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Lemma 3.1 For a self-injective Nakayama algebra \( A \) with the quiver \( Q \) : bounded by \( {ra}{d}^{i}{KQ} = 0, i \geq 2 \) . (1) If \( n > i \), then \[ {C}_{A} = {\left( \begin{matrix} 1 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 1 & 1 \\ 1 & 1 & 0 & \cdots & 0 & 0 & 0 & \cdots & 1 & 1 \\ 1 & 1 & 1 & \cdots & 0 & 0... | Proof (1) The case of \( n > i \) . By the definition of the algebra, we have the following indecomposable projective representation \( P\left( i\right) \) according to the vertex \( i \) :  Then the corresponding Cart... | Yes |
Theorem 3.4 Let \( A \) be a self-injective Nakayama algebra and \( {C}_{A} \) the Cartan matrix of \( A \). Then the determinant \( \left| {C}_{A}\right| = \left\{ \begin{array}{ll} 0 & \left( {n, i}\right) \neq 1 \\ i & \left( {n, i}\right) = 1 \end{array}\right. \), where \( \left( {n, i}\right) \) is the greatest c... | Proof By Lemmas 3.1, 3.2 and 3.3. | No |
Corollary 3.5 Let \( A \) be a self-injective Nakayama algebra and \( {C}_{A} \) be the Cartan matrix. Then \( {C}_{A} \) is not invertible in \( {M}_{n}\left( Z\right) \) . | Proof By Theorem 3.4 and Lemmas 2.5 and 2.6. | No |
Corollary 3.6 Let \( \mathcal{S} \) be the set of isomorphism classes of Nakayama algebras \( A \) . Then there is a surjective map from \( \mathcal{S} \) to the set of integers \( \mathcal{N} = \{ n \mid n \geq 0\} \) via \( f : A \rightarrow \left| {C\left( A\right) }\right| \) . | Proof By Theorem 3.4 and [3, Corollary 7], one can get the desired result. | No |
Example 3.7 Let \( A \) be a self-injective Nakayama algebra with the quiver bounded by \( {\operatorname{rad}}^{i}{KQ} = 0, i \geq 2 \). We only show the case of \( i = 3, i = 4 \) and \( i = {11} \). | (1) The case of \( i = 3 \). Since \( \left( {3,8}\right) = 1 \), then one gets that \[ \left| {C}_{A}\right| = \left| \begin{array}{llllllll} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & ... | No |
Theorem 1.2 Assume that\n\n\[ \rho \left( B\right) < 1 \] | Then \( {T}_{\max } = \infty \) and \( {\left\{ {u}_{i}\right\} }_{i = 1,\ldots, n} \) are bounded in \( \Omega \times \left( {0,\infty }\right) \) . | No |
Lemma 2.1 Suppose that \( {\left\{ {u}_{i0}\right\} }_{i = 1,\ldots, n} \in {C}^{0}\left( \bar{\Omega }\right) \) are nonnegative. Then there exists \( {T}_{\max } \in (0,\infty \rbrack \) and a unique classical solution \( {u}_{1},{u}_{2},...,{u}_{n}, w \) of (1.1)-(1.3) which is nonnegative and belongs to \( \;{C}^{0... | Proof The local existence and regularity of the solution as well as the extensibility criterion (2.1) can be proved by a slight adaption of well-known methods. We thus may confine ourselves with an outline of the proof and refer the reader e.g. to Winkler \( \left( {2010}\right) \left\lbrack {13}\right\rbrack \), where... | Yes |
Lemma 1 Let \( \\omega \) be a regular majorant. There exists a small \( \\epsilon > 0 \), such that\n\n\[ \n\\omega \\left( {s\\delta }\\right) \\lesssim {s}^{\\epsilon }\\omega \\left( \\delta \\right) \n\]\n\nand\n\n\[ \n\\omega \\left( {s\\delta }\\right) \\lesssim {s}^{1 - \\epsilon }\\omega \\left( \\delta \\righ... | Proof Since \( \\frac{\\omega \\left( t\\right) }{t} \) is nonincreasing for \( t > 0 \), we have\n\n\[ \n{\\int }_{0}^{\\delta }\\frac{\\omega \\left( t\\right) }{t}\\mathrm{\\;d}t \\geq \\omega \\left( \\delta \\right) \n\]\n\nNote that \( \\omega \) satisfies\n\n\[ \n{\\int }_{0}^{\\delta }\\frac{\\omega \\left( t\\... | Yes |
Lemma 2 Let \( \omega \) be a regular majorant and \( f \in \mathcal{A} \) . Then \( f \in {\mathcal{A}}_{\omega } \) if and only if\n\n\[ \mathop{\sup }\limits_{{z \in \mathbb{D}}}\frac{1 - {\left| z\right| }^{2}}{\omega \left( {1 - {\left| z\right| }^{2}}\right) }\left| {{f}^{\prime }\left( z\right) }\right| < \infty... | Moreover, we have\n\n\[ \parallel f{\parallel }_{{\mathcal{A}}_{\omega }} \asymp \mathop{\sup }\limits_{{z \in \mathbb{D}}}\frac{1 - {\left| z\right| }^{2}}{\omega \left( {1 - {\left| z\right| }^{2}}\right) }\left| {{f}^{\prime }\left( z\right) }\right| + \mathop{\sup }\limits_{{z \in \mathbb{D}}}\left| {f\left( z\righ... | Yes |
Lemma 3 Let \( \omega \) be a regular majorant and \( \mathop{\limsup }\limits_{{t \rightarrow 0}}\frac{\omega \left( t\right) }{t} < \infty \) . Then\n\n\[ \n{F}_{a}\left( z\right) = {\int }_{0}^{z}\frac{\omega \left( {1 - {\left| a\right| }^{2}}\right) }{1 - \bar{a}s}\mathrm{\;d}s \in {\mathcal{A}}_{\omega }\n\]\n\nw... | Proof By Lemma 1, we know that there exists a small \( \epsilon > 0 \) such that\n\n\[ \n\omega \left( {s\delta }\right) \lesssim {s}^{1 - \epsilon }\omega \left( \delta \right)\n\]\n\nwhere \( s \geq 1 \) . Thus, combine with \( \omega \left( t\right) \) is increasing for \( t > 0 \), we get that\n\n\[ \n\mathop{\sup ... | Yes |
Theorem 2 Let \( \omega \) be a regular majorant and \( \mathop{\limsup }\limits_{{t \rightarrow 0}}\frac{\omega \left( t\right) }{t} < \infty \) . Suppose that \( g \in H\left( \mathbb{D}\right) \), then \( {T}_{g} \) is bounded on \( {\mathcal{A}}_{\omega } \) if and only if \( g \in {\mathcal{A}}_{\omega } \) . More... | Proof Suppose that \( g \in {\mathcal{A}}_{\omega } \) and \( f \in {\mathcal{A}}_{\omega } \) . Notice the fact that\n\n\[{\mathcal{A}}_{\omega } \subseteq \mathcal{A} \subseteq {H}^{\infty }\]\n\nJoining with Lemma 2, we have\n\n\[{\begin{Vmatrix}{T}_{g}f\end{Vmatrix}}_{{\mathcal{A}}_{\omega }} \asymp \mathop{\sup }\... | Yes |
Corollary 1 Let \( \omega \) be a regular majorant and \( \mathop{\limsup }\limits_{{t \rightarrow 0}}\frac{\omega \left( t\right) }{t} < \infty \) . Suppose that \( g \in H\left( \mathbb{D}\right) \), then \( {M}_{g} \) is bounded on \( {\mathcal{A}}_{\omega } \) if and only if \( g \in {\mathcal{A}}_{\omega } \) . | Proof Suppose \( {M}_{g} \) is bounded on \( {\mathcal{A}}_{\omega } \) . Consider the function \( {F}_{a} \) defined as in Lemma 3. We have\n\n\[ \mathop{\sup }\limits_{{z \in \mathbb{D}}}\left| {g\left( z\right) }\right| \lesssim {\begin{Vmatrix}{M}_{g}\end{Vmatrix}}_{{\mathcal{A}}_{\omega }} \]\n\nOn the other hand,... | Yes |
Corollary 1.2 If \( R \) is a 1-FC ring, then any Gorenstein projective left or right \( R \) -module is Ding projective. | We note that Corollary 1.2 has been proved by J. Gillespie in [17]. However, our proof is different from that in [17]. | Yes |
Lemma 2.2 Let \( R \) be a ring and \( M \) a strongly Gorenstein projective left \( R \) - module. If \( {\operatorname{Ext}}_{R}^{1}\left( {M, F}\right) = 0 \) for any flat left \( R \) -module \( F \), then \( M \) is Ding projective. | Proof Since \( M \) is strongly Gorenstein projective, there is an exact sequence of projective left \( R \) -modules  such that \( M \cong \operatorname{im}\left( f\right) \) . By assumption, \( {\operatorname{Ext}}_{... | Yes |
