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Theorem 2 For complex signals \( \{ f\left\lbrack k\right\rbrack {\} }_{k = 0}^{N - 1} \) and \( \{ h\left\lbrack k\right\rbrack {\} }_{k = 0}^{N - 1} \), let \( {z}_{-1} = {\mathcal{F}}_{\alpha }^{-1}\left( {{\mathcal{F}}_{\alpha }h. * }\right. \) \( \left. {{\mathcal{F}}_{\alpha }f}\right) \) with \( \alpha = - 1 \) ... | Proof From (18), (19) and (22), it follows that\n\n\[ \frac{{C}_{H}^{1}f + {C}_{H}^{-1}f}{2} = {L}_{H}f,\;\frac{{C}_{H}^{1}f - {C}_{H}^{-1}f}{2} = {U}_{H}f. \]\n\nBy Lemma 1, we have\n\n\[ \frac{{z}_{-1}\left( n\right) + z\left( n\right) }{2} = {L}_{H}f,\;\frac{z\left( n\right) - {z}_{-1}\left( n\right) }{2} = {U}_{H}f... | Yes |
The ordinary multiplication or the minimum operations is a continuous \( t \) -norm. | In fact, let:\n\n1) \( a * b = {ab} \) ;\n\n2) \( a * b = \min \left( {a, b}\right) \) ,\n\nthen, \( * \) satisfies the condition 1)-4) in Definition 1. So, it is a continuous \( t \) -norm. | No |
Lemma 1 Let \( \\left( {X, M, * }\\right) \) and \( \\left( {Y, N, * }\\right) \) be compact fuzzy metric spaces. Then \( \\left( {X \\times Y,\\widetilde{M}, * }\\right) \) is a compact fuzzy metric space. Here\n\n\[ \n\\widetilde{M}\\left( {\\left( {{x}_{1},{y}_{1}}\\right) ,\\left( {{x}_{2},{y}_{2}}\\right), t}\\rig... | Proof Firstly, we will prove that \( \\left( {X \\times Y,\\widetilde{M}, * }\\right) \) is a fuzzy metric space. Obviously, it only needs to prove that the triangle inequality of fuzzy metric space is established. Let \( \\left( {{x}_{1},{y}_{1}}\\right) ,\\left( {{x}_{2},{y}_{2}}\\right) ,\\left( {{x}_{3},{y}_{3}}\\r... | Yes |
Theorem 1 Let \( \left( {X, M, T}\right) \) be a fuzzy dynamical system. \( \left( {X, M, * }\right) \) is a compact fuzzy metric spaces. Let \( x \in X \) . Then the following are equivalent:\n\n1) \( x \) is an equicontinuous point of fuzzy dynamic system \( \left( {X, M, T}\right) \) .\n\n2) \( \left( {x, x}\right) ... | Proof 1) \( \Rightarrow 2 \) ). Let \( x \) be an equicontinuous point of fuzzy dynamic system \( \left( {X, M, T}\right) \), then for any \( \varepsilon \in \left( {0,1}\right) \), there exists a \( \delta \in \left( {0,1}\right) \), such that \( \forall y \in X \) , satisfying\n\n\[ M\left( {x, y, t}\right) > 1 - \de... | Yes |
Theorem 2 Let \( \\left( {X, M, f}\\right) \) and \( \\left( {Y, N, g}\\right) \) be fuzzy dynamical system. \( \\left( {X, M, * }\\right) \) and \( \\left( {Y, N, \\star }\\right) \) are two compact fuzzy metric spaces and mapping \( \\varphi : X \\rightarrow Y \) is fuzzy uniform topologically conjugate. If fuzzy dyn... | Proof Since \( \\varphi : X \\rightarrow Y \) is fuzzy uniformly continuous mapping. So, for all \( \\varepsilon \\in \\left( {0,1}\\right) \), there is a \( {\\varepsilon }_{1} \\in \\left( {0,1}\\right) \), for each \( t > 0, x, y \\in X \), when \( M\\left( {x, y, t}\\right) > 1 - {\\varepsilon }_{1} \), then \( N\\... | Yes |
Proposition 3 Under Assumption 1, there exists a constant \( \widehat{L} > 0 \) such that\n\n\[ \begin{Vmatrix}{x}_{t}\end{Vmatrix} \leq \widehat{L}\max \left\{ {1,\begin{Vmatrix}{\xi }_{t}\end{Vmatrix}}\right\} \max \left\{ {1,\begin{Vmatrix}{x}_{t - 1}\end{Vmatrix}}\right\} \]\n\nfor any \( {x}_{1} \in {X}_{1},{x}_{t... | Proof Due to Lemma 1, for \( {x}_{t} \in {X}_{t}\left( {{x}_{t - 1},{\xi }_{t}}\right), t = 2,3,\cdots, T \), we have the estimate\n\n\[ \begin{Vmatrix}{x}_{t}\end{Vmatrix} \leq \frac{\max \left\{ {{\left( {h}_{t}\left( {\xi }_{t}\right) - {A}_{t,1}\left( {\xi }_{t}\right) {x}_{t - 1}\right) }_{i},1 \leq i \leq m}\righ... | Yes |
Corollary 1 Under Assumption 1, there exists a constant \( \widetilde{L} > 0 \), such that\n\n\[ \begin{Vmatrix}{x}_{t}\end{Vmatrix} \leq \widetilde{L}\max {\left\{ 1,\begin{Vmatrix}{\xi }^{t}\end{Vmatrix}\right\} }^{t - 1},\;t = 1,2,\cdots, T \]\n\nfor any \( {x}_{1} \in {X}_{1} \) and \( {x}_{t} \in {X}_{t}\left( {{x... | Define \( L = \max \{ \widehat{L},\widetilde{L},1\} \), then we have\n\n\[ \begin{Vmatrix}{x}_{t}\end{Vmatrix} \leq L\max \left\{ {1,\begin{Vmatrix}{\xi }_{t}\end{Vmatrix}}\right\} \max \left\{ {1,\begin{Vmatrix}{x}_{t - 1}\end{Vmatrix}}\right\} ,\;\begin{Vmatrix}{x}_{t}\end{Vmatrix} \leq L\max {\left\{ 1,\begin{Vmatri... | Yes |
Proposition 4 Under Assumption 1 and Assumption 2, \( {Q}_{t}\left( {{x}_{t - 1},{\xi }^{t}}\right) \) is Lipschitz continuous with respect to \( {x}_{t - 1} \), i.e., for \( t = 1,2,\cdots, T \) ,\n\n\[ \left| {{Q}_{t}\left( {{\bar{x}}_{t - 1},{\xi }^{t}}\right) - {Q}_{t}\left( {{x}_{t - 1},{\xi }^{t}}\right) }\right|... | Proof The proof is by induction on \( t \) . Obviously, it holds when \( t = T + 1 \) because \( {Q}_{T + 1} \equiv 0 \) . Assume that the conclusion is true for \( t + 1,\cdots, T \) . We show that it also holds for \( t \) .\n\nWhen \( \left| {{Q}_{t}\left( {{\bar{x}}_{t - 1},{\xi }^{t}}\right) - {Q}_{t}\left( {{x}_{... | Yes |
Corollary 1 If \( a = 0 \), system (3) has a positive periodic solution | \[ {u}^{ * }\left( t\right) = \frac{p \cdot \exp \left( {-b\left( {t - {nT}}\right) }\right) }{1 - \exp \left( {-{bT}}\right) } \] with initial value \[ {u}^{ * }\left( {0}^{ + }\right) = \frac{p}{1 - \exp \left( {-{bT}}\right) },\;t \in ({nT},\left( {n + 1}\right) T\rbrack ,\;n \in N. \] | Yes |
Theorem 1 For each solution \( \left( {{x}_{1}\left( t\right) ,{x}_{2}\left( t\right) ,{x}_{3}\left( t\right) ,{x}_{4}\left( t\right) }\right) \) of system (1) for all \( t \) large enough, there exists a constant \( L > 0 \) such that \( {x}_{1}\left( t\right) \leq L,{x}_{2}\left( t\right) \leq L,{x}_{3}\left( t\right... | Proof Define \( V\left( t\right) = {k}_{1}{x}_{1}\left( t\right) + {k}_{2}{x}_{2}\left( t\right) + {x}_{3}\left( t\right) + {x}_{4}\left( t\right) \) . For \( t \neq \left( {n + l - 1}\right) T, t \neq \) \( {nT} \), take a constant \( \lambda \) such that \( 0 < \lambda < \omega \), then we have\n\n\[ \n{D}^{ + }V\lef... | Yes |
Corollary 2 Let \( x\\left( t\\right) = \\left( {{x}_{1}\\left( t\\right) ,{x}_{2}\\left( t\\right) ,{x}_{3}\\left( t\\right) ,{x}_{4}\\left( t\\right) }\\right) \) be any solution of system (1), then \( {x}_{i}, i = 2,3,4 \) are permanent, \( {x}_{1}\\left( t\\right) \\rightarrow 0 \) as \( t \\rightarrow \\infty \) p... | \[ \\frac{1}{{a}_{2} - {c}_{2}}\\left( {\\ln \\frac{1}{1 - {\\mu }_{2}} + \\frac{{d}_{2}}{\\alpha }\\left( {\\frac{m{p}_{1}}{w} + \\frac{m}{w\\left( {m + w}\\right) }}\\right) }\\right) \\] \\[ < T < \\frac{1}{{a}_{1}}\\left( {\\ln \\frac{1}{1 - {\\mu }_{1}} + \\frac{{d}_{1}}{\\alpha }\\left( {\\frac{m{p}_{1}}{w} + \\f... | Yes |
