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Lemma 3.5. The order of any element in\n\n\[ \n{\left. \operatorname{Ker}\left( {\psi }^{p + 1} - 1\right) \right| }_{B{P}_{q{p}^{j + 1} - 2}\left( {P \land P}\right) }\n\]\n\ndivides \( {p}^{j + 2} \) . | 2304_Topology and Its Applications 2000-03-03_ Vol 101 Iss 3 | 2 | 11 | Algebra | Group theory and generalizations |
Theorem 17.1 Under assumption 1 and 2, the channel capacity of the genetic code is 2.92 Shannon bits. | 9381_Cryptography_ information theory_ and error-correction_ a handbook for the 21st century | 10 | 47 | Computer Science and Engineering | Information and communication, circuits |
Corollary 1.1.9 If \( L, M \) are finite separable extensions of \( k \), their compositum is separable as well. | 1509_Galois groups and fundamental groups | 2 | 5 | Algebra | Field theory and polynomials |
Proposition 1. Let the function \( u = u\left( x\right) \in {C}^{2}\left( {B}_{R}\right) \) be given, and its Fourier coefficients \( {f}_{kl}\left( r\right) \) are defined due to the formula (26). Then the Fourier coefficients \( {\widetilde{f}}_{kl}\left( r\right) \) of \( {\Delta u} \), namely\n\n\[ \n{\widetilde{f}}_{kl}\left( r\right) \mathrel{\text{:=}} {\int }_{\left| \eta \right| = 1}{\Delta u}\left( {r\eta }\right) {H}_{kl}\left( \eta \right) {d\sigma }\left( \eta \right) ,\;k = 0,1,2,\ldots ,\;l = 1,\ldots, N\left( {k, n}\right) ,\n\]\n\nsatisfy the identity\n\n\[ \n{\widetilde{f}}_{kl}\left( r\right) = {L}_{k, n}{f}_{kl}\left( r\right) ,\;k = 0,1,2,\ldots ,\;l = 1,\ldots, N\left( {k, n}\right) , \tag{28}\n\]\n\nwith \( 0 \leq r < R \) . | 22824_Partial Differential Equations_ Foundations and Integral Representations Part 1 | 5 | 29 | Analysis | Harmonic analysis on Euclidean spaces |
Proposition 6.9. For an equilibrium point \( \left( {{x}_{e},{y}_{e}}\right) \) as defined by Prop. 6.2 if the following equations hold:\n\n\[ \left\{ \begin{array}{l} \frac{\partial m}{\partial x} + \left( \frac{{y}_{e} + 1}{{x}_{e} + F}\right) \frac{\partial m}{\partial y} < 0 \\ \frac{\partial m}{\partial x} + \left( \frac{{y}_{e} + 1}{{x}_{e} + F}\right) \frac{\partial m}{\partial y} < 0 \end{array}\right. \tag{18} \]\n\nthen \( \left( {{x}_{e},{y}_{e}}\right) \) is asymptotically stable. | 12899_Analytical and Stochastic Modeling Techniques and Applications_ 16th International Conference_ ASMTA | 6 | 36 | Differential Equations and Dynamical Systems | Partial differential equations |
Lemma 7.5. Let \( m \) be a nonnegative real number.\n\n(a) The following equality holds for every positive real number \( c \) :\n\n\[ 2{c}^{2}{\int }_{0}^{1}r{\left| {J}_{m}\left( cr\right) \right| }^{2}{dr} = {c}^{2}{\left| {J}_{m}^{\prime }\left( c\right) \right| }^{2} + \left( {{c}^{2} - {m}^{2}}\right) {\left| {J}_{m}\left( c\right) \right| }^{2}. \tag{7.5} \] | 11657_Fourier_series_in_control | Unknown | Unknown | Unknown | Unknown |
Lemma 11.3. The operation \( \odot \) on \( {\mathbb{Z}}_{n} \) is associative and commutative and has [1] as an identity element. | 175_Modern Algebra_ An Introduction_ Sixth Edition | 2 | 6 | Algebra | Commutative algebra |