Lemma 2.3 The following are true for any ring \( R \) :\n\n(1) \( \mathcal{D}\mathcal{P}{\mathcal{P}}^{ \bot } = \mathcal{D}{\mathcal{P}}^{ \bot } \) . | Proof (1) It is clear that \( \mathcal{{DP}} \subseteq \mathcal{{DPP}} \) . Then we have \( \mathcal{{DP}}{\mathcal{P}}^{ \bot } \subseteq \mathcal{D}{\mathcal{P}}^{ \bot } \) . Next we prove that \( {\mathcal{{DP}}}^{ \bot } \subseteq {\mathcal{{DPP}}}^{ \bot } \) . Let \( K \in {\mathcal{{DP}}}^{ \bot } \) . Consider... | Yes |
Proposition 2.4 Let \( R \) be a right \( G \) -semihereditary ring and \( M \) a right \( R \) -module. Then the following are equivalent:\n\n(1) \( {\mathrm{{fd}}}_{R}M < \infty \) .\n\n(2) \( {\operatorname{fd}}_{R}M \leq 1 \) .\n\n(3) \( \mathrm{{FP}} - {\mathrm{{id}}}_{R}M < \infty \) .\n\n(4) \( \mathrm{{FP}} - {... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) . \) By \( \left\lbrack {{22}\text{, Theorem 1.1}}\right\rbrack, M \) has Gorenstein flat dimension at most 1. Applying [3, Theorem 2.2], \( {\mathrm{{fd}}}_{R}M = {\mathrm{{Gfd}}}_{R}M \leq 1 \) .\n\n(2) \( \Rightarrow \) (3). Since \( M \) is a right \( R \) -modul... | Yes |
Lemma 2.6 Let \( \left( {e, e}\right) \) be an idempotent of a cross*- regular semigroup \( S \) . Then \( {G}_{\left( e, e\right) } \subseteq {H}_{\left( e, e\right) }^{ * } \), moreover, if \( \left( {u, v}\right) \in {H}_{\left( e, e\right) }^{ * } \) and \( p \) is the smallest integer such that \( {\left( u, v\rig... | Proof Let \( \left( {a, b}\right) \in {G}_{\left( e, e\right) } \) and suppose that \( \left( {s, t}\right) \) is the inverse element of \( \left( {a, b}\right) . \) Then, we deduce that \( \left( {s, t}\right) \left( {a, b}\right) = \left( {e, e}\right) = \left( {a, b}\right) \left( {s, t}\right) . \) This implies \( ... | Yes |
Lemma 2.7 Let \( \left( {c, d}\right) \) be an element of a cross*-regular semigroup \( S \) such that \( {\left( c, d\right) }^{n} \) belongs to a subgroup \( H \) of \( S \) for some positive integer \( n \) . If \( \left( {e, e}\right) \) is the identity of \( H \), then\n\n(1) \( \left( {e, e}\right) \left( {c, d}\... | Proof (1) Let \( \left( {r, l}\right) \) be an inverse of \( {\left( c, d\right) }^{n} \) in \( H \) . Then,\n\n\[ \left( {e, e}\right) \left( {c, d}\right) = \left( {r, l}\right) {\left( c, d}\right) }^{n + 1} = \left( {r, l}\right) \left( {c, d}\right) {\left( c, d\right) }^{n} = \left( {\left( {r, l}\right) \left( {... | Yes |
Lemma 2.8 Let \( {D}_{2n} \) and \( {D}_{2m} \) be dihedral groups. If \( {D}_{2n} \times {D}_{2m} \) is generated by two elements, then both \( n \) and \( m \) are odd \( \left( {n, m > 2}\right) \) . | Proof Let \( {G}_{1} \) and \( {G}_{2} \) be groups. If \( G = {G}_{1} \times {G}_{2} \) is finite, then \( G \) is generated by two elements if and only if each of the groups\n\n\[ \n{G}_{1},{G}_{2},{G}_{1}/N\left( {G}_{1}\right) \times {G}_{2}/N\left( {G}_{2}\right) \n\]\n\nis generated by two elements, where \( N\le... | Yes |
Theorem 2.9 Let \( S \) be a cross*-regular semigroup and \( {D}_{2n} \) a dihedral group. There exist \( \left( {a, b}\right) \in S \) and \( m \in {\mathbb{Z}}^{ + } \) such that \( {\left( a, b\right) }^{m} \) belongs to \( {D}_{2n} \) if and only if \( S \) is a strongly completely cross*-regular semigroup. | Proof Necessity. Let \( \left( {u, s}\right) \) and \( \left( {v, t}\right) \) be in \( {D}_{2n} \) . Then there exist \( \left( {a, b}\right) \in S \) and \( m \in {\mathbb{Z}}^{ + } \) such that \( {\left( a, b\right) }^{m} \) belongs to \( {D}_{2n} \), and the elements of \( {D}_{2n} \) are of the form of \( {\left(... | Yes |
Theorem 2.10 Let \( {D}_{2n} \) and \( {D}_{2m} \) be dihedral groups. Suppose both \( n \) and \( m \) are odd \( \left( {n, m > 2}\right) \) . If \( {D}_{2n} \times {D}_{2m} \) is generated by two elements, then \( {D}_{2n} \times {D}_{2m} \cong S \) , where \( S \) is a cross*-regular semigroup. | Proof In accordance with Lemmas 2.7, 2.8, and Theorem 2.9, the conclusion is obvious. | No |
Theorem 1.1 (Sun [18]) Let \( a, b, c, d, e, f \) be integers with \( a > b \geq 0, c > d \geq 0 \) , \( e > f \geq 0,\;a \equiv b\;\left( {\text{mod}\;2}\right) ,\;c \equiv d\;\left( {\text{mod}\;2}\right) ,\;e \equiv f\;\left( {\text{mod}\;2}\right) ,\;a \geq c \geq e \geq 2,\; \) and \( \;b \geq d \) if \( a = c \),... | In the proof of Theorem 1.1 given in [18], the author first showed that \( {ce} \geq {1000} \) and \( e \leq c \leq a \leq {218} \), or \( {ce} < {1000} \) and \( e \leq c \leq a \leq {185} \), and then used a computer to check the finite remaining cases. | Yes |
Lemma 2.2 We have\n\n\[ \left\{ {\frac{x\left( {{5x} + 1}\right) }{2} + \frac{y\left( {{5y} + 3}\right) }{2} : x, y \in \mathbb{Z}}\right\} \]\n\n(2.1)\n\n\[ = \left\{ {{T}_{x} + {T}_{y} : x, y \in \mathbb{Z}}\right\} = \left\{ {{x}^{2} + 2{T}_{y} : x, y \in \mathbb{Z}}\right\} . \] | Remark 2.2 The second equality in (2.1) was first observed by Euler (cf. [3, p.11]). The first equality in (2.1) is (1.10) of Sun [18]. | No |
Among those tuples \( \left( {a, b, c, d, e, f}\right) \) with \( a = 5 \) listed in the Appendix, only the following 10 tuples have not yet been proved to be universal over \( \mathbb{Z} \): \[ \left( {5,1,2,0,2,0}\right) ,\left( {5,1,4,0,2,0}\right) ,\left( {5,1,4,0,3,1}\right) ,\left( {5,1,5,1,2,0}\right) , \] \[ \l... | Let \( n \in \mathbb{N} \) and \( r \in \{ 1,3\} . \) If \( {40n} + {r}^{2} = 5{x}^{2} + 5{y}^{2} + {z}^{2} \) with \( x, y, z \in \mathbb{Z} \) and \( 2 \nmid z, \) then \( {z}^{2} = {\left( {10}w + r\right) }^{2} \) for some \( w \in \mathbb{Z}, x, y \) and \( x/2 - y/2 \) are even, hence \[ {40n} + {r}^{2} = {20}{\l... | Yes |
Corollary 1.1. (i) For any prime \( p \equiv 1\\left( {\\;\\operatorname{mod}\\;4}\\right) \), we have \( {S}_{p}\\left( {-c}\\right) = {S}_{p}\\left( c}\\right) \) for all \( c \\in \\mathbb{Z} \), in particular \( {S}_{p}\\left( {-1}\\right) = {S}_{p}\\left( 1}\\right) = 0 \) . | Proof. (i) We now prove the first part. Let \( c \\in \\mathbb{Z} \). If \( p \\mid c \), then \( {S}_{p}\\left( {-c}\\right) = {S}_{p}\\left( 0}\\right) = {S}_{p}\\left( c}\\right) \).\n\nNow we assume \( p \\nmid c \). If \( k \) is odd, then \( \\left( {p - 1}\\right) /\\left( {k, p - 1}\\right) \\equiv 0\\left( {\\... | Yes |
Corollary 1.1. There is a circular permutation \( \left( {{q}_{1},\ldots ,{q}_{n}}\right) \) of the first \( n > 2 \) primes \( {p}_{1},\ldots ,{p}_{n} \) with \( {q}_{1} = {p}_{1} = 2 \) and \( {q}_{n} = {p}_{n} \) such that the \( n \) distances \[ \left| {{q}_{1} - {q}_{2}}\right| ,\left| {{q}_{2} - {q}_{3}}\right| ... | Proof. By Theorem 1.1, there is a permutation \( \left( {-{q}_{n}, - {q}_{n - 1},\ldots , - {q}_{2}}\right) \) of \( - {p}_{n}, - {p}_{n - 1},\ldots , - {p}_{2} \) with \( {q}_{n} = {p}_{n} \) such that \( \left| {-{q}_{n} + {q}_{n - 1}}\right| ,\ldots ,\left| {-{q}_{3} + {q}_{2}}\right| \) are pairwise distinct. Set \... | Yes |