Corollary 3 Let \( x\left( t\right) = \left( {{x}_{1}\left( t\right) ,{x}_{2}\left( t\right) ,{x}_{3}\left( t\right) ,{x}_{4}\left( t\right) }\right) \) be any solution of system (1), then \( {x}_{i}, i = 1,3,4 \) are permanent, \( {x}_{2}\left( t\right) \rightarrow 0 \) as \( t \rightarrow \infty \) provided \( {a}_{1... | \[ \frac{1}{{a}_{1} - {c}_{1}}\left( {\ln \frac{1}{1 - {\mu }_{1}} + \frac{{d}_{1}}{\alpha }\left( {\frac{m{p}_{1}}{w} + \frac{m}{w\left( {m + w}\right) }}\right) }\right) \] \[ < T < \frac{1}{{a}_{2}}\left( {\ln \frac{1}{1 - {\mu }_{2}} + \frac{{d}_{2}}{\alpha }\left( {\frac{m{p}_{1}}{w} + \frac{m}{w\left( {m + w}\rig... | Yes |
Example 1 Let \( {a}_{1} = 3,{a}_{2} = 1,{b}_{1} = 1,{b}_{2} = 1,{c}_{1} = {0.7},{c}_{2} = {0.1},{d}_{1} = {0.4} \) , \( {d}_{2} = {0.3}, m = {0.9}, w = {0.2},{\mu }_{1} = {0.4},{\mu }_{2} = {0.2},{p}_{1} = {0.2},{p}_{2} = {0.65},{k}_{1} = {0.95} \) , \( {k}_{2} = {0.95},\alpha = {0.7},\beta = {0.01},\gamma = {0.01}, T... | \n\nFigure 1: Time series of prey-extinction periodic solution | Yes |
Example 2 Let \( {a}_{2} = {1.8},\alpha = {1.25}, T = 2 \) and the other parameters are the same as Example 1. By verification, we get \( T = 2 > {T}_{\max } \approx {0.9165} \), then these parameters satisfy the conditions of Theorem 4, then system (1) is permanent, see Figure 2. | \n\nFigure 2: The permanence of system (1) | No |
Example 3 Let \( {a}_{1} = 2,{a}_{2} = 2,{b}_{1} = 2,{b}_{2} = 2,{c}_{1} = {0.8},{c}_{2} = {0.8},{d}_{1} = {0.1},{d}_{2} = \) \( {0.7}, m = {0.5}, w = {0.18},{\mu }_{1} = {0.4},{\mu }_{2} = {0.2},{p}_{1} = {0.2},{p}_{2} = {0.7},{k}_{1} = {0.65},{k}_{2} = {0.65},\alpha = \) \( 1,\beta = {0.01},\gamma = {0.01} \) . We se... | From Figure 3(a), if the impulsive period \( T < {80.20} \), system(1) is stable. If \( T = {80.20} \) , 98.20 and 102, then bifurcation appears, respectively. And when \( {80.20} < T < {102} \), the periodic behaviors of the prey and the predator appear. If \( {102} < T < {117} \), chaotic phenomenon appears. Therefor... | Yes |
Example 4 Let \( {a}_{1} = {7.9},{a}_{2} = {8.1},{b}_{1} = 2,{b}_{2} = {1.5},{c}_{1} = {0.1},{c}_{2} = {0.1},{d}_{1} = 1 \) , \( {d}_{2} = 1, m = {0.8}, w = {0.3},{\mu }_{1} = {0.1},{\mu }_{2} = {0.2},{p}_{2} = {0.9},{k}_{1} = {0.95},{k}_{2} = {0.95} \) , \( \alpha = {1.93},\beta = {0.03},\gamma = {1.8}, T = 3 \) and \... | On the contrary, the prey will be extinct and (1) has chaotic behaviors when \( {3.31} < {p}_{1} < {18} \) . | Yes |
Proposition 1 The volatility matrices of stocks prices in two markets satisfy:\n\n1) \( {\left( \widetilde{\sigma }\left( t\right) \widetilde{\sigma }{\left( t\right) }^{\prime }\right) }^{-1} = {\left( \sigma \left( t\right) \sigma {\left( t\right) }^{\prime }\right) }^{-1} \) ;\n\n2) \( {\widetilde{\sigma }}^{-1}\lef... | These results are evident in view of equations (11), (14) and (15), thus we omit the detailed proof. | No |
Proposition 2 Let \( W\left( t\right) = {\left( {W}_{1}\left( t\right) ,{W}_{2}\left( t\right) ,\cdots ,{W}_{d}\left( t\right) \right) }^{\prime } \) and \( \widetilde{W}\left( t\right) = \left( {{\widetilde{W}}_{1}\left( t\right) ,{\widetilde{W}}_{2}\left( t\right) }\right. \) , \( \cdots ,{\widetilde{W}}_{n}\left( t\... | Please see Appendix A. 1 for the proof. | No |
Proposition 3 Define \( g\left( {t,\pi }\right) = f{\left( t, z\right) }^{1 - k} \) and \( z = {\pi }^{-1} \) . Then, with the boundary condition given by \( f\left( {T, z}\right) = 1, f\left( {t, z}\right) \) has a solution of the form \( f\left( {t, z}\right) = D\left( t\right) + F\left( t\right) z \) with the bounda... | Please see Appendix A. 2 for the proof. | No |
Proposition 4 If equation (27) has a solution of the form \( h\left( {t,\pi }\right) = U\left( t\right) + \frac{M\left( t\right) }{\pi } \) with the boundary conditions given by \( U\left( T\right) = 0 \) and \( M\left( T\right) = 0 \), then | \[ U\left( t\right) = 0,\;M\left( t\right) = {\int }_{T}^{t}{\xi }_{2}\left( s\right) {\mathrm{e}}^{{\int }_{t}^{s}{q}_{2}\left( l\right) \mathrm{d}l}\mathrm{\;d}s, \] where \[ {q}_{2}\left( t\right) = {\widetilde{\sigma }}_{\Pi }\left( t\right) {\widetilde{\sigma }}^{-1}\left( t\right) \widetilde{B}\left( t\right) - L... | Yes |
Theorem 1 The optimal strategy and the expected actual wealth of problem (20) under the power utility function are given by\n\n\[ \n{\theta }^{ * }\left( t\right) = \frac{y - \frac{M\left( t\right) }{\pi }}{y\left( {1 - k}\right) }{\left( \widetilde{\sigma }{\widetilde{\sigma }}^{\prime }\right) }^{-1}\widetilde{B} - \... | Please see Appendix A. 3 for the proof. | No |
Theorem 2 The optimal strategy and the expected actual wealth of problem (8) under the power utility function are given by\n\n\[ \n{\theta }^{ * }\left( t\right) = \underset{\mathrm{I}}{\underbrace{\frac{\left( {{\pi y} - M\left( t\right) }\right) {\left( \sigma {\sigma }^{\prime }\right) }^{-1}B}{{\pi y}\left( {1 - k}... | This theorem tells us that the optimal strategy is composed of three parts. The first part I is the speculative demand of stocks. The investor obtains the risk premium through holding this demand. This is consistent with the result obtained in [14]. The second part \( \Pi \) is the hedging demand of the contribution. H... | Yes |
Theorem 2 The CSE problem (general model) is strongly NP-complete even for the digraph with only one terminal and only one compound vertex. | Proof First, as discussed in Theorem 1, a solution \( {G}^{\prime } \) itself is a concise certificate, so the problem is in NP. Second, we have shown that the basic model is strongly NP-complete and it is equivalent to a general model with only one terminal and only one compound vertex. To see this, given a basic mode... | Yes |
Proposition 1 The CSE problem (basic model) has polynomial-time algorithms for the case of \( \left| T\right| = 1 \) (single terminal) or \( M = \varnothing \) (no intermediate vertices). | Proof For the case of \( \left| T\right| = 1 \), the problem is equivalent to finding a shortest path from a source \( s \) to a sink \( t \), which is a well-known polynomially solvable problem (see Section 7.1 of [12] for Dijkstra’s algorithm). For the case of \( M = \varnothing \) (viz. \( V = \{ s\} \cup T \) ), th... | Yes |
Proposition 2 The feasibility procedure decides whether \( {G}_{X} \) is feasible in \( O\left( \left| V\right| \right) \) time. Moreover, if \( {G}_{X} \) is feasible, then \( {G}_{X} \) contains a feasible subgraph \( {G}^{\prime } = \left( {{V}^{\prime },{E}^{\prime }}\right) \) of the basic model. | Proof We first show the correctness of the procedure. Suppose that the procedure terminates with the conclusion that every vertex \( v \in V \smallsetminus X \) has a label. If \( v \) has a label \( l\left( v\right) = i > 0 \), then its label must come from a predecessor \( u \in {N}^{ - }\left( v\right) \) with label... | Yes |
Theorem 3 The B & B Algorithm solves the basic model in polynomial time provided that the number of the intermediate vertices is a given constant. | Proof We first show the correctness of the algorithm. The algorithm enumerates all subsets \( X \subseteq M \) apart from those of infeasible \( {G}_{X} \) and those dominated by a chosen feasible solution \( {G}^{ * } \) . So when the algorithm terminates, the current solution \( {G}^{ * } \) is the best among all con... | Yes |
