begin{array}{lll} 1 & 0 & b \\ 0 & 1 & c \\ 0 & 0 & 0 \end{array}\right) \;\text{ or }\;\left( \begin... | Yes |
Theorem 1.5 Let \( R \) be a commutative local ring. Then \( R \) is a strong 2-sum ring if and only if so is \( \mathcal{L}\left( R\right) \) . | Proof ( \( \Leftarrow \) ) Let \( A = \left( {a}_{ij}\right) \in \mathcal{L}\left( R\right) \) . Note that \( \phi : \mathcal{L}\left( R\right) \rightarrow R \) with \( \phi \left( A\right) = {a}_{11} \) is an epimorphism of rings. Then \( R \) is a strong 2 -sum ring by [4, Lennna 2].\n\n\( \left( \Rightarrow \right) ... | Yes |
Theorem 1.6 Let \( R \) be a commutative local ring. Then the following statements are equivalent.\n\n(1) \( R \) is a strong 2-sum ring.\n\n(2) \( \mathcal{L}\left( R\right) \) is a strong 2-sum ring.\n\n(3) \( {\mathbb{T}}_{2}\left( R\right) \) is a strong 2-sum ring. | Proof (1) \( \Leftrightarrow \) (2) by Theorem 1.5. (1) \( \Rightarrow \) (3) by Theorem 1.4 as \( {\mathbb{T}}_{2}\left( R\right) \cong e\mathcal{L}\left( R\right) e \), where\n\n\[ e = \left( \begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right) \in \mathcal{L}\left( R\right) \]\n\n\( \left( 3\right)... | Yes |
Theorem 1.7 Let \( R \) be a commutative local ring with \( R/J\left( R\right) ≆ {\mathbb{Z}}_{2} \) . Then \( e\mathfrak{L}\left( R\right) e \) is a strong 2-sum ring for any idempotent matrix \( e \) of \( \mathfrak{L}\left( R\right) \) . | Proof The proof is similar to that of Theorem 1.4 and we just find all idempotents in \( \mathfrak{L}\left( R\right) \) . Let \( A = \left( {a}_{ij}\right) \in \mathfrak{L}\left( R\right) \) . If \( {A}^{2} = A \) . Then it is equivalent to saying that the equations \( {a}_{11}^{2} = {a}_{11},{a}_{22}^{2} = {a}_{22},{a... | Yes |
Theorem 1.9 Let \( R \) be a commutative local ring. Then the following statements are equivalent.\n\n(1) \( R \) is a strong 2-sum ring.\n\n(2) \( \mathfrak{L}\left( R\right) \) is a strong 2-sum ring.\n\n(3) \( {\mathbb{T}}_{2}\left( R\right) \) is a strong 2-sum ring. | From [3, Theorem 8], for a commutative local ring \( R,{\mathbb{M}}_{3}\left( R\right) \) is a strong 2-sum ring if and only if so is \( R \) . We prove that two subrings of \( {\mathbb{M}}_{3}\left( R\right) ,\mathcal{L}\left( R\right) \) and \( \mathfrak{L}\left( R\right) \) are strong 2-sum rings if and only if so i... | Yes |
Let \( {\mathbb{Z}}_{p} \) denote the ring of integers modulo \( p \) where \( p \) is a prime number. Then \( {\mathbb{Z}}_{p} \) is a finite field. Consider the ring \( {\mathbb{Z}}_{p}\left\lbrack x\right\rbrack /\left( {x}^{n + 1}\right) \) where \( \left( {x}^{n + 1}\right) \) is the ideal generated by \( {x}^{n +... | Now we show that \( {\mathbb{Z}}_{p}\left\lbrack u\right\rbrack = {\mathbb{Z}}_{p}\left\lbrack x\right\rbrack /\left( {x}^{n + 1}\right) \) is not a strongly nil clean ring when \( p \neq 2 \). Note that \( {\mathbb{Z}}_{p}\left\lbrack u\right\rbrack = {\mathbb{Z}}_{p}\left\lbrack x\right\rbrack /\left( {x}^{n + 1}\rig... | Yes |
Proposition 1.1 If \( R \) is a finite ring, then for any positive integer \( n \), the \( n \times n \) matrix ring \( {M}_{n}\left( R\right) \) is a strongly \( \pi \) nil clean ring. | Proof Suppose that \( R \) is a finite ring which contains \( m \) elements. Then for any positive integer \( n \), the \( n \times n \) matrix ring \( {M}_{n}\left( R\right) \) is also a finite ring which contains \( {m}^{{n}^{2}} \) elements. So every \( A \in {M}_{n}\left( R\right) \) is a periodic matrix. Therefore... | Yes |
Proposition 1.2 Every strongly \( \\pi \) nil clean ring is strongly clean. | Proof Suppose that \( R \) is a strongly \( \\pi \) nil clean ring. Then for any \( x \\in R \), there exist a periodic element \( p \\in R \) and a nilpotent element \( n \\in N\\left( R\\right) \) such that \( x = p + n \) and \( {pn} = {np} \). Let \( k \) and \( l\\left( {k > l}\\right) \) be two distinct positive ... | Yes |