Lemma 2.1. Let \( q \) be an odd prime power and set \( S = \{ {a}^{2} : a \in {\mathbb{F}}_{q} \smallsetminus \{ 0\} \} \) .\n\n(i) The field \( {\mathbb{F}}_{q} \) has a primitive element \( g \) with \( {g}^{2} - 1 \in S \) if and only if \( q \notin \) \( \{ 3,5,9,{25}\} \) .\n\n(ii) The field \( {\mathbb{F}}_{q} \... | Proof. For an odd prime \( p \) let \( {G}_{p} \) be the set of those integers \( g \in \{ \pm 1,\ldots , \pm \left( {p - 1}\right) /2\} \) which are primitive roots modulo \( p \) . Then\n\n\[ \n{G}_{3} = \{ - 1\} ,\;{G}_{5} = \{ \pm 1\} ,\;{G}_{7} = \{ - 2,3\} ,\;{G}_{13} = \{ \pm 2, \pm 6\} \n\] \n\nand \n\n\[ \n{G}... | No |
Theorem 3.1 Let \( f : C\\left( {0,1}\\right) \\rightarrow \\mathbb{C} \) be an L-Lipschitzian mapping on the circle \( C\\left( {0,1}\\right) \). Then we have the inequality\n\n\[ \n\\left| {\\left( {t - a}\\right) \\left\\lbrack {f\\left( {e}^{ia}\\right) + f\\left( {e}^{ib}\\right) }\\right\\rbrack + \\left( {a + b ... | Proof It is known that if \( p : \\left\\lbrack {c, d}\\right\\rbrack \\rightarrow \\mathbb{C} \) is a Riemann integrable function and \( v : \\left\\lbrack {c, d}\\right\\rbrack \\rightarrow \\mathbb{C} \) is Lipschitzian with the constant \( L > 0 \), then the Riemann-Stieltjes integral \( {\\int }_{c}^{d}p\\left( t\... | Yes |
Theorem 2.1 If the parameters in the original system (1.1) satisfy the conditions: \( {\left( \mathbf{H}\right) }_{1} : - 2\sqrt{{N}_{1} - {n}_{1}} < M < 0 \) . Then for each \( l = 0,1,2,\cdots \) ,(2.3) possesses a pair of which implies that | \[ {\left( \operatorname{Re}\lambda \right) }^{\prime }\left( {\tau }_{j}\right) = \operatorname{Re}\left( {{\lambda }^{\prime }\left( {\tau }_{j}\right) }\right) = : {\left. \operatorname{Re}\frac{d\lambda }{d\tau }\right| }_{\tau = {\tau }_{j}} \] \[ = {\left. \frac{1}{{\left| \{ \frac{d\lambda }{d\tau }{\} }^{-1}\ri... | Yes |
Lemma 3.1 Let \( \left( {\mathcal{H}, m, u,\Delta ,\epsilon }\right) \) be a graded connected bialgebra, and \( {\left\{ {b}_{\gamma }\right\} }_{\gamma \in \Gamma } \) an algebraic basis of \( \mathcal{H} \) . If there is a homomorphism \( \alpha : \mathcal{H} \rightarrow \mathcal{H} \) satisfing the following conditi... | Proof Since \( {\left\{ {b}_{\gamma }\right\} }_{\gamma \in \Gamma } \) is an algebraic basis of \( \mathcal{H} \), condition (a) implies for al- 1 \( {h}_{1},{h}_{2} \in \mathcal{H},\alpha \circ m\left( {{h}_{1} \otimes {h}_{2}}\right) = m\left( {\alpha \left( {h}_{1}\right) \otimes \alpha \left( {h}_{2}\right) }\righ... | Yes |
For \( \left( {\mathcal{H}, m}\right) \) an associative algebra and \( \alpha : \mathcal{H} \mapsto \mathcal{H} \) a homomorphism, if \( \alpha \) satisfies \( \alpha \circ m = m \circ \left( {\alpha \otimes \alpha }\right) \), then \( {m}_{\alpha } \mathrel{\text{:=}} \alpha \circ m \) satisfies that\n\n\[ \n{m}_{\alp... | Proof Using the definition of \( {m}_{\alpha } \) and the associativity property of multiplication \( m \), we have\n\n\[ \n{LHS} = \alpha \circ m\left( {\alpha \otimes \left( {\alpha \circ m}\right) }\right) \n\]\n\n\[ \n= \alpha \circ m\left( {\alpha \otimes \alpha }\right) \left( {{id} \otimes m}\right) ) \n\]\n\n\[... | Yes |
Lemma 3.3 For \( \left( {\mathcal{H}, m, u,\Delta ,\epsilon }\right) \) a bialgebra and \( \alpha : \mathcal{H} \mapsto \mathcal{H} \) an isomorphism satisfying \( \alpha \circ m = m \circ \left( {\alpha \otimes \alpha }\right) \), then \( \left( {\mathcal{H},{m}_{\alpha },\alpha, u}\right) \) is a unital Hom-associati... | Proof Since \( {m}_{\alpha }\left( {1 \otimes h}\right) = \alpha \left( h\right) \) and the isomorphism \( \sim \) is given by \( \mathcal{H}\overset{\alpha }{ \rightleftarrows }\mathcal{H}\overset{Id}{ \rightleftarrows } \) \( \mathbb{K} \otimes \mathcal{H} \), it is straightforward to prove the following diagram comm... | No |
For \( \left( {\mathcal{H},\Delta }\right) \) a coalgebra and \( \alpha : \mathcal{H} \mapsto \mathcal{H} \) a homomorphism, if \( \alpha \) satisfies \( \Delta \circ \alpha = \left( {\alpha \otimes \alpha }\right) \circ \Delta \), then \( {\Delta }_{\alpha } \mathrel{\text{:=}} \Delta \circ \alpha \) satisfies that\n\... | Proof From the definition of \( {\Delta }_{\alpha } \) and the condition that \( \alpha \) satisfies, we have\n\n\[ {LHS} = \left( {\alpha \otimes \left( {\alpha \otimes \alpha }\right) }\right) \circ \left( {{id} \otimes \Delta }\right) \circ \left( {\Delta \circ \alpha }\right) \]\n\n\[ = \left( {\left( {\alpha \otim... | Yes |
Lemma 3.5 For \( \left( {\mathcal{H}, m, u,\Delta ,\epsilon }\right) \) a graded connected bialgebra and \( \alpha : \mathcal{H} \mapsto \mathcal{H} \) an isomorphism satisfying:\na) \( \Delta \circ \alpha = \left( {\alpha \otimes \alpha }\right) \circ \Delta \) ,\nb) for all \( h \in {\mathcal{H}}_{ + } \mathrel{\text... | Proof By Lemma 3.4, it is sufficient to prove the following diagram commutes:\n\n\n\n(3.3)\n\nFor \( {1}_{\mathcal{H}} \in {\mathcal{H}}_{0} \), since \( \alpha \left( {1}_{\mathcal{H}}\right) = \alpha \left( {m\left( ... | Yes |
Lemma 2. Let \( \mathbf{s} = \left( {{s}_{1},\ldots ,{s}_{q}}\right) ,\Phi \left( \mathbf{s}\right) = \left( {{t}_{1},\ldots ,{t}_{q}}\right) \) and \( w = {s}_{1}\ldots {s}_{q} \) . Let \( {\mathrm{T}}_{w} \) be the set of elements of \( \dot{\mathrm{T}} \) such that \( \eta \left( {w, t}\right) = - 1 \) . Then \( \ma... | Clearly \( {\mathrm{T}}_{w} \subset \left\{ {{t}_{1},\ldots ,{t}_{q}}\right\} \) . Taking \( \mathbf{s} \) to be reduced, it follows that \( \operatorname{Card}\left( {\mathrm{T}}_{w}\right) \leq l\left( w\right) \) . Moreover, if the \( {t}_{i} \) are distinct, then \( n\left( {\mathbf{s}, t}\right) \) is equal to \( ... | Yes |
Lemma 3. Let \( w \in \mathrm{W} \) and \( s \in \mathrm{S} \) be such that \( l\left( {sw}\right) \leq l\left( w\right) \) . For any sequence \( s = \left( {{s}_{1},\ldots ,{s}_{q}}\right) \; \) of elements of \( \;\mathrm{S}\; \) with \( \;w = {s}_{1}\ldots {s}_{q},\; \) there \( \; \) exists an integer \( \;j \) suc... | Let \( p \) be the length of \( w \) and \( {w}^{\prime } = {sw} \) . By the Remark of no. 3, \( l\left( {w}^{\prime }\right) \equiv \) \( l\left( w\right) + 1{\;\operatorname{mod.}\;2} \) . The hypothesis \( l\left( {w}^{\prime }\right) \leq l\left( w\right) \) and the relation\n\n\[ \left| {l\left( w\right) - l\left(... | Yes |