Theorem 4 For the basic model with symmetric cost function \( c \) satisfying the triangle inequality, Algorithm 1-MWA is a 2-factor approximation algorithm. | Proof Let \( {G}^{ * } = \left( {{V}^{ * },{E}^{ * }}\right) \) be an optimal subgraph of the basic model, which is a rooted tree with root \( s \) . So \( {OPT} = c\left( {E}^{ * }\right) \) . Now we change each edge \( \left( {u, v}\right) \in {E}^{ * } \) into a pair reverse edges \( \left( {u, v}\right) \) and \( \... | Yes |
Theorem 1 Suppose that conditions (A1)-(A3) hold. Then\n\n\[ \ell \left( \beta \right) \overset{d}{ \rightarrow }{\chi }_{r}^{2} \]\n\nwhere \( {\chi }_{r}^{2} \) is a chi-squared distributed random variable with \( r \) degrees of freedom. | Let \( {z}_{\alpha, r} \) satisfy \( P\left( {{\chi }_{r}^{2} \leq {z}_{\alpha, r}}\right) = 1 - \alpha \) for \( 0 < \alpha < 1 \) . It follows from Theorem 1 that an EL-based confidence region for \( \beta \) with asymptotically correct coverage probability \( 1 - \alpha \) can be constructed as\n\n\[ \left\{ {\beta ... | Yes |
Lemma 1 Let \( \\left\\{ {{\\eta }_{i}, i \\geq 1}\\right\\} \) be a \( \\alpha \) -mixing process and \( {\\mathcal{F}}_{s}^{t} \) denote the \( \\sigma \) -algebra generated by \( \\left\\{ {{\\eta }_{i}, s \\leq i \\leq t}\\right\\} \) for \( s \\leq t \) . Suppose that \( \\xi \) and \( \\eta \) are random variable... | Proof See Lemma 1 in [13]. | No |
Lemma 2 Let \( \\left\\{ {{\\eta }_{i}, i \\geq 1}\\right\\} \) be a \( \\alpha \) -mixing process and \( {\\mathcal{F}}_{s}^{t} \) denote the \( \\sigma \) -algebra generated by \( \\left\\{ {{\\eta }_{i}, s \\leq i \\leq t}\\right\\} \) for \( s \\leq t \) . Suppose that \( \\left\\{ {{\\xi }_{i},1 \\leq i \\leq n}\\... | Proof See Lemma 1.1 in [14]. Note that (9) is also true for complex random variables \( \\left\\{ {{\\xi }_{i},1 \\leq i \\leq n}\\right\\} \) with \( \\left| a\\right| \) replaced by the norm of a complex number \( a \) . | Yes |
Lemma 3 Let \( 2 < {p}_{0} < {r}_{0} < \infty \), and \( \left\{ {{\eta }_{i}, i \geq 1}\right\} \) be a \( \alpha \) -mixing process with \( E\left( {\eta }_{i}\right) = 0 \) and \( E{\left| {\eta }_{i}\right| }^{{r}_{0}} < \infty \) for \( i \geq 1 \) . Suppose that \( \alpha \left( n\right) \leq C{n}^{-\theta } \) f... | Proof See Theorem 4.1 in [15]. | No |
Lemma 4 Suppose that Assumptions (A2), (A3) of the (i)-(iii) and (vi) are satisfied, \( \\left\\{ {{\\epsilon }_{i},1 \\leq i \\leq n}\\right\\} \) is a \( \\alpha \) -mixing sequence, there exists \( \\delta > 0 \) such that\n\n\[ \n\\mathop{\\max }\\limits_{{1 \\leq i \\leq n}}E{\\left| {\\epsilon }_{i}\\right| }^{4 ... | Proof of Lemma 4 To prove (11), we only need to show, for any given \( l \\in {R}^{r} \) with \( \\parallel l\\parallel = 1 \), that\n\n\[ \n{l}^{\\tau }{S}_{n}\\overset{d}{ \\rightarrow }N\\left( {0,1}\\right) \n\]\n\n(12)\n\nWrite \( {l}^{\\tau }{S}_{n} = {l}^{\\tau }{S}_{n}^{\\prime } + {l}^{\\tau }{S}_{n}^{\\prime ... | Yes |
Theorem 1 Let \( {D}_{C} \) be a super corona distance matrix of order \( n\left( {{n}_{1} + 1}\right) + m\left( {{n}_{2} + }\right. \) 1) as defined above. Then the spectrum of \( {D}_{C} \) consists of:\n\n(i) All the roots of the following equation\n\n\[ \n{\lambda }^{4} - \left\lbrack {{a}_{i}\left( {1 + {n}_{1}}\r... | Proof (a) To prove (i), we suppose that the vector\n\n\[ \n\phi = \left\lbrack \begin{matrix} {k}_{1}{X}_{i} \\ {k}_{2}{Y}_{i} \\ {k}_{3}{1}_{{n}_{1}} \otimes {X}_{i} \\ {k}_{4}{1}_{{n}_{2}} \otimes {Y}_{i} \end{matrix}\right\rbrack \n\] \n\nis an eigenvector of \( {D}_{C} \) corresponding to the eigenvalue \( \lambda ... | Yes |
Example 2 Consider the subdivision double corona graph \( {K}_{4}^{\left( S\right) } \circ \left\{ {{K}_{3},{P}_{2}}\right\} \) . Then the distance matrix of \( {K}_{4}^{\left( S\right) } \circ \left\{ {{K}_{3},{P}_{2}}\right\} \) is\n\n\[ \n{D}_{C}\left( {{K}_{4}^{\left( S\right) } \circ \left\{ {{K}_{3},{P}_{2}}\righ... | Solution 1 Through a Matlab program, the distance spectrum of \( {K}_{4}^{\left( S\right) } \circ \left\{ {{K}_{3},{P}_{2}}\right\} \)\n\nis\n\n\[ \n\left\{ {-{22.5248}^{\left( 3\right) }, - {7.2678}, - {3.5831}^{\left( 3\right) }, - {3}^{\left( 2\right) }, - {2.694},}\right. \n\]\n\n\[ \n\left. {-{1}^{\left( {14}\righ... | Yes |
Proposition 1 Let \( G \) be a complete graph on \( n \) vertices and \( m \) edges. Let \( {G}_{1} \) be a \( {r}_{1} \) -regular graph on \( {n}_{1} \) vertices with an adjacency matrix \( A\left( {G}_{1}\right) \) and \( {\operatorname{spec}}_{A}\left( {G}_{1}\right) = \) \( \left\{ {{r}_{1} = {\lambda }_{1}\left( {... | Proof By a harmonious labeling of vertex set, the distance matrix of \( {G}^{\left( S\right) } \circ \) \( \left\{ {{G}_{1},{G}_{2}}\right\} \) can be expressed in the form\n\n\[ \n{D}_{C}\left( {{G}^{\left( S\right) } \circ \left\{ {{G}_{1},{G}_{2}}\right\} }\right)\n\]\n\n<table><tr><td rowspan=\ | Yes |
Proposition 2 Let \( G \) be a complete graph on \( n \) vertices and \( m \) edges. Let \( {G}_{1} \) be a \( {r}_{1} \) -regular graph on \( {n}_{1} \) vertices with an adjacency matrix \( A\left( {G}_{1}\right) \) and \( {\operatorname{spec}}_{A}\left( {G}_{1}\right) = \) \( \left\{ {{r}_{1} = {\lambda }_{1}\left( {... | (i) All the roots of the equation\n\n\[ \n{\lambda }^{4} - \left\lbrack {\left( {2 - n}\right) \left( {{n}_{1} + 1}\right) - 2{n}_{1} - {r}_{1} - {r}_{2} - 6}\right\rbrack {\lambda }^{3} - \left\lbrack {\left( {2 - n}\right) \left( {1 + {n}_{1} + {n}_{2} + {n}_{1}{n}_{2}}\right) }\right. \n\] \n\n\[ \n\left. {-n\left( ... | Yes |
Proposition 3 Let \( G \) be a complete graph on \( n \) vertices and \( m \) edges. Let \( {G}_{1} \) be a \( {r}_{1} \) -regular graph on \( {n}_{1} \) vertices with an adjacency matrix \( A\left( {G}_{1}\right) \) and \( {\operatorname{spec}}_{A}\left( {G}_{1}\right) = \) \( \left\{ {{r}_{1} = {\lambda }_{1}\left( {... | (i) All the roots of the equation\n\n\[ \n{\lambda }^{4} - \left\lbrack {\left( {1 - n}\right) \left( {{n}_{2} + 1}\right) - {n}_{1} - {r}_{1} - {r}_{2} - 5}\right\rbrack {\lambda }^{3} + \left\lbrack \left( {1 + {n}_{1} + {n}_{2} + {n}_{1}{n}_{2}}\right) \right.\n\]\n\n\[ \n\left. {+n\left( {{r}_{1} + {r}_{2} + 4}\rig... | Yes |
Proposition 4 Let \( G \) be a complete graph on \( n \) vertices and \( m \) edges. Let \( {G}_{1} \) be a \( {r}_{1} \) -regular graph on \( {n}_{1} \) vertices with an adjacency matrix \( A\left( {G}_{1}\right) \) and \( {\operatorname{spec}}_{A}\left( {G}_{1}\right) = \) \( \left\{ {{r}_{1} = {\lambda }_{1}\left( {... | (i) All the roots of the equation\n\n\[ \n{\lambda }^{4} - \left\lbrack {\left( {2 - n}\right) \left( {{n}_{2} + 1}\right) - {n}_{1} - {r}_{1} - {r}_{2} - 5}\right\rbrack {\lambda }^{3} - \left\lbrack {\left( {{r}_{1} + {r}_{2} + 4}\right) \left( {1 - n}\right) }\right.\n\]\n\n\[ \n\left. {+\left( {2 - n}\right) \left(... | Yes |