Lemma 9.1.8. [28] Let \( \left( {X, q{p}_{b}}\right) \) be a quasi-partial \( b \) -metric space and \( \left( {X,{d}_{q{p}_{b}}}\right) \) be the corresponding \( b \) -metric space. Then \( \left( {X,{d}_{q{p}_{b}}}\right) \) is complete if \( \left( {X, q{p}_{b}}\right) \) is complete. | 25453_Advances in Mathematical Analysis and its Applications | 5 | 28 | Analysis | Sequences, series, summability |
Problem 12.3. Let the process \( X\\left( t\\right), t \\in \\left\\lbrack {0, T}\\right\\rbrack \\), be the geometric Lévy pro-\n\nprocess\n\n\\[ \n{dX}\\left( t\\right) = X\\left( {t}^{ - }\\right) \\left\\lbrack {\\alpha \\left( t\\right) {dt} + \\beta \\left( t\\right) {dW}\\left( t\\right) + {\\int }_{{\\mathbb{R}}_{0}}\\gamma \\left( {t, z}\\right) \\widetilde{N}\\left( {{dt},{dz}}\\right) }\\right\\rbrack \n\\]\n\nwhere the involved coefficients are deterministic. Find \( {D}_{t, z}X\\left( T\\right) \) for \( t \\leq T \) . | 35342_Malliavin calculus for Lévy processes with applications to finance _ Lévy过程的Malliavin分析及其在金融学中的应用 | 9 | Unknown | Probability and Statistics | Unknown |
Corollary 7.1 (Odd prime \( p \) ) For idempotents \( e \in {Z}_{p - 1} \) as naturals \( e < p : {e}^{p} \equiv \) \( e{\;\operatorname{mod}\;{p}^{3}} \Rightarrow e = 1 \) . | 22305_Associative Digital Network Theory_ An Associative Algebra Approach to Logic_ Arithmetic and State M | 3 | 13 | Number Theory | Number theory |
Theorem 7. If (W2), (W3) and (W4) hold, then eq. (50) has a nontrivial, finite energy solution. This solution has radial symmetry, namely\n\n\[ \n\mathbf{A}\left( x\right) = {g}^{-1}\mathbf{A}\left( {gx}\right) \;\forall g \in O\left( 3\right) \n\]\n\nwhere \( O\left( 3\right) \) is the orthogonal group in \( {\mathbf{R}}^{3} \) . | 12335_Contributions to Nonlinear Analysis_ A Tribute to D_G_ de Figueiredo on the Occasion of his 70th Bir | 6 | 36 | Differential Equations and Dynamical Systems | Partial differential equations |
Example 20.1.2 (Replacing a variable by a constant) Consider again the skeleton class declaration in Table 20.1. If none of \( {\mathrm{C}}_{1},\ldots ,{\mathrm{C}}_{k} \) contains a command of the form \( \mathrm{X} \mathrel{\text{:=}} \mathrm{E} \), then replacing each occurrence of the expression \( \mathrm{X} \) in \( {\mathrm{C}}_{1},\ldots ,{\mathrm{C}}_{k} \) by 0 (the initial value of \( \mathrm{X} \) ) and deleting the declaration of \( \mathrm{X} \) should not change the behaviour of objects of the class, and hence should not affect the behaviour of any program in which the class appears. | 18074_The Pi Calculus | Unknown | Unknown | Unknown | Unknown |
Proposition 7.5. The Lie algebras \( \overline{gl}\left( \infty \right) \) and \( \widehat{gl}\left( \infty \right) \) with the above introduced degree are graded Lie algebras. The algebra \( \overline{gl}\left( \infty \right) \) is also graded as associative algebra. | 8322_Krichever-Novikov Type Algebras Theory and Applications | 2 | 9 | Algebra | Nonassociative rings and algebras |