Example 1.2 Let \( \mathbb{R} \) be the field of real numbers. Then \( \mathbb{R} \) is strongly clean. Now we show that \( \mathbb{R} \) is not strongly \( \pi \) nil clean. | Choose any \( r \in \mathbb{R} \) with \( r \neq 0 \) and \( r \neq 1 \) . If \( r \) is a strongly \( \pi \) nil clean element, then \( r \) must be a periodic element since \( N\left( \mathbb{R}\right) = \{ 0\} \) . This is impossible. | "No" |
Proposition 1.3 Let \( R \) be a \( p \) -ring and let \( a \in R \) . Then the following statements are equivalent:\n\n(1) \( a \) is a strongly \( \pi \) nil clean element;\n\n(2) \( {a}^{k} \) is a strongly \( \pi \) nil clean element for every positive integer \( k \) ;\n\n(3) \( {a}^{k} \) is a strongly \( \pi \) ... | Proof (1) \( \Rightarrow \) (2). Suppose that \( a \in R \) is a strongly \( \pi \) nil clean element. Then there exist a periodic element \( p \in R \) and a nilpotent element \( u \in N\left( R\right) \) such that \( a = p + u \) and \( {pu} = {up} \) . Then for any positive integer \( k \), we have\n\n\[ \n{a}^{k} =... | Yes |
(1) Every homomorphic image of a strongly \( \pi \) nil clean ring is strongly \( \pi \) nil clean; | Proof (1) is directly verified. | No |
Corollary 1.1 If \( a \in {R}_{1} \) and \( b \in {R}_{2} \) are strongly \( \pi \) nil clean in \( {R}_{1} \) and \( {R}_{2} \), respectively, then \( \left( {a, b}\right) \) is strongly \( \pi \) nil clean in \( {R}_{1} \oplus {R}_{2} \) . | Proof By the same way as the proof of Proposition 1.4, we complete the proof. | No |
Corollary 1.2 Let \( R \) be a ring and \( e \) a central idempotent in \( R \) . Then the following statements are equivalent:\n\n(1) \( R \) is strongly \( \pi \) nil clean;\n\n(2) Both \( {eRe} \) and \( \left( {1 - e}\right) R\left( {1 - e}\right) \) are strongly \( \pi \) nil clean. | Proof Since \( e \) is a central idempotent in \( R \), we have \( R = {eRe} \oplus \left( {1 - e}\right) R\left( {1 - e}\right) \) . Thus \( \left( 1\right) \Leftrightarrow \left( 2\right) \) follows from Proposition 1.4. | No |
Proposition 1.5 Let \( R \) be a ring. Then the following statements are equivalent:\n\n(1) \( R \) is strongly \( \pi \) nil clean;\n\n(2) Every factor ring of \( R \) is strongly \( \pi \) nil clean;\n\n(3) Every indecomposable factor ring of \( R \) is strongly \( \pi \) nil clean. | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) . It follows by Proposition 1.4.\n\n\( \left( 2\right) \Rightarrow \left( 3\right) \) . It is clear.\n\n\( \left( 3\right) \Rightarrow \left( 1\right) \) . Suppose that every indecomposable factor ring is strongly \( \pi \) nil clean. We show that \( R \) is stron... | Yes |
Now we show that \( {\mathbb{Z}}_{p}\left\lbrack x\right\rbrack \) is not strongly \( \pi \) nil clean. | Let \( f = {a}_{0} + {a}_{1}x + \cdots + {a}_{n}{x}^{n} \in {\mathbb{Z}}_{p}\left\lbrack x\right\rbrack \) . We first show that \( f \) is a periodic element if and only if \( {a}_{i} = 0 \) for all \( 1 \leq i \leq n \) . If \( {a}_{i} = 0 \) for all \( 1 \leq i \leq n \), then clearly, \( f = {a}_{0} \) is a periodic... | Yes |
Theorem 2.1 Let \( E = \operatorname{end}\left( {{}_{R}M}\right) \) be a \( p \) -ring, and let \( \alpha \in E \) . Then the following statements are equivalent:\n\n(1) \( \alpha \in E \) is strongly \( \pi \) nil clean;\n\n(2) There exist a periodic element \( p \in E \) and a positive integer \( k \in \mathbb{N} \) ... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) . Since \( \alpha \) is strongly \( \pi \) nil clean in \( E \), by definition, there exist a periodic element \( p \in E \) and a \( u \in N\left( E\right) \) such that \( \alpha = p + u \) and \( {pu} = {up} \) . Thus \( {p\alpha } = {\alpha p} \) . Since \( p \... | Yes |
Corollary 2.1 Let \( R \) be a \( p \) -ring and let \( \alpha \in R \) . Then the following statements are equivalent:\n\n(1) \( \alpha \in R \) is strongly \( \pi \) nil clean;\n\n(2) There exist a periodic element \( p \in R \) and a positive integer \( k \in \mathbb{N} \) such that \( {p}^{k} \) is an idempotent, \... | Proof By analogy with the proof of Corollary 2.2 in [5], we complete the proof. | No |
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