Lemma 1 (Abdirishit) \( {}^{\left\lbrack {13}\right\rbrack } \) Express the matrix \( D \) in the form of \( D = \mathop{\sum }\limits_{{i = 1}}^{K}{D}_{i}, K \in \) \( {N}^{ + } \), we have\n\n\[ \n{\mathrm{e}}^{\left( \tau D\right) } = \mathop{\lim }\limits_{{\sigma \rightarrow \infty }}\mathop{\prod }\limits_{{i = 1... | According to Lemma 1, we have\n\n\[ \n{\mathrm{e}}^{C\tau } \approx \mathop{\prod }\limits_{{i = 1}}^{K}{\mathrm{e}}^{\tau {C}_{i}}\n\] | No |
Theorem 2 There exist \( {u}^{n} \in {Z}_{h}^{0} \) which satisfies (2)-(4). | Proof The theorem is proved by mathematical induction, suppose \( {u}^{0},{u}^{1},\cdots ,{u}^{n} \) , \( n \leq N - 1 \) satisfies (2)-(4). Next, we proof \( {u}^{n + 1} \) also satisfies (2)-(4). Let\n\n\[ g\left( v\right) = {2v} - 2{u}^{n} + 2{v}_{{xx}\bar{x}\bar{x}} - 2{u}_{{xx}\bar{x}\bar{x}}^{n} - \tau {v}_{x\bar... | Yes |
Consider the initial boundary problem of Rosenau-Burgers equation\n\n\\[ \n\\left\\{ \\begin{array}{l} {u}_{t} + {u}_{xxxxt} - {u}_{xx} + {u}_{x} + u{u}_{x} = 0,\\;0 \\leq x \\leq 1,\\;0 \\leq t \\leq {10}, \\\\ u\\left( {x,0}\\right) = {x}^{4}{\\left( 1 - x\\right) }^{4},\\;0 \\leq x \\leq 1, \\\\ u\\left( {0, t}\\rig... | Since we don't know the exact solution of problem. In order to obtain the error estimation, we consider the solution on mesh \\( h = \\frac{1}{160} \\) as the reference solution. The maximum errors from the two schemes in the article are presented in Table 1.\n\nTable 1: The comparison of numerical results of (28) and ... | No |
Theorem 1 The optimal utility is given by\n\n\[ \nV\left( {t, r,\widehat{w}}\right) = \frac{{\widehat{w}}^{1 - \gamma }}{1 - \gamma }{\mathrm{e}}^{{q}_{0}^{L}\left( t\right) + {q}_{1}^{L}\left( t\right) r}, \]\n\n(23)\n\nand the optimal investment strategy is given by\n\n\[ \n{\widehat{u}}^{{L}^{ * }}\left( t\right) = ... | Proof See Appendix A. | No |
Theorem 2 The optimal utility is given by\n\n\[ \nV\left( {t, r,\widehat{w}}\right) \triangleq {V}^{B}\left( {t, r,\widehat{w}}\right) = \frac{{\widehat{w}}^{1 - \gamma }}{1 - \gamma }{\mathrm{e}}^{{q}_{0}^{B}\left( t\right) + {q}_{1}^{B}\left( t\right) r}, \n\]\n\nand the optimal investment strategy is given by\n\n\[ ... | Proof The proof is similar to that of Theorem 1 so it is omitted here. | No |
Theorem 3 Let \( c\left( t\right) \) be a deterministic function of time \( t \) . The optimal utility has the form of\n\n\[ \nV\left( {t, r,\widehat{w}}\right) = \frac{{\widehat{w}}^{1 - \gamma }}{1 - \gamma }{\mathrm{e}}^{{q}_{0}^{L}\left( t\right) + {q}_{1}^{L}\left( t\right) r}, \]\n\n(49)\n\nand the optimal invest... | Proof See Appendix B. | No |
Theorem 4 Let \( c\left( t\right) \) be a deterministic function of time \( t \) . The optimal utility has the form of \[ V\left( {t, r,\widehat{w}}\right) \triangleq {V}^{B}\left( {t, r,\widehat{w}}\right) = \frac{{\widehat{w}}^{1 - \gamma }}{1 - \gamma }{\mathrm{e}}^{{q}_{0}^{B}\left( t\right) + {q}_{1}^{B}\left( t\r... | Proof The proof is similar to that of Theorem 3 so we omit it here. | No |
Theorem 5 When \( \lambda > - 2{\nabla }_{R}^{B}\left( {{\lambda }_{R}{\sigma }_{R} + {\sigma }_{R}^{2}{q}_{1}\left( t\right) }\right) \), we have \( {V}^{L} > {V}^{B} \) . | \( \lambda > - 2{\nabla }_{R}^{B}\left( {{\lambda }_{R}{\sigma }_{R} + {\sigma }_{R}^{2}{q}_{1}\left( t\right) }\right) \) is a sufficient condition to make sure that the value function in the market with the longevity bond is higher than the value function in the market without it, which means that the maximum expecte... | Yes |
Theorem 6 When \( \lambda > - 2{\nabla }_{R}^{B}\left( {{\lambda }_{R}{\sigma }_{R} + {\sigma }_{R}^{2}{q}_{1}\left( t\right) }\right) \), we have \( {V}^{L} > {V}^{B} \) . | When \( \lambda > - 2{\nabla }_{R}^{B}\left( {{\lambda }_{R}{\sigma }_{R} + {\sigma }_{R}^{2}{q}_{1}\left( t\right) }\right) \), the value function in the market with a longevity bond is higher than the value function in the market without it, which means that the maximum expected utility of the terminal wealth in the ... | Yes |
Proposition 1 The problem (16)-(17) and the problem (18)-(19) are equivalent. | Proof First, the problem (18)-(19) and the variational inequality (22) have a unique solution([21], Theorem 1.25). Next, the equivalence proof between the problem (16)-(17) and the problem (18)-(19) ([21], page195, Theorem 5.1). Therein, applying some results from [21] as follows: if \( v \in {H}^{1}\left( \Omega \righ... | Yes |
Theorem 1 Let \( u \in {H}_{g}^{1}\left( \Omega \right) \) be the unique solution of (20), and \( w \in V \) an approximation of \( u \) . Then the following estimate holds for any \( {\mathbf{r}}^{ * } \in {Q}^{ * } \) : | \[ \frac{1}{2}a\left( {u - w, u - w}\right) \] \[ \leq {\int }_{\Omega }{\left| \nabla w + {\mathbf{r}}_{\mathbf{1}}^{ * }\right| }^{2} + \mathop{\inf }\limits_{{{\mathbf{q}}^{ * } \in {Q}_{c}^{ * }}}\left\{ {{\int }_{\Omega }\left\lbrack {{\left| {\mathbf{q}}^{ * } - {\mathbf{r}}^{ * }\right| }^{2} + f\left| w\right| ... | Yes |
Theorem 2 Let \( u \in V \) be the unique solution of (20), and \( w \in V \) an approximation of \( u \) . Then the following estimate holds for any \( {\mathbf{r}}^{ * } \in {Q}^{ * } \) :\n\n\[ \n\frac{1}{2}a\left( {u - w, u - w}\right) \leq {\int }_{\Omega }{\left| \nabla w + {\mathbf{r}}^{ * }\right| }^{2} + \math... | With the special selection \( {\mathbf{r}}^{ * } = - \nabla w \), the error bound (39) leads to the following error estimate\n\n\[ \n\frac{1}{2}a\left( {u - w, u - w}\right) \n\]\n\n\[ \n\leq \mathop{\inf }\limits_{{{\int }_{\Omega }{\mathbf{q}}^{ * } \cdot \nabla v = {\int }_{\Omega }f\left| {v + g}\right| }}\left\{ {... | Yes |
For any optimal schedule of problem \( 1 \mid s \) -batch, \( b < n \mid \left( {{C}_{\max }^{A},\mathop{\sum }\limits_{j}{C}_{j}^{B}}\right) \) , all \( A \) -jobs are processed contiguously in \( \left\lceil \frac{{n}_{A}}{b}\right\rceil \) batches and the \( B \) -jobs are processed in the SPT order. | Without loss of generality, we may regard the batches of agent \( A \) as a single big batch \( {\mathcal{B}}^{A} \) with the processing time\n\n\[ \n{P}_{A} = \left( {\left\lceil \frac{{n}_{A}}{b}\right\rceil - 1}\right) s + \mathop{\sum }\limits_{{j = 1}}^{{n}_{A}}{p}_{j}^{A} \n\]\n\nby Lemma 1. Let \( \sigma \) be a... | Yes |
Lemma 4 For each Pareto optimal point \( \left( {C, T}\right) \) of problem \( P\left( {{C}_{\max }^{A},\mathop{\sum }\limits_{j}{C}_{j}^{B}}\right) \) , there is a corresponding effective Pareto optimal schedule \( \sigma \) such that \( \left( {\sigma ;\left( {C, T}\right) }\right) \in \overrightarrow{\Gamma } \) . | To attain the goal, we need some preparations. We call the algorithm that solves the problem \( P\left( {{C}_{\max }^{B},\mathop{\sum }\limits_{j}{C}_{j}^{B}}\right) \) presented by He et al \( {}^{\left\lbrack {10}\right\rbrack } \) Algorithm \( \mathcal{A}\left( H\right) \) . So we may get all Pareto optimal points b... | Yes |