Lemma 4. Let \( w \in \mathrm{W} \) have length \( q \geq 1 \), let \( \mathrm{D} \) be the set of reduced decompositions of \( w \), and let \( \mathrm{F} \) be a map from \( \mathrm{D} \) to a set \( \mathrm{E} \) . Assume that \( \mathrm{F}\left( \mathbf{s}\right) = \mathrm{F}\left( {\mathbf{s}}^{\prime }\right) \) ... | A) Let \( \mathbf{s},{\mathbf{s}}^{\prime } \in \mathrm{D} \) and put \( \mathbf{t} = \left( {{s}_{1}^{\prime },{s}_{1},\ldots ,{s}_{q - 1}}\right) \) . We are going to show that if \( \mathrm{F}\left( \mathbf{s}\right) \neq \mathrm{F}\left( {\mathbf{s}}^{\prime }\right) \) then \( \mathbf{t} \in \mathrm{D} \) and \( \... | Yes |
Lemma 1. Let \( {s}_{1},\ldots ,{s}_{q} \in \mathrm{S} \) and let \( w \in \mathrm{W} \) . We have\n\n\[ \n\mathrm{C}\left( {{s}_{1}\ldots {s}_{q}}\right) .\mathrm{C}\left( w\right) \subset \mathop{\bigcup }\limits_{\left( {i}_{1},\ldots ,{i}_{p}\right) }\mathrm{C}\left( {{s}_{{i}_{1}}\ldots {s}_{{i}_{p}}w}\right) ,\n\... | We argue by induction on \( q \), the case \( q = 0 \) being trivial. If \( q \geq 1 \), we have \( \mathrm{C}\left( {{s}_{1}\ldots {s}_{q}}\right) .\mathrm{C}\left( w\right) \subset \mathrm{C}\left( {s}_{1}\right) .\mathrm{C}\left( {{s}_{2}\ldots {s}_{q}}\right) .\mathrm{C}\left( w\right) \) . By the induction hypothe... | Yes |
Lemma 2. Let \( \mathrm{H} \) be a normal subgroup of \( \mathrm{G} \). There exists a subset \( \mathrm{X} \) of \( \mathrm{S} \) such that \( \mathrm{{BH}} = {\mathrm{G}}_{\mathrm{X}} \) and such that every element of \( \mathrm{X} \) commutes with every element of \( \mathrm{S} - \mathrm{X} \). | Since \( {BH} \) is a subgroup of \( G \) containing \( B \), there exists a unique subset \( X \) of \( \mathrm{S} \) such that \( \mathrm{{BH}} = {\mathrm{G}}_{\mathrm{X}} \) (Th. 3).\n\nLet \( {s}_{1} \in \mathrm{X} \) and \( {s}_{2} \in \mathrm{S} - \mathrm{X} \) ; let \( {n}_{1} \) and \( {n}_{2} \) be representat... | Yes |
Lemma 1. Any two non-parallel hyperplanes have non-empty intersection. | Let \( \mathrm{H} \) and \( {\mathrm{H}}^{\prime } \) be two non-parallel hyperplanes, \( a \in \mathrm{H} \) and \( {a}^{\prime } \in {\mathrm{H}}^{\prime } \) ; there exist two hyperplanes \( \mathrm{M} \) and \( {\mathrm{M}}^{\prime } \) of the vector space \( \mathrm{T} \) such that \( \mathrm{H} = \mathrm{M} + a \... | Yes |
Lemma 2. Let \( \mathrm{H} \) and \( {\mathrm{H}}^{\prime } \) be two distinct hyperplanes of \( \mathrm{E} \), and \( f,{f}^{\prime } \) two affine functions on \( \mathrm{E} \) such that \( \mathrm{H} \) (resp. \( {\mathrm{H}}^{\prime } \) ) consists of the points a in \( \mathrm{E} \) such that \( f\left( a\right) =... | The lemma being trivial when \( \mathrm{L} = \mathrm{H} \), we assume that there exists a point \( a \) in \( \mathrm{L} \) with \( a \notin \mathrm{H} \) . Put \( \lambda = {f}^{\prime }\left( a\right) ,{\lambda }^{\prime } = - f\left( a\right) \) and\n\n\[ g = \lambda \cdot f + {\lambda }^{\prime } \cdot {f}^{\prime ... | Yes |
Lemma 1. The set of hyperplanes \( \mathfrak{H} \) is locally finite. | Indeed, let \( \mathrm{K} \) be a compact subset of \( \mathrm{E} \) . If a hyperplane \( \mathrm{H} \in \mathfrak{H} \) meets \( \mathrm{K} \) , the set \( {s}_{\mathrm{H}}\left( \mathrm{K}\right) \) also meets \( \mathrm{K} \), since every point of \( \mathrm{K} \cap \mathrm{H} \) is fixed by \( {s}_{\mathrm{H}} \) .... | Yes |
Lemma 2. Let \( \mathrm{C} \) be a chamber.\n\n(i) For any \( x \in \mathrm{E} \), there exists an element \( w \in \mathrm{W} \) such that \( w\left( x\right) \in \overline{\mathrm{C}} \) . | (i) Let \( x \in \mathrm{E} \) and let \( \mathrm{J} \) be the orbit of \( x \) under the group \( {\mathrm{W}}_{\mathfrak{M}} \) . It suffices to prove that \( \mathrm{J} \) meets \( \overline{\mathrm{C}} \) .\n\nLet \( a \) be a point of \( \mathrm{C} \) ; there is a closed ball \( \mathrm{B} \) with centre \( a \) m... | Yes |
Lemma 3. Let \( q \) be a positive quadratic form on a real vector space \( \mathrm{V} \) and let \( \mathrm{B} \) be the associated symmetric bilinear form. Let \( {a}_{1},\ldots ,{a}_{n} \) be elements of \( \mathrm{V} \) such that \( \mathrm{B}\left( {{a}_{i},{a}_{j}}\right) \leq 0 \) for \( i \neq j \). (i) If \( {... | The relation \( \mathrm{B}\left( {{a}_{i},{a}_{j}}\right) \leq 0 \) for \( i \neq j \) immediately implies that \[ q\left( {\mathop{\sum }\limits_{i}\left| {c}_{i}\right| {a}_{i}}\right) \leq q\left( {\mathop{\sum }\limits_{i}{c}_{i}{a}_{i}}\right) \] hence (i). If \( q \) is non-degenerate, the relation \( \mathop{\su... | Yes |
Lemma 4. Let \( \mathrm{Q} = \left( {q}_{ij}\right) \) be a real, symmetric, square matrix of order \( n \) such that:\na) \( {q}_{ij} \leq 0 \) for \( i \neq j \) ;\nb) there does not exist a partition of \( \{ 1,2,\ldots, n\} \) into two non-empty subsets \( \mathrm{I} \) and \( \mathrm{J} \) such that \( \left( {i, ... | Since \( q \) is a positive quadratic form, the kernel \( \mathrm{N} \) of \( q \) is the set of isotropic vectors for \( q \) (Algebra, Chap. IX, \( \underline{§}7 \), no. 1, Cor. of Prop. 2). Let \( {a}_{1},\ldots ,{a}_{n} \) be the canonical basis of \( {\mathbf{R}}^{n} \) . If \( \mathop{\sum }\limits_{i}{c}_{i}{a}... | Yes |
Lemma 5. Let \( {e}_{1},\ldots ,{e}_{n} \) be vectors generating \( \mathrm{T} \) such that:\n\na) \( \left( {{e}_{i} \mid {e}_{j}}\right) \leq 0 \) for \( i \neq j \) ;\n\nb) there does not exist a partition of \( \{ 1,\ldots, n\} \) into two non-empty subsets \( \mathrm{I} \) and \( \mathrm{J} \) such that \( \left( ... | Put \( {q}_{ij} = \left( {{e}_{i} \mid {e}_{j}}\right) \) . The matrix \( \mathrm{Q} = \left( {q}_{ij}\right) \) then satisfies the hypotheses of Lemma 4: conditions \( a \) ) and \( b \) ) of Lemma 4 are the same as conditions \( a \) ) and \( b \) ) above, and \( c \) ) is satisfied since \( \mathop{\sum }\limits_{{i... | Yes |
Lemma 6. Let \( \\left( {{e}_{1},\\ldots ,{e}_{n}}\\right) \) be a basis of \( \\mathrm{T} \) such that \( \\left( {{e}_{i} \\mid {e}_{j}}\\right) \\leq 0 \) for \( i \\neq j \). (i) If \( x = \\mathop{\\sum }\\limits_{i}{c}_{i}{e}_{i} \\in \\mathrm{T} \) is such that \( \\left( {x \\mid {e}_{i}}\\right) \\geq 0 \) for... | Under the hypotheses of (i), assume that \( {c}_{i} < 0 \) for some \( i \). Let \( f \) be the linear form on \( \\mathrm{T} \) defined by \( f\\left( {e}_{i}\\right) = 1 \) and \[ f\\left( {e}_{j}\\right) = - {c}_{i}/\\left( {\\mathop{\\sum }\\limits_{{k = 1}}^{n}\\left| {c}_{k}\\right| }\\right) \\text{ for }j \\neq... | Yes |
Lemma 7. Let \( \mathrm{A} \) be a set of unit vectors in \( \mathrm{T} \) . If there exists a real number \( \lambda < 1 \) such that \( \left( {a \mid {a}^{\prime }}\right) \leq \lambda \) for \( a,{a}^{\prime } \in \mathrm{A} \) and \( a \neq {a}^{\prime } \), then the set \( \mathrm{A} \) is finite. | For \( a,{a}^{\prime } \in \mathrm{A} \) such that \( a \neq {a}^{\prime } \), we have\n\n\[
{\begin{Vmatrix}a - {a}^{\prime }\end{Vmatrix}}^{2} = 2 - 2\left( {a \mid {a}^{\prime }}\right) \geq 2 - {2\lambda }.