Theorem 1.10. A finitistic ANRU \( X \) is dominated in uniform homotopy by a uniform polyhedron \( P \) . Moreover, if \( \Delta \mathrm{d}X \leq n \), then \( P \) can be chosen so that \( \dim P \leq n \) . Here \( \Delta \mathrm{d}X \) denotes the uniform dimension of \( X \) (see [3]). | 23563_Topology and Its Applications 2001-06-29_ Vol 113 Iss 1-3 | 4 | 21 | Geometry and Topology | Manifolds and cell complexes |
Corollary 2.11. For any invariant point selection \( \mathbf{X} \), the adjoint functors \( \mathcal{C} \) and \( \mathcal{S} \) restrict to an equivalence between the category XSO of X-sober spaces and the category XVS of X- \( \bigvee \) -spatial lattices. | 23554_Topology and its Applications 2004-02-28_ Vol 137 Iss 1-3 | 2 | 4 | Algebra | General algebraic systems |
Theorem 6.26. Let \( u,{u}^{\prime } \) be primitive elements of \( F\left( {a, b}\right) \), and set \( \bar{u} = {u}^{\pi } \) , \( \overline{{u}^{\prime }} = {u}^{\prime \pi } \) for the projections. Then\n\n\[ \n\bar{u} = \overline{{u}^{\prime }} \Leftrightarrow u,{u}^{\prime }\text{ are conjugate in }F\left( {a, b}\right) .\n\] | 27350_Markov_s theorem and 100 years of the uniqueness conjecture_ a mathematical journey from irrational | 2 | 5 | Algebra | Field theory and polynomials |
Lemma 4.1. Let \( C \in {\mathcal{M}}_{d}\left( \mathbb{R}\right) \) be a symmetric positive definite matrix, let \( G \sim \) \( {\mathcal{N}}_{d}\left( {0, C}\right) \), and let \( \phi ,\psi : {\mathbb{R}}^{d} \rightarrow \mathbb{R} \) be two Lipschitz and \( {\mathcal{C}}^{1} \) functions. Then\n\n\[ \n\operatorname{Cov}\left( {\phi \left( G\right) ,\psi \left( G\right) }\right) = {\int }_{0}^{1}E{\left\langle \sqrt{C}\nabla \phi \left( {G}_{\alpha }\right) ,\sqrt{C}\nabla \psi \left( {H}_{\alpha }\right) \right\rangle }_{{\mathbb{R}}^{d}}{d\alpha }, \tag{4.7} \n\] \n\nwhere \n\n\[ \n\left( {{G}_{\alpha },{H}_{\alpha }}\right) \sim {\mathcal{N}}_{2d}\left( {0,\left( \begin{matrix} C & {\alpha C} \\ {\alpha C} & C \end{matrix}\right) }\right) ,\;0 \leq \alpha \leq 1. \n\] | 27353_Selected Aspects of Fractional Brownian Motion | 9 | 44 | Probability and Statistics | Probability theory and stochastic processes |
Example 5.17 Consider again the Bayesian network in Figure 5.8a. Any of the cluster-trees in Figure 5.9 describes a partition of variables into clusters. We can now place each input function into a cluster that contains its scopes, and verify that each is a legitimate tree decomposition. For example, Figure 5.9c shows a cluster-tree decomposition with two vertices, and labeling \( \chi \left( 1\right) = \) \( \{ G, F\} \) and \( \chi \left( 2\right) = \{ A, B, C, D, F\} \) . Any function with scope \( \{ G\} \) must be placed in vertex 1 because vertex 1 is the only vertex that contains variable \( G \) (placing a function having \( G \) in its scope in another vertex will force us to add variable \( G \) to that vertex as well). Any function with scope \( \{ A, B, C, D\} \) or one of its subsets must be placed in vertex 2, and any function with scope \( \{ F\} \) can be placed either in vertex 1 or 2 . Notice that the tree-decomposition at Figure 5.9a is actually a bucket-tree. | 16173_Reasoning with probabilistic and deterministic graphical models_ exact algorithms | 8 | 43 | Optimization and Control | Operations research, mathematical programming |