Theorem 1 For problem \( P\left( {{C}_{\max }^{A},\mathop{\sum }\limits_{j}{C}_{j}^{B}}\right) \), Algorithm POP generates a set of Pareto optimal pairs covering all Pareto optimal points in \( O\left( {{n}_{B}^{3} + {n}_{A}}\right) \) time. | Proof The correctness of Algorithm POP is guaranteed by the above discussion. Step 0 takes \( O\left( {n}_{A}\right) \) time. Step 1 has \( O\left( {n}_{B}\right) \) rounds and each round needs \( O\left( {n}_{B}^{2}\right) \) . So Step 1 needs \( O\left( {n}_{B}^{3}\right) \) time. Step 2 has \( O\left( {n}_{B}\right)... | Yes |
The disease-free steady state \( {E}_{0} = \left( {\frac{\Lambda }{\mu },0,0}\right) \) is locally asymptotically stable if \( {\mathcal{R}}_{0} < 1 \), and unstable if \( {\mathcal{R}}_{0} > 1 \) | By (14), the characteristic equation at \( {E}_{0} \) is\n\n\[ \left( {\lambda + \mu }\right) \left( {f\left( \lambda \right) - 1}\right) = 0 \]\n\nwhere\n\n\[ f\left( \lambda \right) = {\int }_{0}^{\infty }k\left( c\right) {\rho }_{2}\left( c\right) {e}^{-{\lambda c}}\mathrm{\;d}c{\int }_{0}^{\infty }p\left( a\right) ... | Yes |
Theorem 1 The terminal sliding mode dynamics (5) is stable and its state trajectories converge to zero in the finite time \( {T}_{1} \), given by\n\n\[ \n{T}_{1} = \frac{1}{2\lambda }\ln \left\{ {1 + \frac{\lambda }{\mu }\left\lbrack {\mathop{\sum }\limits_{{i = 1}}^{3}{\left( {e}_{i}^{2}\left( 0\right) \right) }^{\mu ... | Proof Choose the Lyapunov function as \( {V}_{1}\left( t\right) = \mathop{\sum }\limits_{{i = 1}}^{3}{e}_{i}^{2}\left( t\right) \) . Its derivative is\n\n\[ \n{\dot{V}}_{1}\left( t\right) = 2\mathop{\sum }\limits_{{i = 1}}^{3}{e}_{i}{\dot{e}}_{i} = 2\mathop{\sum }\limits_{{i = 1}}^{3}{e}_{i}\left\lbrack {{D}_{t}^{1 - q... | Yes |
Theorem 3 Let \( A = \left( {a}_{ij}\right) \in {\mathbb{R}}^{n \times n} \) be a stochastic matrix, taking \( {d}_{i} = {a}_{ii}, i \in N \) , if \( \lambda \neq 1 \) is an eigenvalue of \( A \), then:\n\n(I) If \( {N}_{1} = N \), then \( \left| \lambda \right| \leq \bar{r}\left( A\right) = \operatorname{trace}\left( ... | Proof Since \( {d}_{i} = {a}_{ii}, i \in N \), and \( B = \operatorname{diag}\left\{ {{d}_{1},{d}_{2},\cdots ,{d}_{n}}\right\} e{e}^{T} - {A}^{T} \), we have that for any \( i \in N \), and \( j \neq i \) ,\n\n\[ {b}_{ij} = {a}_{ii} - {a}_{ji} \]\n\n(3)\n\n\[ {b}_{ii} = 0 \]\n\n(4)\n\nFirstly, we consider the following... | Yes |
Theorem 4 Let \( A = \left( {a}_{ij}\right) \in {\mathbb{R}}^{n \times n} \) be a positive stochastic matrix, and for the radii \( {\bar{r}}_{m}\left( A\right) \), we have\n\n\[ \mathop{\lim }\limits_{{m \rightarrow \infty }}{\bar{r}}_{m}\left( A\right) = 0. \] | Proof The proof is similar to that of the Theorem 3.5 in [10]. Since \( A \) is a positive stochastic matrix, then 1 is the spectral radius of \( A \), which is the unique simple dominant eigenvalue. As well known, there exists a limit matrix \( {A}^{\infty } = \left( {\alpha }_{ij}\right) = \) \( \mathop{\lim }\limits... | Yes |
Example 1 Stochastic matrices \( {A}_{1},{A}_{2} \) are the same as in \( \left\lbrack {{10},{12}}\right\rbrack \) . | \[ \n{A}_{1} = \left( \begin{matrix} {0.27} & {0.18} & {0.18} & {0.10} & {0.27} \\ {0.20} & {0.40} & {0.20} & 0 & {0.20} \\ {0.11} & {0.22} & {0.22} & {0.34} & {0.11} \\ {0.06} & {0.25} & {0.31} & {0.19} & {0.19} \\ {0.08} & {0.17} & 0 & {0.42} & {0.33} \end{matrix}\right) ,\;{A}_{2} = \left( \begin{matrix} \frac{1}{4}... | Yes |
\[ {A}_{3} = \left( \begin{matrix} {0.1} & {0.2} & {0.3} & {0.4} \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}\right) . \] | Matrix \( {A}_{3} \) is the example in [10] with \( {x}_{i} = i/{10}, i \in \{ 1,2,3,4\} \) . By computations, \( {v}_{m}\left( {A}_{3}\right) ,{\widetilde{v}}_{m}\left( {A}_{3}\right) \) and \( {\bar{r}}_{m}\left( {A}_{3}\right) \) are shown in Table 3.\n\nTable 3 The values of \( {v}_{m}\left( {A}_{3}\right) ,{\widet... | Yes |
Lemma 1 Let \( {\mathcal{F}}_{t}^{Y} \) denote the filtration generated by \( Y\left( t\right) \) for \( 0 \leq t \leq T \) . For\n\n\[ Y\left( t\right) = \mathop{\sum }\limits_{{j = 0}}^{{{N}_{2}\left( t\right) }}{U}_{2j} \]\n\nwe introduce a new measure \( {Q}_{1} \) equivalent to \( Q \) on \( {\mathcal{F}}_{t}^{Y} ... | Proof For \( 0 \leq t \leq T \), note that\n\n\[ {\mathbb{E}}_{Q}\left\lbrack {\left. \frac{{\Lambda }_{T}}{{\Lambda }_{t}}\right| \;{\mathcal{F}}_{t}^{Y}}\right\rbrack = {\mathbb{E}}_{Q}\left\lbrack {\left. \frac{{\mathrm{e}}^{\beta \left( {Y\left( T\right) - Y\left( t\right) }\right) }{\mathbb{E}}_{Q}\left( {\mathrm{... | Yes |
Theorem 1 Let \( X \) be an rv with pdf \( f \) and twice-differentiable cdf \( F \) . Further, let \( A \) be the event \( \{ a \leq X \leq b\} \) . Then:\n\n(a) The conditional extropy \( J\left( {X \mid A}\right) \) is partially increasing in \( b \) if \( F\left( x\right) \) is log-concave; | Proof (a) We have from (3) that\n\n\[ \frac{\partial J\left( {X \mid A}\right) }{\partial b} = - \frac{f\left( b\right) }{2{\left\lbrack F\left( b\right) - F\left( a\right) \right\rbrack }^{3}}\left\{ {\left\lbrack {F\left( b\right) - F\left( a\right) }\right\rbrack f\left( b\right) - 2{\int }_{a}^{b}{f}^{2}\left( x\ri... | Yes |
Theorem 2 Let \( X \) be an rv with pdf \( f \) and twice-differentiable cdf \( F \) . Further, let \( A \) be the event \( \{ a \leq X \leq b\} \) . Then the conditional Shannon’s differential entropy \( H\left( {X \mid A}\right) \) is a partially increasing function of interval \( \left\lbrack {a, b}\right\rbrack \) ... | Proof If \( f\left( x\right) \) is log-concave, then \( F \) is log-concave. Thus, \( H\left( {X \mid A}\right) \) is a partially increasing function of \( b \) by Lemma 2.2 in Shangari and Chen \( {}^{\left\lbrack {18}\right\rbrack } \) . Moreover, if \( f\left( x\right) \) is \( \log \) -concave, then \( \bar{F} \) i... | Yes |
Lemma 1 Let \( X \) be an rv with pdf \( f \) and twice-differentiable cdf \( F \) . Further, let \( A \) be the event \( \{ a \leq X \leq b\} \) and suppose that \( F\left( b\right) = 1 \) . Then \( \operatorname{CRE}\left( {X \mid A}\right) \) given in (6) is decreasing in \( a \) if \( v\left( {t;b}\right) = {\int }... | Proof Note that (6) can be rewritten as\n\n\[ \n{CRE}\left( {X \mid A}\right) = - \frac{1}{\bar{F}\left( a\right) }\left\lbrack {{\int }_{a}^{b}\bar{F}\left( x\right) \log \bar{F}\left( x\right) \mathrm{d}x - \log \bar{F}\left( a\right) {\int }_{a}^{b}\bar{F}\left( x\right) \mathrm{d}x}\right\rbrack .\n\]\n\nThus\n\n\[... | Yes |