\]\n\nNow, the unit sphere \( S \) of \( T \) being compact, there exists a finite covering of \( \mathrm{S} ... | Yes |
Lemma 2. The orthogonal complement \( {\mathrm{E}}^{0} \) of \( \mathrm{E} \) for \( {\mathrm{B}}_{\mathrm{M}} \) is of dimension 1; it is generated by an element \( v = \mathop{\sum }\limits_{{s \in \mathrm{S}}}{v}_{s}{e}_{s} \) with \( {v}_{s} > 0 \) for all \( s \) . | This follows from Lemma 4 of \( §3 \), no. 5, applied to the matrix with entries \( {B}_{\mathrm{M}}\left( {{e}_{s},{e}_{{s}^{\prime }}}\right) \). | No |
Lemma 2. Let \( \mathrm{K} \) be a commutative field, \( \mathrm{V} \) a finite dimensional vector space over \( \mathrm{K},\mathrm{S} = {\bigoplus }_{n \geq 0}{\mathrm{\;S}}_{n} \) the symmetric algebra of \( \mathrm{V}, s \) an endomorphism of \( \mathrm{V} \), and \( {s}^{\left( n\right) } \) the canonical extension... | Extending the base field if necessary, we can assume that \( \mathrm{K} \) is algebraically closed. Let \( \left( {{e}_{1},\ldots ,{e}_{r}}\right) \) be a basis of \( \mathrm{V} \) with respect to which the matrix of \( s \) is lower triangular, and let \( {\lambda }_{1},\ldots ,{\lambda }_{r} \) be the diagonal elemen... | Yes |
Lemma 3. Let \( \mathrm{K},\mathrm{V} \) and \( \mathrm{S} \) be as in Lemma 2, \( \mathrm{G} \) a finite group of automorphisms of \( \mathrm{V}, q \) the order of \( \mathrm{G} \), and \( \mathrm{R} \) the graded subalgebra of \( \mathrm{S} \) consisting of the elements invariant under \( \mathrm{G} \) . Assume that ... | Indeed, the endomorphism \( f = {q}^{-1}\mathop{\sum }\limits_{{g \in \mathrm{G}}}{g}^{\left( n\right) } \) is a projection of \( {\mathrm{S}}_{n} \) onto \( {\mathrm{R}}_{n} \) , so \( \operatorname{Tr}\left( f\right) = {\dim }_{\mathrm{K}}{\mathrm{S}}_{n}^{\mathrm{G}} \) . Thus, the Poincaré series of \( \mathrm{R} \... | Yes |
Lemma 4. Let \( \mathrm{K} \) be a commutative field, \( \mathrm{V} \) a finite dimensional vector space over \( \mathrm{K},\mathrm{G} \) a finite group of automorphisms of \( \mathrm{V} \) whose order \( q \) is invertible in \( \mathrm{K},\mathrm{S} \) the symmetric algebra of \( \mathrm{V} \), and \( \mathrm{R} \) t... | To say that B is ramified over R means that its inertia group \( {\mathcal{G}}^{\mathrm{T}}\left( \mathrm{B}\right) \) does not reduce to the identity, in other words that there exists \( g \neq 1 \) in \( \mathrm{G} \) such that \( g\left( z\right) \equiv z \) (mod. B) for all \( z \in \mathrm{S} \) . Since \( \mathrm... | Yes |
Lemma 1. Let \( \mathrm{X} \) be a finite forest, and \( x \mapsto {g}_{x} \) a map from \( \mathrm{X} \) to a group \( \Gamma \) such that \( {g}_{x} \) and \( {g}_{y} \) are conjugate whenever \( x \) and \( y \) are not linked in \( \mathrm{X} \) . Let \( \mathcal{T} \) be the set of total orderings on \( \mathrm{X}... | 1) We proceed by induction on \( n = \operatorname{Card}\mathrm{X} \) . The case \( n = 1 \) is immediate, so assume that \( n \geq 2 \) . There exists in \( \mathrm{X} \) a terminal vertex \( a \) (Chap. IV, Appendix, no. 3, Prop. 2). Let \( b \in \mathrm{X} - \{ a\} \) be a vertex linked to \( a \) if one exists; if ... | Yes |
Lemma 1. Let \( \\mathrm{V} \) be a vector space over \( k,\\mathrm{R} \) a finite subset of \( \\mathrm{V} \) generating V. For any \( \\alpha \\in \\mathrm{R} \) such that \( \\alpha \\neq 0 \), there exists at most one reflection \( s \) of \( \\mathrm{V} \) such that \( s\\left( \\alpha \\right) = - \\alpha \) and ... | Let \( \\mathrm{G} \) be the group of automorphisms of \( \\mathrm{V} \) leaving \( \\mathrm{R} \) stable. Since \( \\mathrm{R} \) generates \( \\mathrm{V},\\mathrm{G} \) is isomorphic to a subgroup of the symmetric group of \( \\mathrm{R} \) , and hence is finite. Let \( s,{s}^{\\prime } \) be reflections of \( \\math... | Yes |
Lemma 2. Let \( \mathrm{R} \) be a root system in \( \mathrm{V} \). Let \( \left( {x \mid y}\right) \) be a symmetric bilinear form on \( \mathrm{V} \), non-degenerate and invariant under \( \mathrm{W}\left( \mathrm{R}\right) \). Identify \( \mathrm{V} \) with \( {\mathrm{V}}^{ * } \) by means of this form. If \( \alph... | This follows from formula (4) of Chap. V,§ 2, no. 3. | No |
Lemma 3. Let \( \mathrm{C} \) be a chamber of \( \mathrm{R} \) and \( \mathrm{P} \) a closed subset of \( \mathrm{R} \) containing \( {\mathrm{R}}_{ + }\left( \mathrm{C}\right) \) (in the notation of no. 6). Let \( \sum = \mathrm{B}\left( \mathrm{C}\right) \cap \left( {-\mathrm{P}}\right) \), and let \( \mathrm{Q} \) b... | It is enough to show that \( \mathrm{P} \cap \left( {-{\mathrm{R}}_{ + }\left( \mathrm{C}\right) }\right) = \mathrm{Q} \) . Let \( - \alpha \in \mathrm{Q} \) . Then \( \alpha \) is the sum of \( n \) elements of \( \sum \) . We show, by induction on \( n \), that \( - \alpha \in \mathrm{P} \) . This is clear if \( n = ... | Yes |
Lemma 4. For all \( w \in \mathrm{W} \), the characteristic polynomial of \( w \) has integer coefficients. | We know (no. 6, Th. 3) that \( \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{l}}\right\} \) is a basis of the subgroup Q(R) of \( \mathrm{V} \) generated by \( \mathrm{R} \) . Since \( w \) leaves \( \mathrm{Q}\left( \mathrm{R}\right) \) stable, its matrix with respect to \( \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{l}}\ri... | Yes |
Lemma 1. Let \( \\mathrm{X} \) be a locally compact space countable at infinity, \( \\mathrm{G} \) a group acting continuously and properly on \( \\mathrm{X},\\mu \) a measure on \( \\mathrm{X} \) invariant under \( \\mathrm{G},\\mathrm{G}^{\\prime } \) a subgroup of \( \\mathrm{G} \) , \( \\mathrm{U} \) and \( \\mathr... | Let \( \\left( s_{\\lambda }\\right)_{\\lambda \\in \\mathit{\\Lambda}} \) be a family of representatives of the right cosets of \( \\mathrm{G}^{\\prime } \) in \( \\mathrm{G} \). Let \( \\mathrm{U}_{1} \) be the union of the \( s_{\\lambda }\\mathrm{U} \). Then the \( s^{\\prime }\\mathrm{U}_{1} \), for \( s^{\\prime ... | Yes |
Lemma 2. Let \( x \in \mathrm{A}\left\lbrack \mathrm{P}\right\rbrack \) and let \( {\left( {x}_{p}{e}^{p}\right) }_{p \in \mathrm{X}} \) be the family of maximal terms of \( x \) . Let \( q \in \mathrm{P} \) and let \( y \in \mathrm{A}\left\lbrack \mathrm{P}\right\rbrack \) be such that \( {e}^{q} \) is the unique maxi... | Put \( x = \mathop{\sum }\limits_{p}{x}_{p}{e}^{p}, y = \mathop{\sum }\limits_{r}{y}_{r}{e}^{r} \) and \( {xy} = \mathop{\sum }\limits_{t}{z}_{t}{e}^{t}. \) Then \( r \leq q \) for all \( r \in \mathrm{P} \) such that \( {y}_{r} \neq 0 \) and \( {z}_{t} = \mathop{\sum }\limits_{{p + r = t}}{x}_{p}{y}_{r} \) . If \( t =... | Yes |