Theorem 9 [35,36] Suppose \( c \in {\mathbb{R}}^{\ell }\langle \langle X\rangle \rangle \) has coefficients which satisfy\n\n\[ \left| \left( {c,\eta }\right) \right| \leq {K}_{c}{M}_{c}^{\left| \eta \right| },\;\forall \eta \in {X}^{ * }.\]\n\nThen there exists a real number \( \widehat{R} > 0 \) such that for each \( \widehat{u} \in {B}_{\infty }^{m + 1}\left\lbrack 1\right\rbrack \left( \widehat{R}\right) \), the series (57) converges absolutely for any \( N \geq 1 \) . | 1529_Discrete Mechanics_ Geometric Integration and Lie_Butcher Series_ DMGILBS_ Madrid_ May 2015 | 5 | 28 | Analysis | Sequences, series, summability |
Theorem 11.8 Let \( f \) be a self-map and let \( \left( {X, d}\right) \) be a complete metric space. If for each \( k \neq l \in X \)\n\n\[{\int }_{0}^{d\left( {{fk},{fl}}\right) }\omega \left( s\right) {ds} \leq \gamma \left( {d\left( {k, l}\right) }\right) {\int }_{0}^{m\left( {k, l}\right) }\omega \left( s\right) {ds} \tag{11.1}\]\n\nand\n\n\[m\left( {k, l}\right) = \max \left\{ {\frac{d\left( {k,{fk}}\right) \cdot d\left( {l,{fl}}\right) }{1 + d\left( {k, l}\right) }, d\left( {k, l}\right) }\right\} , \tag{11.2}\]\n\nwhere \( \omega \in \Psi ,\gamma : {R}^{ + } \rightarrow \lbrack 0,1) \) is a function with\n\n\[\mathop{\lim }\limits_{{\delta \rightarrow t > 0}}\sup \gamma \left( \delta \right) < 1 \tag{11.3}\]\n\nThen \( f \) has a unique fixed point. | 6842_Advances in Applied Mathematical Analysis and Applications | 5 | 33 | Analysis | Operator theory |
Exercise 5.16. Suppose that \( \mathbb{K} \) denotes any field with characteristic not equal to 2 or 3, and \( E : {y}^{2} = {x}^{3} + {ax} + b\left( {a, b \in \mathbb{K}}\right) \). Assuming the binary operation defined before makes \( E\left( \mathbb{K}\right) \) into a group, prove that \( P = \left( {x, y}\right) \) has order 2 if and only if \( y = 0 \). | 6066_An Introduction to Number Theory | 4 | 14 | Geometry and Topology | Algebraic geometry |
Example 5.8.12. To compute the variance of a standard normal distributed random variable \( X \), we compute \( {\int }_{-\infty }^{\infty }{x}^{2}{f}_{X}\left( x\right) {dx} = 1 \) . This can be done with partial integration. Hence \( E\left( {X}^{2}\right) = 1 \), and since \( E\left( X\right) = 0 \), it follows that the variance of \( X \) is 1 . | 23684_A Natural Introduction to Probability Theory_ Second Edition | 9 | 44 | Probability and Statistics | Probability theory and stochastic processes |
Corollary 8.3.7. Let \( R \) be a ring and \( S \) is a multiplicative closed subset of \( R \). (i) If \( R \) is a directed union of Artinian subrings, then so is \( {S}^{-1}R \) . | 30537_Non-Associative and Non-Commutative Algebra and Operator Theory_ NANCAOT_ Dakar_ Senegal_ May 23_25_ | 2 | 6 | Algebra | Commutative algebra |