Theorem 3 Let \( X \) be an rv with pdf \( f \) and twice-differentiable cdf \( F \) . Further, let \( A \) be the event \( \{ a \leq X \leq b\} \) . Then \( \operatorname{CRE}\left( {X \mid A}\right) \) given in (5) is a partially decreasing function of \( a \) if \( F\left( x\right) \) is log-concave. | Proof Let \( {X}_{b} \) be an rv with survival function\n\n\[ \bar{G}\left( x\right) = \frac{F\left( b\right) - F\left( x\right) }{F\left( b\right) - F\left( a\right) },\;a \leq x \leq b. \]\n\nNote that \( G\left( b\right) = 1 - \bar{G}\left( b\right) = 1 \) and \( {\int }_{t}^{b}\bar{G}\left( x\right) \mathrm{d}x \) ... | Yes |
Let \( {X}_{1} \) and \( {X}_{2} \) be exponentially distributed rv’s with pdf \( f\left( x\right) = \) \( {\mathrm{e}}^{-x}, x \geq 0 \), which is log-concave. Using (8), we have | \[ g\left( {x, b}\right) = - \frac{{\mathrm{e}}^{x - {2b}} - {\mathrm{e}}^{-x}}{2{\left( 1 - {\mathrm{e}}^{-b}\right) }^{2}},\;x \in \left( {0, b}\right) . \] This implies \[ J\left( {U \mid 0 \leq {X}_{1},{X}_{2} \leq b}\right) = - \frac{1}{2}{\int }_{0}^{b}{g}^{2}\left( {x;b}\right) \mathrm{d}x \] \[ = \frac{1}{{16}{... | Yes |
Theorem 1 Assume that \( s > 5/2, T > 0 \), and \( z = \left( {u, v}\right) \in C\left( {\left\lbrack {0, T}\right\rbrack ;{H}^{s}}\right) \times \) \( C\left( {\left\lbrack {0, T}\right\rbrack ;{H}^{s}}\right) \) is a solution of the initial value problem (1). If the initial date \( {z}_{0}\left( x\right) = \) \( \lef... | \[ \left| {{u}_{0}\left( x\right) }\right| \sim O\left( {\mathrm{e}}^{-\theta \left| x\right| }\right) ,\;\left| {{\partial }_{x}{u}_{0}\left( x\right) }\right| \sim O\left( {\mathrm{e}}^{-\theta \left| x\right| }\right) , \] \[ \left| {{v}_{0}\left( x\right) }\right| \sim O\left( {\mathrm{e}}^{-\theta \left| x\right| ... | Yes |
Theorem 1 Let \( R, S \) be two connected graphs other than one of \( \left\{ {{P}_{4},{P}_{3}}\right\} \) . Then every connected (2-edge-connected or 2-connected) \( \{ R, S\} \) -free graph \( G \) implies \( \chi \left( G\right) = \) \( \varphi \left( G\right) \) if and only if \( \{ R, S\} \preccurlyeq \left\{ {{P}... | ## 2 The proof of Theorem 1\n\nLet's start with the following result.\n\nLemma 1 Every connected | No |
Theorem 4 \( \varphi \left( {I{G}_{n,2}}\right) \geq n - 2 \) . | Proof We give the graph \( I{G}_{n,2} \) a \( \left( {n - 2}\right) \) -coloring \( f : {V}_{1} = \{ \{ 1,3\} ,\{ 1,4\} ,\cdots ,\{ 1, n - \) \( 1\} \} ;{V}_{2} = \{ \{ 2,4\} ,\{ 2,5\} ,\cdots ,\{ 2, n\} \} ;\cdots ;{V}_{n - 2} = \{ \{ n - 2, n\} \} . \n\nFirstly, for any \( A, B \in {V}_{i} \), we have \( A \cap B = \... | Yes |
Theorem 5 \( \varphi \left( {I{G}_{n,3}}\right) \geq n - 4 \) . | Proof We give the graph \( I{G}_{n,3} \) a \( \left( {n - 4}\right) \) -coloring \( f \) :\n\n\[ \n{V}_{1} = \left\{ {\left\{ {1,3,{t}_{1}^{1}}\right\} : 5 \leq {t}_{1}^{1} \leq n - 1}\right\} \cup \left\{ {\left\{ {1,4,{t}_{2}^{1}}\right\} : 6 \leq {t}_{2}^{1} \leq n - 1}\right\} \cup \cdots \cup \{ 1, n - 3, n - 1\} ... | Yes |
Consider the following function: \( \varphi : \mathbf{R} \rightarrow \mathbf{R} \), defined by\n\n\[ \varphi \left( x\right) = \left\{ \begin{matrix} {2x}, & \text{ if }x \geqq 0 \\ x - {x}^{2}, & \text{ if }x < 0 \end{matrix}\right. \]\n\nIt is easy to verify that \( \varphi \) is 1-order strong pseudoconvex of type I... | does not hold for any \( c > 0 \) . | Yes |
Example 2 Consider the function \( f : \mathbf{R} \rightarrow {\mathbf{R}}^{2}, f\left( x\right) = \left( {{f}_{1}\left( x\right) ,{f}_{2}\left( x\right) }\right) \) for \( x \in \mathbf{R} \) , defined as \[ {f}_{1}\left( x\right) = \left\{ {\begin{matrix} - {3x}, & \text{ if }x \leqq 0, \\ - \sin x, & \text{ if }x > ... | By a direct calculation, we get that \( {\partial }^{c}{f}_{1}\left( 0\right) = \left\lbrack {-3, - 1}\right\rbrack \) and \( {\partial }^{c}{f}_{2}\left( 0\right) = \left\lbrack {-2,0}\right\rbrack \) . Then, we verify that \( f \) is 1-order strong pseudoconvex of type I at \( \bar{x} = 0 \) with \( \alpha = \left( {... | Yes |
Theorem 1 Let \( \bar{x} \in X \) and \( {f}_{i}, i = 1,2,\cdots, p \), be regular on \( X \) . If \( \bar{x} \) is a strict minimizer of order \( m \) to the (NMOP), then \( \bar{x} \) solves the (WVVIP). | Proof We proceed by contradiction. Suppose that \( \bar{x} \) is a strict minimizer of order \( m \) to the (NMOP) but is not a solution of (WVVIP). Therefore, there exists an \( \widehat{x} \in X \) such that\n\n\[ \langle \xi ,\widehat{x} - \bar{x}\rangle < 0,\;\forall \xi \in {\partial }^{c}f\left( \bar{x}\right) ,\... | Yes |
Theorem 2 Let \( \bar{x} \in X \) . If \( f \) is \( m \) -order strong pseudoconvex function of type I at \( \bar{x} \) and \( \bar{x} \) solves the (WVVIP), then \( \bar{x} \) is a strict minimizer of order \( m \) to the (NMOP). | Proof Suppose to the contrary that \( \bar{x} \) is not a strict minimizer of order \( m \) to the (NMOP). Then, for any \( \alpha \in \operatorname{int}\left( {\mathbf{R}}_{ + }^{p}\right) \), there exists \( x \in X \) such that\n\n\[ f\left( x\right) < f\left( \bar{x}\right) + \alpha \parallel x - \bar{x}{\parallel ... | Yes |
Theorem 3 Let \( \bar{x} \in X \) be an vector critical point to the (NMOP). If \( f : X \rightarrow {\mathbf{R}}^{p} \) is \( m \) -order strong pseudoconvex function of type I at \( \bar{x} \), then \( \bar{x} \) is a strict minimizer of order \( m \) to the (NMOP). | Proof Since \( \bar{x} \) is a vector critical point to the (NMOP), there exists \( \beta \in {\mathbf{R}}^{p} \) with \( \beta \geq 0 \) and a vector \( \xi \in {\partial }^{c}f\left( \bar{x}\right) \) such that\n\n\[ \langle \beta ,\xi \rangle = 0 \]\n\n(4)\n\n\n\nIf \( \bar{x} \) is not a strict minimizer of order \... | Yes |
Theorem 4 Any vector critical point is a strict minimizer of order \( m \) to the (NMOP), if and only if \( f : X \rightarrow {\mathbf{R}}^{p} \) is a \( m \) -order strongly pseudoconvex type I function at that point. | Proof The sufficiency yields from Theorem 3. It is only needed to prove that if every vector critical point is a strict minimizer of order \( m \) to the (NMOP), then the vector valued function \( f \) satisfies the condition for \( m \) -order strongly pseudoconvexity of type I at that point.\n\nSuppose that \( \bar{x... | Yes |
Corollary 2 If \( {f}_{i}, i = 1,2,\cdots, p \), are regular and \( m \) -order strongly pseudoconvex functions of type I on \( X \), then the vector critical points, the strict minimizer of order \( m \) to the (NMOP), and the solutions of (WVVIP) are equivalent. | ## 4 Conclusions\n\nWe have introduced a new generalization of higher-order strong pseudoconvexity for the Lischitz functions, called \( m \) -order strongly pesudoconvex functions of type I. Examples are presented to illustrate its existence. We have examined the relations between vector variational inequality problem... | No |