Lemma 4. Let \( \mathrm{I} \) be an ordered set satisfying the following condition:\n\n(MIN) Every non-empty subset of I contains a minimal element.\n\nLet \( \mathrm{E} \) be an A-module, \( {\left( {e}_{i}\right) }_{i \in \mathrm{I}} \) a basis of \( \mathrm{E} \) and \( {\left( {x}_{i}\right) }_{i \in \mathrm{I}} \)... | For any subset \( J \) of \( I, \) let \( {E}_{J} \) be the submodule of \( E \) with basis \( {\left( {e}_{i}\right) }_{i \in J}. \) Let \( \mathfrak{S} \) be the set of subsets \( \mathrm{J} \) of \( \mathrm{I} \) with the following properties:\n\n(a) If \( {i}^{\prime } \leq i \) and \( i \in \mathrm{J} \), then \( ... | Yes |
Lemma 1. For all \( i,\mathop{\sum }\limits_{{j \neq i}}{q}_{ij}^{2} < 1 \) . | Let \( \mathrm{J} \) be the set of \( j \in \mathrm{I} \) such that \( {q}_{ij} \neq 0 \), in other words such that \( \{ i, j\} \) is an edge of X. If \( j,{j}^{\prime } \in \mathrm{J} \) and \( j \neq {j}^{\prime },\left\{ {j,{j}^{\prime }}\right\} \) is not an edge (otherwise \( i, j,{j}^{\prime } \) would form a ci... | Yes |
Lemma 2. Any vertex of \( \mathrm{X} \) belongs to at most 3 edges. | Indeed, if \( i \) is linked to \( h \) other vertices, the relations \( {q}_{ij}^{2} \geq \frac{1}{4} \) for these other vertices implies that \( \frac{h}{4} < 1 \) by Lemma 1, so \( h \leq 3 \) . | Yes |
Lemma 3. If \( i \) belongs to 3 edges, these edges are of order 3. | If not, we would have, in view of the relation \( \cos \frac{\pi }{4} = \frac{\sqrt{2}}{2} \), \[ \mathop{\sum }\limits_{{j \neq i}}{q}_{ij}^{2} \geq \frac{1}{4} + \frac{1}{4} + {\left( \frac{\sqrt{2}}{2}\right) }^{2} = 1 \] which is impossible (Lemma 1). | Yes |
Lemma 4. If there exists an edge of order \( \geq 6 \), then \( l = 2 \) . | Indeed, let \( \{ i, j\} \) be such an edge. If \( l > 2 \), one of the edges \( i, j \) (say \( i \) ) would be linked to a third vertex \( {j}^{\prime } \), since \( \mathrm{X} \) is connected. In view of the relation \( \cos \frac{\pi }{6} = \frac{\sqrt{3}}{2} \) we would have\n\n\[ \mathop{\sum }\limits_{{k \neq i}... | Yes |
Lemma 5. A vertex cannot belong to two distinct edges of order \( \geq 4 \) . | Let \( i \) be such a vertex. We would have \( \mathop{\sum }\limits_{{j \neq i}}{q}_{ij}^{2} \geq {\left( \frac{\sqrt{2}}{2}\right) }^{2} + {\left( \frac{\sqrt{2}}{2}\right) }^{2} = 1 \), which is impossible (Lemma 1). | Yes |
Lemma 6. If \( \{ i, j\} \) is of order 3, \( \mathrm{W}\left( {\mathrm{M}}^{\prime }\right) \) is a finite Coxeter group. | Indeed, \( {q}_{ij} = - \frac{1}{2} \), so (1) becomes\n\n\[\n\mathop{\sum }\limits_{{k,{k}^{\prime } \in {\mathrm{I}}^{\prime }}}{q}_{k{k}^{\prime }}^{\prime }{\xi }_{k}{\xi }_{{k}^{\prime }} = \mathop{\sum }\limits_{{k,{k}^{\prime } \in \mathrm{I}}}{q}_{k{k}^{\prime }}{\xi }_{k}{\xi }_{{k}^{\prime }}\n\]\n\nand \( {\... | Yes |
Lemma 7. We have one of the following:\n\na) X has a unique ramification point (Chap. IV, Appendix, no. 1), and all the edges of \( \mathrm{X} \) are of order 3.\n\nb) \( \mathrm{X} \) is a chain and has at most one edge of order \( \geq 4 \) . | We argue by induction on \( l \) .\n\na) Assume that \( \mathrm{X} \) has a ramification point \( i \) . Then \( i \) belongs to 3 edges of order \( 3,\left\{ {i,{k}_{1}}\right\} ,\left\{ {i,{k}_{2}}\right\} ,\left\{ {i,{k}_{3}}\right\} \) (Lemmas 2 and 3). If \( l = 4 \) the lemma is proved. If not, then \( {k}_{1} \)... | Yes |
Lemma 8. Let \( {i}_{1},{i}_{2},\ldots ,{i}_{p} \) be vertices of \( \mathrm{X} \) such that \( \left\{ {{i}_{1},{i}_{2}}\right\} ,\left\{ {{i}_{2},{i}_{3}}\right\} ,\ldots \) \( \ldots ,\left\{ {{i}_{p - 1},{i}_{p}}\right\} \) are edges of order 3 . Then \( q\left( {\mathop{\sum }\limits_{{r = 1}}^{p}r{e}_{{i}_{r}}}\r... | We have \( \left( {{e}_{{i}_{r}}|{e}_{{i}_{r}}}\right) = 1,\left( {{e}_{{i}_{r}}|{e}_{{i}_{r + 1}}}\right) = - \frac{1}{2},\left( {{e}_{{i}_{r}}|{e}_{{i}_{s}}}\right) = 0 \) if \( s > r + 1. \) Thus\n\n\[ q\left( {\mathop{\sum }\limits_{{r = 1}}^{p}r{e}_{{i}_{r}}}\right) = \mathop{\sum }\limits_{{r = 1}}^{p}{r}^{2} - 2... | Yes |
Lemma 9. Assume that \( \mathrm{X} \) is a chain with vertices \( 1,2,\ldots, l \) and edges \( \{ 1,2\} ,\{ 2,3\} ,\ldots ,\{ l - 1, l\} \) . (i) If one of the edges \( \{ 2,3\} ,\{ 3,4\} ,\ldots ,\{ l - 2, l - 1\} \) is of order \( \geq 4 \), this edge is of order 4 and the graph is the following: ![40ce2b08-6f4d-481... | We can assume that \( l > 2 \) (Lemma 4). Assume that \( \{ i, i + 1\} \) is of order \( \geq 4 \), with \( 1 \leq i \leq l - 1 \) . Put \[ x = {e}_{1} + 2{e}_{2} + \cdots + i{e}_{i}, y = {e}_{i} + 2{e}_{l - 1} + \cdots + \left( {l - i}\right) {e}_{i + 1},\text{ and }j = l - i. \] By Lemma 8, \( \parallel x{\parallel }... | Yes |
Theorem 1.1 Let \( A \in {\mathbb{C}}^{n \times n} \). Then \( A \) can be written as the sum of matrices \( {A}_{1} \) and \( {A}_{2} \), i.e., \( A = {A}_{1} + {A}_{2} \), where: | Proof Let \( A \) be the form of (1.5). Then\n\n\[ A = U\left\lbrack \begin{matrix} \sum & 0 \\ 0 & 0 \end{matrix}\right\rbrack {V}^{ * }U{U}^{ * } \]\n\n\[ = U\left\lbrack \begin{matrix} \sum & 0 \\ 0 & 0 \end{matrix}\right\rbrack \left\lbrack \begin{matrix} K & L \\ M & N \end{matrix}\right\rbrack {U}^{ * } \]\n\n\[ ... | No |
Theorem 1.3 Let \( A \in {\mathbb{C}}^{n \times n} \). Then there exists unitary matrix \( U \) such that\n\n\[ A = U\\left\\lbrack \\begin{matrix} D & L \\\\ 0 & N \\end{matrix}\\right\\rbrack {U}^{ * },\]\n\nwhere \( D \) is invertible, and \( N \) is nilpotent. | Proof Let \( A = {U}_{1}\\left\\lbrack \\begin{matrix} {K}_{1} & {L}_{1} \\\\ 0 & 0 \\end{matrix}\\right\\rbrack {U}_{1}^{ * } \) be the Hartwig-Spindelböck decomposition of \( A \), and let \( {K}_{1} = {U}_{11}\\left\\lbrack \\begin{matrix} {K}_{2} & {L}_{2} \\\\ 0 & 0 \\end{matrix}\\right\\rbrack {U}_{11}^{ * } \) b... | Yes |
Lemma 1.6 Let \( A \in {\mathbb{C}}^{n \times n} \). Then\n\n\[ \operatorname{ind}\left( A\right) = \min \left\{ {s \in \mathbb{N} : \mathcal{R}\left( {A}^{s}\right) = \mathcal{R}\left( {A}^{s + 1}\right) }\right\} \]\n\n\[ = \min \left\{ {t \in \mathbb{N} : \mathcal{N}\left( {A}^{t}\right) = \mathcal{N}\left( {A}^{t +... | Proof Let \( A = {P}^{-1}\left\lbrack \begin{matrix} D & 0 \\ 0 & N \end{matrix}\right\rbrack P \) be the Jordan normal form of \( A \), where \( P, D \) are invertible and \( N \) is nilpotent. Then\n\n\[ \operatorname{ind}\left( A\right) = \min \left\{ {s \in \mathbb{N} : \mathcal{R}\left( {A}^{s}\right) = \mathcal{R... | Yes |