Theorem 11.12 (Final Value Theorem) If the Laplace transforms of \( f : \lbrack 0,\infty ) \rightarrow \) \( \mathbb{R} \), its derivative \( {f}^{\prime } \) exist and \( F\left( s\right) = \mathcal{L}f\left( t\right) \) then\n\n\[ \mathop{\lim }\limits_{{s \rightarrow 0}}{sF}\left( s\right) = \mathop{\lim }\limits_{{t \rightarrow \infty }}f\left( t\right) \]\n\nprovided the two limits exit. | 21999_Fundamentals of Partial Differential Equations | 6 | 37 | Differential Equations and Dynamical Systems | Dynamical systems and ergodic theory |
Theorem 5.2. Let \( \alpha \in \left( {0,1}\right), T > 0 \) and \( \Omega \) be a bounded domain in \( {\mathbb{R}}^{d} \) . Let \( {u}_{0} \in {L}_{\infty }\left( \Omega \right) \) and suppose that the assumptions (HA) and (Hf) are satisfied. Let \( u \in {W}_{\alpha } \) be a bounded weak solution of (16) in \( \left( {0, T}\right) \times \Omega \) . Then there holds for any \( Q \subset \left( {0, T}\right) \times \Omega \) separated from the parabolic boundary \( \left( {\{ 0\} \times \Omega }\right) \cup \left( {\left( {0, T}\right) \times \partial \Omega }\right) \) by a positive distance \( D \) ,\n\n\[{\left\lbrack u\right\rbrack }_{{C}^{\frac{\alpha \epsilon }{2},\epsilon }\left( \bar{Q}\right) } \leq C\left( {{\left| u\right| }_{{L}_{\infty }\left( {\left( {0, T}\right) \times \Omega }\right) } + {\left| {u}_{0}\right| }_{{L}_{\infty }\left( \Omega \right) } + {\left| f\right| }_{{L}_{r}\left( {\left( {0, T}\right) ;{L}_{q}\left( \Omega \right) }\right) }}\right)\]\n\nwith positive constants \( \epsilon = \epsilon \left( {{\left| A\right| }_{\infty }, v,\alpha, r, q, d,\operatorname{diam}\Omega ,\mathop{\inf }\limits_{{\left( {\tau, z}\right) \in Q}}\tau }\right) \) and \( C = C\left( {{\left| A\right| }_{\infty }, v,\alpha ,}\right. \) \( r, q, d,\operatorname{diam}\Omega ,{\lambda }_{d + 1}\left( Q\right), D) \) . | 20333_Handbook of Fractional Calculus with Applications_ Volume 2_ Fractional Differential Equations | 5 | 32 | Analysis | Functional analysis |
Example 3.20 For the random variable \( \widetilde{X} \), the range set and probability mass function are given as\n\n\[ \n{R}_{\widetilde{X}} = \{ a, b, c\}\n\]\n\n\[ \np\left( {x = a}\right) = \frac{1}{4}\;p\left( {x = b}\right) = \frac{2}{4}\;p\left( {x = c}\right) = \frac{1}{4}.\n\]\n\n(a) Let \( {\widetilde{X}}_{1}^{N} \sim p\left( x\right) \) . Find a strongly typical sequence \( {x}_{1}^{N} \) for \( N = {20} \), and verify the inequality\n\n\[ \n{2}^{-N\left( {H\left( \widetilde{X}\right) + \epsilon }\right) } \leq p\left( {{x}_{1},{x}_{2},\ldots ,{x}_{N}}\right) \leq {2}^{-N\left( {H\left( \widetilde{X}\right) - \epsilon }\right) } \tag{3.44}\n\]\n\nfor your sequence. | 3010_Information Theory for Electrical Engineers | 9 | 44 | Probability and Statistics | Probability theory and stochastic processes |