Lemma 1. Suppose \( \mathcal{F} \) is a nonempty collection of subsets of a set \( X \) such that the union of every subchain of \( \mathcal{F} \) belongs to \( \mathcal{F} \) . Suppose \( g \) is a function which associates to each \( A \in \mathcal{F} \) a set \( g\left( A\right) \in \mathcal{F} \) such that \( A \su... | Proof. Let \( {A}_{0} \in \mathcal{F} \) . Call a subcollection \( {\mathcal{F}}^{\prime } \subset \mathcal{F} \) a tower if \( {A}_{0} \in {\mathcal{F}}^{\prime } \), the union of every subchain of \( {\mathcal{F}}^{\prime } \) is in \( {\mathcal{F}}^{\prime } \), and \( g\left( A\right) \in {\mathcal{F}}^{\prime } \)... | Yes |
Lemma 2. For each \( s \in \left( {0,1}\right) \) there exists a constant \( {k}_{s} \) such that\n\n\[ \n{P}_{r}\left( {\theta - \phi }\right) \leq {k}_{s}{P}_{r}\left( {-\phi }\right) \n\]\n\nfor all \( \left( {r,\theta }\right) \) such that \( r{e}^{i\theta } \in {\Gamma }_{s} \) . | Proof. By elementary arithmetic,\n\n\[ \n{P}_{r}\left( {\theta - \phi }\right) = \frac{1 - {r}^{2}}{{\left| {e}^{i\phi } - r{e}^{i\theta }\right| }^{2}} \n\]\n\nand\n\n\[ \n{P}_{r}\left( {-\phi }\right) = \frac{1 - {r}^{2}}{{\left| {e}^{i\phi } - r\right| }^{2}}. \n\]\n\n(This alternate formula can be found in any comp... | Yes |
Lemma 3. Let \( {\left\{ {b}_{n}\right\} }_{n = 0}^{\infty } \) be a sequence of complex numbers. Then\n\n\[ \sqrt{\mathop{\sum }\limits_{{n = 0}}^{\infty }{\left| {b}_{n}\right| }^{2}} = \mathop{\sup }\limits_{{\sum {\left| {a}_{n}\right| }^{2} \leq 1}}\left| {\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{b}_{n}}\r... | Proof. If \( \sum {\left| {b}_{n}\right| }^{2} < \infty \) this follows from the Cauchy-Schwarz inequality applied to \( {\ell }^{2}\left( \mathbb{N}\right) \), where equality is achieved when \( \left\{ {a}_{n}\right\} \) is the unit vector in the same direction (actually, the conjugate) as \( \left\{ {b}_{n}\right\} ... | Yes |
Lemma 4. If \( {\mu }_{F} \) is the Borel measure corresponding to the increasing, right-continuous function \( F \), then for any \( \mu \) -measurable set \( E \) , | \[ \mu \left( E\right) = \mathop{\inf }\limits_{{E \subset \cup \left( {{a}_{j},{b}_{j}}\right) }}\sum \mu \left( \left( {{a}_{j},{b}_{j}}\right) \right) . \] In words, this lemma says that it is equivalent to use coverings of open intervals instead of half-open intervals. This is nice because it enables us to use theo... | No |
Lemma 1. Let \( x, y \in \operatorname{End}\left( \mathrm{V}\right) \). (i) Assume that \( \mathrm{V} \) is finite dimensional. Then \( x \) is triangularizable if and only if \( \mathrm{V} = \mathop{\sum }\limits_{{a \in k}}{\mathrm{\;V}}^{a}\left( x\right) \) . | Part (i) follows from Algebra, Chap. VII, §5, no. 2, Prop. 3. | No |
Lemma 2. Assume that \( k \) is of characteristic 0 . Let \( \mathfrak{g} \) be a semi-simple Lie algebra over \( k \), B the Killing form of \( \mathfrak{g},\mathfrak{m} \) a subalgebra of \( \mathfrak{g} \) . Assume that the following conditions are satisfied:\n\n1) the restriction of \( \mathrm{B} \) to \( \mathfrak... | By Chap. I, \( §6 \), no. 4, Prop. \( {5d} \) ), \( \mathfrak{m} \) is reductive. Let \( \mathfrak{c} \) be the centre of \( \mathfrak{m} \) . If \( x \in \mathfrak{c} \) is nilpotent, then \( x = 0 \) ; indeed, for all \( y \in \mathfrak{m} \), ad \( x \) and ad \( y \) commute, their composition ad \( x \circ \operat... | Yes |
PROPOSITION 3. Let \( \mathfrak{g} \) be a Lie algebra, \( \mathfrak{h} \) a subalgebra of \( \mathfrak{g} \), and \( {k}^{\prime } \) an extension of \( k \) . Then \( \mathfrak{h} \) is a Cartan subalgebra of \( \mathfrak{g} \) if and only if \( \mathfrak{h}{ \otimes }_{k}{k}^{\prime } \) is a Cartan subalgebra of \(... | Indeed, \( \mathfrak{h} \) is nilpotent if and only if \( \mathfrak{h}{ \otimes }_{k}{k}^{\prime } \) is (Chap. I,§4, no. 5). On the other hand, if \( \mathfrak{n} \) is the normalizer of \( \mathfrak{h} \) in \( \mathfrak{g} \), the normalizer of \( \mathfrak{h}{ \otimes }_{k}{k}^{\prime } \) in \( \mathfrak{g}{ \otim... | No |
Lemma 2. (ii) Put \( \mathrm{R} = \left\{ {{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{p}}\right\} \) where the \( {\lambda }_{i} \) are mutually distinct. Let \( \mathrm{F} \) be the map from \( {\mathfrak{g}}^{0}\left( \mathfrak{h}\right) \times {\mathfrak{g}}^{{\lambda }_{1}}\left( \mathfrak{h}\right) \times \... | We prove (ii). Let \( n = \dim \mathfrak{g} \) . If \( \lambda \in \mathrm{R} \) and \( x \in {\mathfrak{g}}^{\lambda }\left( \mathfrak{h}\right) \) , we have \( {\left( \operatorname{ad}x\right) }^{n} = 0 \) . It follows that \( \left( {y, x}\right) \mapsto {e}^{\operatorname{ad}x}y \) is a polynomial map from \( \mat... | Yes |
Lemma 3. Let \( \mathfrak{g} \) be a Lie algebra, \( \mathfrak{r} \) its radical, \( \varphi \) the canonical homomorphism from \( \mathfrak{g} \) to \( \mathfrak{g}/\mathfrak{r}, v \) an elementary automorphism of \( \mathfrak{g}/\mathfrak{r} \) . There exists an elementary automorphism \( u \) of \( \mathfrak{g} \) s... | We can assume that \( v \) is of the form \( {e}^{\operatorname{ad}b} \), where \( b \in \mathfrak{g}/\mathfrak{r} \) and ad \( b \) is nilpotent. Let \( \mathfrak{s} \) be a Levi subalgebra of \( \mathfrak{g} \) (Chap. I,§6, no. 8, Def. 7) and let \( a \) be the element of \( \mathfrak{s} \) such that \( \varphi \left... | Yes |
(i) The function \( {r}_{a} \) is upper semi-continuous. | (i) If \( {r}_{a}\left( x\right) = i \), then \( {a}_{i}\left( x\right) \neq 0 \) and, for all \( y \) in a neighbourhood of \( x \), we have \( {a}_{i}\left( y\right) \neq 0 \), so \( {r}_{a}\left( y\right) \leq i \) . | Yes |
Lemma 2. Let \( 0 \rightarrow {\mathrm{V}}^{\prime } \rightarrow \mathrm{V} \rightarrow {\mathrm{V}}^{\prime \prime } \rightarrow 0 \) be an exact sequence of \( \mathrm{G} \) -modules defined by analytic linear representations \( {\rho }^{\prime },\rho ,{\rho }^{\prime \prime } \) of \( \mathrm{G} \), respectively. Th... | Indeed, for all \( g \in \mathrm{G} \), there is \( \left( {§1\text{, no. 1, Cor. 3 of Th. 1 }}\right) \) an exact sequence\n\n\[ \n0 \rightarrow {\left( {\mathrm{V}}^{\prime }\right) }^{1}\left( {{\rho }^{\prime }\left( g\right) }\right) \rightarrow {\mathrm{V}}^{1}\left( {\rho \left( g\right) }\right) \rightarrow {\l... | Yes |
Lemma 3. Let \( a \in \mathrm{G} \) and let \( \mathfrak{m} \) be a complement of \( {\mathfrak{g}}^{1}\left( a\right) \) in \( \mathfrak{g} \) . Let \( \mathrm{U} \) be a neighbourhood of 0 in \( \mathfrak{g} \) and \( \exp \) an exponential map from \( \mathrm{U} \) to \( \mathrm{G} \) . The map \[ f : \left( {x, y}\... | The tangent linear maps at 0 of the maps \( x \mapsto a\left( {\exp x}\right) \) and \( y \mapsto \) \( \left( {\exp y}\right) a{\left( \exp y\right) }^{-1} \) are the maps \( x \mapsto {ax} \) and \( y \mapsto {ya} - {ay} = a\left( {{a}^{-1}{ya} - y}\right) \) from \( \mathfrak{g} \) to \( {\mathrm{T}}_{a}\mathrm{G} =... | Yes |