Theorem 1.7 Let \( A \in {\mathbb{C}}^{n \times n} \) . Then the following statements are equivalent:\n\n(1) \( \operatorname{ind}\left( A\right) = 1 \) .\n\n(2) \( \mathcal{R}\left( A\right) = \mathcal{R}\left( {A}^{2}\right) \) .\n\n(3) \( \mathcal{N}\left( A\right) = \mathcal{N}\left( {A}^{2}\right) \) .\n\n(4) Ther... | Proof The proof is left to the readers. | No |
Theorem 1.9 Let \( A \in {\mathbb{C}}^{n \times n} \) be of rank \( r \) . Then the following statements are equivalent:\n\n(1) \( A \) is a projection matrix.\n\n(2) There is a unitary matrix \( U \in {\mathbb{C}}^{n \times n} \) such that \( A = U\left\lbrack \begin{matrix} {I}_{r} & 0 \\ 0 & 0 \end{matrix}\right\rbr... | Proof (1) \( \Rightarrow \) (2). By Theorem 1.3, there exists a unitary matrix \( U \) such that\n\n\[ A = U\left\lbrack \begin{array}{ll} D & L \\ 0 & N \end{array}\right\rbrack {U}^{ * },\]\n\nwhere \( D \) is invertible, \( N \) is nilpotent. Since \( A \) is Hermitian, we obtain\n\n\[ {A}^{ * } = U\left\lbrack \beg... | Yes |
Given any nonempty set \( X \) of \( R \), the set of all left annihilators of \( X \) in \( R \) is \[ {\mathbf{l}}_{R}\left( X\right) = \{ r \in R : {rx} = 0\text{ for all }x \in X\} . \] | It is easy to determine that \( {\mathbf{l}}_{R}\left( X\right) \) is a left ideal of \( R \) . For convenience, \( {\mathbf{l}}_{R}\left( X\right) \) is also written as \( \mathbf{l}\left( X\right) \), and in case \( X = \{ x\} \), we usually mark \( {\mathbf{l}}_{R}\left( {\{ x\} }\right) \) as \( {\mathbf{l}}_{R}\le... | No |
Lemma 2.17 Let \( e \in R \) be an idempotent. Then: | \[ {}^{ \circ }e = R\left( {1 - e}\right) ,\;{e}^{ \circ } = \left( {1 - e}\right) R. \] | No |
Theorem 2.19 (Jacobson’s Lemma) Let \( a, b \in R \). (1) If \( 1 - {ab} \) is left invertible, then so is \( 1 - {ba} \). In this case, if \( x \in R \) is a left inverse of \( 1 - {ab} \), then \( 1 + {bxa} \) is a left inverse of \( 1 - {ba} \). (2) If \( 1 - {ab} \) is right invertible, then so is \( 1 - {ba} \). I... | Proof It can be verified directly. | No |
Lemma 2.21([44, Lemma 2.5,2.6]) Let \( a, b \in R \) . Then:\n\n(1) If \( {aR} \subseteq {bR} \) then \( {}^{ \circ }b \subseteq {}^{ \circ }a \) . | Proof (1) Suppose that \( {aR} \subseteq {bR} \) and \( {ub} = 0 \) for some \( u \in R \) . There exists \( x \in R \) such that \( a = {bx} \) so \( {ua} = {ubx} = 0 \) . | Yes |
Example 3.5 Let \( {RG} \) be the group ring of \( G \) over a commutative ring \( R \) (see Example 2.15). Then there is a classical involution in \( {RG} \) defined as follows: for any \( \alpha = \mathop{\sum }\limits_{{g \in G}}{a}_{g}g \in {RG} \), set | \[ {\alpha }^{ * } = \mathop{\sum }\limits_{{g \in G}}{a}_{g}{g}^{-1} \] | Yes |
Proposition 3.12 Let \( R \) be a \( * \) -ring. Then \( * \) is \( n \) -proper if and only if the induced involution \( * \) in \( {R}^{n \times n} \) is proper. | Proof It is clear. | No |
Example 4.2 Let \( D \) be a division ring. Then \( {D}^{n \times n} \) is unit-regular. | Proof For any \( A \in {D}^{n \times n} \), there are invertible \( P, Q \in {D}^{n \times n} \) such that\n\n\[ A = P\left\lbrack \begin{matrix} {I}_{r} & 0 \\ 0 & 0 \end{matrix}\right\rbrack Q. \]\n\nLet \( U = {Q}^{-1}{P}^{-1} \) . Then\n\n\[ {AUA} = P\left\lbrack \begin{matrix} {I}_{r} & 0 \\ 0 & 0 \end{matrix}\rig... | Yes |
Example 4.3 Let \( D \) be a division ring. Then \( {CFM}\left( D\right) \) is regular. | Proof Let \( {V}_{D} \) be an infinite dimensional vector space over \( D \) . Since \( {CFM}\left( D\right) \cong \) \( \operatorname{End}\left( {V}_{D}\right) \), it suffices to show \( \operatorname{End}\left( {V}_{D}\right) \) is regular. Let \( f \in \operatorname{End}\left( {V}_{D}\right) \) . Then\n\n\[ \n{V}_{D... | Yes |
Proposition 4.4 Let \( n \) be a positive integer. Then a ring \( R \) is regular (resp., unit-regular) if and only if \( {R}^{n \times n} \) is regular (resp., unit-regular). | Proof It follows by [13, Corollary 4.7]. | No |
Theorem 4.11 ([14, Theorem 1]) Let \( R \) be a \( * \) -proper ring. Then \( R \) is regular if and only if \( R \) is right (resp., left) P-injective. | Proof It suffices to prove the sufficiency. Suppose that \( R \) be right P-injective and * is proper. For any \( a, b \in R \) with \( {a}^{ * }{ab} = 0 \), we have \( {\left( ab\right) }^{ * }{ab} = {b}^{ * }{a}^{ * }{ab} = 0 \), and hence \( {ab} = 0 \) since \( * \) is proper. It follows that \( \mathbf{r}\left( a\... | Yes |
Theorem4.18( \( {}^{\left( \left\lbrack 7,\text{Proposition 2.11}\right\rbrack ,\left\lbrack {10},\text{Theorem 3.8}\right\rbrack \right) } \) Let \( R \) be a \( * \) -ring and \( n \geq 2 \) . Then the following statements are equivalent:\n\n(1) \( {R}^{n \times n} \) is \( * \) -unit regular.\n\n(2) \( R \) is unit-... | Proof It follows by Proposition 4.4 and Theorem 4.14. | No |
Theorem 7.7 Let \( A \in {\mathbb{C}}^{m \times n} \) and \( b \in {\mathbb{C}}^{m} \). Then \( {A}^{ \dagger }b \) is the minimum-norm least-squares solution of \( {Ax} = b \). | Proof Since \( {A}^{ \dagger }A{A}^{ \dagger }b = {A}^{ \dagger }b,{A}^{ \dagger }b \) is a least-squares solution of \( {Ax} = b \) by Theorem 7.5.\n\nIt is easy to check that \( \mathcal{N}\left( A\right) = \mathcal{N}\left( {{A}^{ \dagger }A}\right) = \left\{ {\left( {I - {A}^{ \dagger }A}\right) h : h \in {\mathbb{... | Yes |
Lemma 8.5 Let \( a \in S \) . Then:\n\n(1) \( a \) is \( \{ 1,3\} \) -invertible if and only if there exists a unique projection \( p \) such that \( {aS} = {pS} \) . | Proof (1) If \( a \) is \( \{ 1,3\} \) -invertible, suppose that \( {a}^{\left( 1,3\right) } \) is a \( \{ 1,3\} \) -inverse of \( a \) . It is easy to verify that \( p = a{a}^{\left( 1,3\right) } \) is a projection satisfying \( {aS} = {pS} \) . If \( {p}_{1}R = {p}_{2}R = {aR} \) , then \( {p}_{1} = {p}_{2} \) by Lem... | Yes |
Corollary 8.9 Let \( a \in {R}^{\{ 1\} } \) and \( {a}^{\left( 1\right) } \in a\{ 1\} \) . Then\n\n\[ a\{ 1\} = \left\{ {{a}^{\left( 1\right) } + z - {a}^{\left( 1\right) }{aza}{a}^{\left( 1\right) } : z \in R}\right\} . \] | Proof The above set is obtained by writing \( y = {a}^{\left( 1\right) } + z \) in the set of solution of \( {axa} = a \) as given by Theorem 8.8. | No |