Lemma 4.73. Let \( w \in C\left( {\left\lbrack {0, T}\right\rbrack \;;\;{H}_{\mathrm{{per}}}^{s}}\right) \cap {C}^{1}\left( {\left\lbrack {0, T}\right\rbrack \;;\;{H}_{\mathrm{{per}}}^{s - 1}}\right) \; \) satisfy \( \;{\partial }_{t}^{2}w\left( t\right) = \) \( {c}^{2}{\partial }_{x}^{2}w\left( t\right) \) . Define\n\n\[ \n{E}_{s}\left( t\right) = {E}_{s}\left( {t;w}\right) = {\begin{Vmatrix}\frac{1}{c}{\partial }_{t}w\left( t\right) \end{Vmatrix}}_{s - 2}^{2} + {\begin{Vmatrix}{\partial }_{x}w\left( t\right) \end{Vmatrix}}_{s - 2}^{2}. \tag{4.137} \n\]\n\nThen\n\n\[ \n{\partial }_{t}{E}_{s}\left( t\right) = 0\forall t \in \left\lbrack {0, T}\right\rbrack , \tag{4.138} \n\]\n\nso that\n\n\[ \n{E}_{s}\left( t\right) = {E}_{s}\left( 0\right) \;\forall t \in \left\lbrack {0, T}\right\rbrack . \tag{4.139} \n\] | 13050_Fourier Analysis and Partial Differential Equations | Unknown | Unknown | Unknown | Unknown |
Theorem 6.2 If \( \lambda \in \Lambda \), then any solution of \( J{u}^{\prime } + {qu} = {\lambda wu} \) is identically equal to 0 . | 25784_From Complex Analysis to Operator Theory_ A Panorama_ In Memory of Sergey Naboko _Operator Theory_ A | 6 | 35 | Differential Equations and Dynamical Systems | Ordinary differential equations |
Lemma 9.5.29. The following statements are true.\n\n(b1) Equation (9.5.89) has solutions of prime period 2 if and only if \( \alpha = 1 \) . | 26736_Discrete oscillation theory | 6 | 37 | Differential Equations and Dynamical Systems | Dynamical systems and ergodic theory |
Theorem 3.2. Let \( \\left( {\\mu }_{n}\\right) \) be a sequence of probability measures on \( \\mathbb{Z} \) and for \( f : \\mathbb{Z} \\rightarrow \\mathbb{R} \) define the maximal operator\n\n\[ \n\\left( {Mf}\\right) \\left( x\\right) = \\mathop{\\sup }\\limits_{n}\\left| {\\left( {{\\mu }_{n}f}\\right) \\left( x\\right) }\\right|, x \\in \\mathbb{Z}. \n\]\n\nWe assume\n\n(*) (Regularity of coefficients). There is \( 0 < \\alpha \\leq 1 \) and \( C > 0 \) such that, for each \( n \\geq 1 \) ,\n\n\[ \n\\left| {{\\mu }_{n}\\left( {x + y}\\right) - {\\mu }_{n}\\left( x\\right) }\\right| \\leq C\\frac{{\\left| y\\right| }^{\\alpha }}{{\\left| x\\right| }^{1 + \\alpha }}\\text{ for }x, y \\in \\mathbb{Z},0 < 2\\left| y\\right| \\leq \\left| x\\right| . \n\]\n\nThen the maximal operator \( M \) is weak-type \( \\left( {1,1}\\right) \) ; i.e., there is a \( {C}^{\\prime } > 0 \) such that for any \( \\lambda > 0 \)\n\n\[ \n\\left| \\left\\{ {x \\in \\mathbb{Z} : \\left( {Mf}\\right) \\left( x\\right) > \\lambda }\\right\\} \\right| \\leq \\frac{{C}^{\\prime }}{\\lambda }\\parallel f{\\parallel }_{1}\\text{ for all }f \\in {\\ell }^{1} = {\\ell }^{1}\\left( \\mathbb{Z}\\right) . \n\] | 18335_Harmonic Analysis and Partial Differential Equations_ Essays in Honor of Alberto P_ Calderon _Chicag | 9 | 44 | Probability and Statistics | Probability theory and stochastic processes |
Theorem 6.14. Euler’s Theorem. If \( m \) is a positive integer and \( a \) is an integer with \( \left( {a, m}\right) = 1 \), then \( {a}^{\phi \left( m\right) } \equiv 1\left( {\;\operatorname{mod}\;m}\right) \) . | 31750_Elementary Number Theory and Its Applications_Fourth Edition | 3 | 13 | Number Theory | Number theory |