Lemma 4. Let \( \mathrm{U} \) be a neighbourhood of 0 in \( \mathfrak{g} \) and \( \exp \) an exponential map from \( \mathrm{U} \) to \( \mathrm{G} \) ,étale at every point of \( \mathrm{U} \) and such that \( {\mathfrak{g}}^{1}\left( {\exp x}\right) = {\mathfrak{g}}^{0}\left( x\right) \) for all \( x \in \mathrm{U} \... | Let \( l = \operatorname{rk}\left( \mathfrak{g}\right) \) . If \( x \in \mathrm{U} \) is a regular element of \( \mathfrak{g} \) , \[ {r}_{\mathrm{{Ad}}}\left( {\exp x}\right) = \dim {\mathfrak{g}}^{1}\left( {\exp x}\right) = \dim {\mathfrak{g}}^{0}\left( x\right) = l. \] Since the regular elements of \( \mathfrak{g} \... | Yes |
Lemma 1. Let \( \mathrm{X} \) be a connected topological space and \( \Omega \) a dense open subset of \( \mathrm{X} \). If, for any \( x \in \mathrm{X} \), there exists a neighbourhood \( \mathrm{V} \) of \( x \) such that \( \mathrm{V} \cap \Omega \) is connected, then \( \Omega \) is connected. | Indeed, let \( {\Omega }_{0} \) be a non-empty open and closed subset of \( \Omega \) . Let \( x \in \mathrm{X} \) and let \( \mathrm{V} \) be a neighbourhood of \( x \) such that \( \mathrm{V} \cap \Omega \) is connected. If \( x \in {\bar{\Omega }}_{0} \) ,\n\n\[ \left( {\mathrm{V} \cap \Omega }\right) \cap {\Omega }... | Yes |
Lemma 2. Let \( \mathrm{U} \) be an open ball in \( {\mathbf{C}}^{n} \) and \( f : \mathrm{U} \rightarrow \mathbf{C} \) a holomorphic function, not identically zero. Let \( \mathrm{A} \) be a subset of \( \mathrm{U} \) such that \( f = 0 \) on \( \mathrm{A} \) . Then \( \mathrm{U} - \mathrm{A} \) is dense in \( \mathrm... | The density of U- A follows from Differentiable and Analytic Manifolds, Results,3.2.5. Assume first that \( n = 1 \) . If \( a \in \mathrm{A} \), the power series expansion of \( f \) about a (Differentiable and Analytic Manifolds, Results,3.2.1) is not reduced to 0, and it follows that there exists a neighbourhood \( ... | Yes |
Lemma 3. Let \( \mathrm{X} \) be a finite dimensional connected complex-analytic manifold and let \( \mathrm{A} \) be a subset of \( \mathrm{X} \) satisfying the following condition:\n\nFor any \( x \in \mathrm{X} \), there exists an analytic function germ \( {f}_{x} \), not vanishing at \( x \), such that the germ of ... | The density of X-A follows from Differentiable and Analytic Manifolds, Results, 3.2.5. We can assume that A is closed (General Topology, Chap. I,\n\n\( §{11} \), no. 1, Prop. 1). For any \( x \in \mathrm{X} \), there exists an open neighbourhood \( \mathrm{V} \) of \( x \) and an isomorphism \( c \) from \( \mathrm{V} ... | Yes |
Lemma 1. Let \( \mathrm{A} \) be an associative algebra over \( k, H \) and \( X \) elements of \( \mathrm{A} \) such that \( \left\lbrack {H, X}\right\rbrack = {2X} \). (i) \( \left\lbrack {H,{X}^{n}}\right\rbrack = {2n}{X}^{n} \) for any integer \( n \geq 0 \). (ii) If \( Z \) is an element of \( \mathrm{A} \) such t... | The map \( T \mapsto \left\lbrack {H, T}\right\rbrack \) from A to A is a derivation, which implies (i). With the assumptions in (ii), \[ \left\lbrack {Z,{X}^{n}}\right\rbrack = \mathop{\sum }\limits_{{i + j = n - 1}}{X}^{i}H{X}^{j} \] \[ = \mathop{\sum }\limits_{{i + j = n - 1}}\left( {{X}^{i}{X}^{j}H + {X}^{i}{2j}{X}... | Yes |
Lemma 2. In the enveloping algebra of \( \mathfrak{{sl}}\left( {2, k}\right) \) , \n\n\[ \n\left\lbrack {H,{X}_{ + }^{n}}\right\rbrack = {2n}{X}_{ + }^{n}\;\left\lbrack {H,{X}_{ - }^{n}}\right\rbrack = - {2n}{X}_{ - }^{n} \n\] \n\nfor any integer \( n \geq 0 \), and \n\n\[ \n\left\lbrack {{X}_{ - },{X}_{ + }^{n}}\right... | The first and third relations follow from Lemma 1. The others can be deduced from them by using the canonical involution of \( \mathfrak{{sl}}\left( {2, k}\right) \) . | No |
Lemma 3. If \( x \) is an element of weight \( \lambda \), then \( {X}_{ + }x \) is an element of weight \( \lambda + 2 \) and \( {X}_{ - }x \) is an element of weight \( \lambda - 2 \) . | Indeed, \( H{X}_{ + }x = \left\lbrack {H,{X}_{ + }}\right\rbrack x + {X}_{ + }{Hx} = 2{X}_{ + }x + {X}_{ + }{\lambda x} = \left( {\lambda + 2}\right) {X}_{ + }x \), and similarly \( H{X}_{ - }x = \left( {\lambda - 2}\right) {X}_{ - }x \) (cf. also Chap. VII,§1, no. 3, Prop. 10 (ii)). | Yes |
Lemma 4. Let \( \mathrm{E} \) be a non-zero finite dimensional \( \mathfrak{{sl}}\left( {2, k}\right) \) -module. Then \( \mathrm{E} \) has primitive elements. | Since \( {X}_{ + } \) is a nilpotent element of \( \mathfrak{{sl}}\left( {2, k}\right) ,{X}_{+\mathrm{E}} \) is nilpotent. Assume that \( {X}_{+\mathrm{E}}^{m - 1} \neq 0 \) and \( {X}_{+\mathrm{E}}^{m} = 0 \) . By Lemma 2,\n\n\[ m\left( {{H}_{\mathrm{E}} - m + 1}\right) {X}_{+\mathrm{E}}^{m - 1} = \left\lbrack {{X}_{-... | Yes |
Lemma 1. (i) For all \( h \in \mathfrak{h},{\theta }_{\alpha }\left( t\right) \) . \( h = h - \alpha \left( h\right) {H}_{\alpha } \) . | Let \( h \in \mathfrak{h} \) . If \( \alpha \left( h\right) = 0,\left\lbrack {{X}_{\alpha }, h}\right\rbrack = \left\lbrack {{X}_{-\alpha }, h}\right\rbrack = 0 \), so \( {\theta }_{\alpha }\left( t\right) .h = h \) . On the other hand, the formulas (5) of \( §1 \), no. 5 show that \( {\theta }_{\alpha }\left( t\right)... | Yes |
Lemma 2. There exists a family \( {\left( {X}_{\alpha }\right) }_{\alpha \in \mathrm{R}} \) such that, for all \( \alpha \in \mathrm{R} \), \[ {X}_{\alpha } \in {\mathfrak{g}}^{\alpha }\text{ and }\left\lbrack {{X}_{\alpha },{X}_{-\alpha }}\right\rbrack = - {H}_{\alpha }.\] | Let \( {\mathrm{R}}_{1} \) be a subset of \( \mathrm{R} \) such that \( \mathrm{R} = {\mathrm{R}}_{1} \cup \left( {-{\mathrm{R}}_{1}}\right) \) and \( {\mathrm{R}}_{1} \cap \left( {-{\mathrm{R}}_{1}}\right) = \varnothing \) . For \( \alpha \in {\mathrm{R}}_{1} \), choose an arbitrary non-zero element \( {X}_{\alpha } \... | Yes |
Lemma 3. For all \( \alpha \in \mathrm{R} \) , \n\n\[ \n\left\langle {{X}_{\alpha },{X}_{-\alpha }}\right\rangle = - \frac{1}{2}\left\langle {{H}_{\alpha },{H}_{\alpha }}\right\rangle \n\] | Indeed, \n\n\[ \n2\left\langle {{X}_{\alpha },{X}_{-\alpha }}\right\rangle = \left\langle {\alpha \left( {H}_{\alpha }\right) {X}_{\alpha },{X}_{-\alpha }}\right\rangle = \left\langle {\left\lbrack {{H}_{\alpha },{X}_{\alpha }}\right\rbrack ,{X}_{-\alpha }}\right\rangle \n\] \n\n\[ \n= \left\langle {{H}_{\alpha },\left... | Yes |
Lemma 4. Let \( \alpha ,\beta \in \mathrm{R} \) be such that \( \alpha + \beta \in \mathrm{R} \) . Let \( p \) (resp. \( q \) ) be the largest integer \( j \) such that \( \beta + {j\alpha } \in \mathrm{R} \) (resp. \( \beta - {j\alpha } \in \mathrm{R} \) ). Then,\n\n\[ \n{\mathrm{N}}_{\alpha ,\beta }{\mathrm{N}}_{-\al... | Let \( \rho \) be the representation of \( \mathfrak{{sl}}\left( {2, k}\right) \) on \( \mathfrak{g} \) defined by \( {X}_{\alpha } \) . The element \( e = {X}_{\beta + {p\alpha }} \) is primitive of weight \( p + q \) (Prop. 4 (i)). Put\n\n\[ \n{e}_{n} = \frac{{\left( -1\right) }^{n}}{n!}\rho {\left( {X}_{ - }\right) ... | Yes |
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