Corollary 8.10 Let \( a \in R \) .\n\n(1) If \( a \in {R}^{\{ 1,3\} } \) with \( {a}^{\left( 1,3\right) } \in a\{ 1,3\} \), then\n\n\[ a\{ 1,3\} = \left\{ {{a}^{\left( 1,3\right) } + \left( {1 - {a}^{\left( 1,3\right) }a}\right) z : z \in R}\right\} .\n\] | Proof (1) From the proof of Lemma 8.5, we know that \( x \) is a \( \{ 1,3\} \) -inverse of \( a \) if and only if \( {ax} = a{a}^{\left( 1,3\right) } \) . By Theorem 8.8, we have the general solution of \( {ax} = a{a}^{\left( 1,3\right) } \) is\n\n\[ x = {a}^{\left( 1,3\right) }a{a}^{\left( 1,3\right) } + s - {a}^{\le... | Yes |
Theorem 8.11 ([15, Proposition 5]) Let \( a \in R \) . Then:\n\n(1) \( a \) is \( \{ 1,3\} \) -invertible if and only if \( R = {aR} \oplus {\left( {a}^{ * }\right) }^{ \circ } \) . In this case\n\n\[ a\{ 1,3\} = \{ r + \left( {1 - {ra}}\right) w : w \in R\} \]\n\nwhere \( 1 = {ar} + u \) for some \( r \in R \) and \( ... | Proof (1) If \( a \) is \( \{ 1,3\} \) -invertible, then there exists a unique projection \( p \) such that \( {aR} = {pR} \) by Lemma 8.5. So \( {pa} = a \) and \( p = a{a}^{\prime } \) for some \( {a}^{\prime } \in R \) . And for any idempotent \( p \), we have \( R = {pR} \oplus \left( {1 - p}\right) R \) . It follo... | Yes |
Corollary 8.19 Let \( R \) be with the GN-property. Then every idempotent has a unique range projection given by (8.33). | Proof If \( R \) has the GN-property, then for any idempotent \( f \) ,\n\n\[ \n{\left( f + {f}^{ * } - 1\right) }^{2} = 1 + \left( {{f}^{ * } - f}\right) \left( {{f}^{ * } - f}\right) \in {R}^{-1}, \n\]\n\nwhich implies that \( f + {f}^{ * } - 1 \) is invertible in \( R \) . | No |
Theorem 8.27([14, Theorem 6]) Let \( A \in {R}^{m \times n} \). Then:\n\n(1) If \( A\{ 1,3\} \neq \varnothing \), then \( {A}^{ * }A \) is invertible if and only if \( \mathcal{N}\left( A\right) = 0 \) . | Proof (1) Assume that \( {A}^{ * }A \) is invertible, it is clear that \( \mathcal{N}\left( A\right) = 0 \). Conversely, if \( \mathcal{N}\left( A\right) = 0, X \in A\{ 1,3\} \). From the equation \( A\left( {{XA} - I}\right) = 0 \), we obtain\n\n\[ I = {XA} = {XAXA} = X{\left( AX\right) }^{ * }A = X{X}^{ * }{X}^{ * }A... | Yes |
Theorem 9.5 Let \( p, a, q \in R \) with \( {p}^{\prime }{pa} = a = {aq}{q}^{\prime } \) for some \( {p}^{\prime },{q}^{\prime } \in R \) . If \( a \in {R}^{ \dagger } \) , then the following statements are equivalent:\n\n(1) \( {paq} \in {R}^{ \dagger } \) .\n\n(2) \( u = {p}^{ * }{pa}{a}^{ \dagger } + 1 - a{a}^{ \dag... | Proof It follows by the above discussion. | No |
Theorem 9.11 Let \( p, a, q \in R \) . If \( a \) is regular with \( {a}^{ - } \in a\{ 1\} \) . Then the following conditions are equivalent:\n\n(1) \( u = {aq}{\left( paq\right) }^{ * }{pa}{a}^{ - } + 1 - a{a}^{ - } \in {R}^{-1} \) .\n\n(2) \( v = {a}^{ - }{aq}{\left( paq\right) }^{ * }{pa} + 1 - {a}^{ - }a \in {R}^{-... | Proof (1) \( \Leftrightarrow \) (2). By Jacobson’s Lemma. | No |
(1) If \( p\bar{q} \in {R}^{ \dagger } \), then \( p - a = p\bar{q}p \in {R}^{ \dagger } \) and \( \left( {p - a}\right) {\left( p - a\right) }^{ \dagger }b = b \) . | Proof (1). Since \( p\bar{q} \in {R}^{ \dagger } \), we have \( p - a = p\bar{q}p = \left( {p\bar{q}}\right) {\left( p\bar{q}\right) }^{ * } \in {R}^{ \dagger } \) . Moreover,\n\n\[ \left( {p - a}\right) {\left( p - a\right) }^{ \dagger }b = \left( {p\bar{q}}\right) {\left( p\bar{q}\right) }^{ * }{\left\lbrack \left( p... | Yes |
Corollary 11.12 If \( a \) is regular with an inner inverse \( {a}^{ - } \), then the following conditions are equivalent:\n\n(1) \( a \in {R}^{ \dagger } \) .\n\n(2) \( a{a}^{ - } \in {R}^{\{ 1,3\} } \) and \( {a}^{ - }a \in {R}^{\{ 1,4\} } \) . | In this case, \( a{a}^{ \dagger } \in \left( {a{a}^{ - }}\right) \{ 1,3\} ,{a}^{ \dagger }a \in \left( {{a}^{ - }a}\right) \{ 1,4\} \) and\n\n\[ \n{a}^{ \dagger } = {\left( {a}^{ - }a\right) }^{\left( 1,4\right) }{a}^{ - }a{a}^{ - }{\left( a{a}^{ - }\right) }^{\left( 1,3\right) }\n\]\n\nfor any \( {\left( a{a}^{ - }\ri... | Yes |
Example 11.13 Let \( {\mathbb{C}}^{2 \times 2} \) be the ring of all \( 2 \times 2 \) matrices over the complex field \( \mathbb{C} \), with transpose as the involution. Suppose that \( a = \left\lbrack \begin{matrix} 1 & - i \\ i & - 1 \end{matrix}\right\rbrack \) and \( b = \left\lbrack \begin{matrix} 0 & 0 \\ 0 & - ... | From Example 11.13, we can see that \( 1 - {ba} \) may not be \( \{ 1,4\} \) -invertible when \( 1 - {ab} \) is. Dually, \( 1 - {ab} \in {R}^{\{ 1,3\} } \) does not imply \( 1 - {ba} \in {R}^{\{ 1,3\} } \), either. | "No" |
Lemma 11.14( \( {}^{\left( {45},\text{Lemma 3.2}\right) } \) 1 Let \( e \in R \) be an idempotent. Then \( e \in {R}^{\{ 1,3\} } \) if and only if \( 1 - e \in {R}^{\{ 1,4\} } \) . In this case, \( 1 - e{e}^{\left( 1,3\right) } \in \left( {1 - e}\right) \{ 1,4\} \) and \( 1 - {\left( 1 - e\right) }^{\left( 1,4\right) }... | Proof If \( e \in {R}^{\{ 1,3\} } \), in order to prove that \( 1 - e \in {R}^{\{ 1,4\} } \), we only need to check that \( 1 - e{e}^{\left( 1,3\right) } \in \left( {1 - e}\right) \{ 1,4\} \) . Indeed, we have\n\n\[ \left( {1 - e{e}^{\left( 1,3\right) }}\right) \left( {1 - e}\right) = 1 - e{e}^{\left( 1,3\right) } - e ... | No |
Theorem 11.15 \( \left( \left\lbrack {{45},\text{Theorem 5.6}}\right\rbrack \right) \) Let \( a, b \in R \) . If \( \alpha = 1 - {ab} \in {R}^{\{ 1,3\} } \) with a \( \{ 1,3\} \) -inverse \( {\alpha }^{\left( 1,3\right) } \), then the following conditions are equivalent:\n\n(1) \( \beta = 1 - {ba} \in {R}^{\{ 1,3\} } \... | Proof As \( \alpha = 1 - {ab} \in {R}^{\{ 1,3\} } \) with a \( \{ 1,3\} \) -inverse \( {\alpha }^{\left( 1,3\right) } \), it is clear that \( \alpha {\alpha }^{\left( 1,3\right) } \in \) \( {R}^{\{ 1,3\} } \) with \( \alpha {\alpha }^{\left( 1,3\right) } \in \left( {\alpha {\alpha }^{\left( 1,3\right) }}\right) \{ 1,3\... | No |
Theorem 11.17 \( {}^{\left( \left\lbrack {45},\text{Theorem 5.8}\right\rbrack \right) } \) Let \( a, b \in R \) . If \( \alpha = 1 - {ab} \in {R}^{ \dagger } \), then the following conditions are equivalent:\n\n(1) \( \beta = 1 - {ba} \in {R}^{ \dagger } \) .\n\n(2) \( 1 - b{\alpha }_{r}^{\pi }a \in {R}^{\{ 1,3\} } \) ... | Proof The equivalence of conditions (1)-(4) clearly follows from Theorem 11.15, Theorem 11.16 and Lemma 8.2. Next we give a formula for \( {\beta }^{ \dagger } \) .\n\nIt is clear that \( \beta \) is regular with an inner inverse \( {\beta }^{ - } = 1 + b{\alpha }^{ \dagger }a \) . If (4) holds, according to the proof ... | Yes |
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