Example 6.3.3. The balance equations for the \( M/M/2/3 \) queueing system with \( \lambda = {2\mu } \)\n\nare:\n\nstate \( j \) departure rate from \( j = \) arrival rate to \( j \)\n\n\[ \n{2\mu }{\pi }_{0} = \mu {\pi }_{1} \n\]\n\n\[ \n1\;\left( {{2\mu } + \mu }\right) {\pi }_{1} = {2\mu }{\pi }_{0} + \left( {2 \times \mu }\right) {\pi }_{2} \n\]\n\n\[ \n2\;\left( {{2\mu } + 2 \times \mu }\right) {\pi }_{2} = {2\mu }{\pi }_{1} + \left( {2 \times \mu }\right) {\pi }_{3} \n\]\n\n\[ \n\left( {2 \times \mu }\right) {\pi }_{3} = {2\mu }{\pi }_{2} \n\]\n\nThat is,\n\n\[ \n2{\pi }_{0}\overset{\left( 0\right) }{ = }{\pi }_{1} \n\]\n\n\[ \n3{\pi }_{1}\overset{\left( 1\right) }{ = }2{\pi }_{0} + 2{\pi }_{2} \n\]\n\n\[ \n2{\pi }_{2}\overset{\left( 2\right) }{ = }{\pi }_{1} + {\pi }_{3} \n\]\n\n\[ \n{\pi }_{3}\overset{\left( 3\right) }{ = }{\pi }_{2} \n\] | 8535_Basic Probability Theory with Applications | 6 | 37 | Differential Equations and Dynamical Systems | Dynamical systems and ergodic theory |
Problem 472. Describe the sample space, if two distinguishable coins are rolled simultaneously. | 24820_Discrete Mathematics with Cryptographic Applications_ A Self-Teaching Introduction | 9 | 44 | Probability and Statistics | Probability theory and stochastic processes |
Automobile Emissions. We can apply Theorem 7.3.3 to answer the question at the end of Example 7.3.7. In the notation of the theorem, we have \( n = {46},{\sigma }^{2} = {0.5}^{2} = {0.25} \) , \( {\mu }_{0} = 2 \), and \( {v}^{2} = {1.0} \) . The average of the 46 measurements is \( {\bar{x}}_{n} = {1.329} \) . The posterior distribution of \( \theta \) is then the normal distribution with mean and variance given by | 7422_Probability and Statistics 4 | 9 | 44 | Probability and Statistics | Probability theory and stochastic processes |
Corollary 26.6.9 Suppose that \( f \) is a meromorphic function on a domain \( U \), and that \( {S}_{f} \) is the disjoint union of \( A \) and \( B \) . Then there exist meromorphic functions \( g \) and \( h \) such that \( f = g + h,{S}_{g} = A \) and \( {S}_{h} = B \) . | 30436_A Course in Mathematical Analysis_ Volume III_ Complex Analysis_ Measure and Integration | 5 | 24 | Analysis | Functions of a complex variable |
Example 4.4.1. Let \( f\left( x\right) = {x}^{2},0 \leq x \leq {10} \) . The function is continuous and strictly monotone increasing. Therefore, its inverse \( g\left( x\right) = \sqrt{x},0 \leq x \leq {100} \) is continuous strictly monotone increasing as well. | 27302_Calculus Light | 5 | 22 | Analysis | Real functions |
Example 3.4.5. Let \( {\bar{B}}_{{c}_{0}}\left( {0,1}\right) \) be the closed unit ball of \( {c}_{0} \), the space of all null sequences \( \mathrm{x} = \left\{ {{x}_{n} : n \in \mathbb{N}}\right\} ,\mathop{\lim }\limits_{n}{x}_{n} = 0 \), endowed with the norm \( \parallel \mathrm{x}\parallel = \mathop{\sup }\limits_{i}\left| {x}_{i}\right| \) . We define \( f : {\bar{B}}_{{c}_{0}}\left( {0,1}\right) \rightarrow {\bar{B}}_{{c}_{0}}\left( {0,1}\right) \), by \( f\left( \mathrm{x}\right) = f\left( {{x}_{1},{x}_{2},\ldots }\right) = \left( {1,{x}_{1},{x}_{2},\ldots }\right) \) . Then \( \parallel f\left( \mathrm{x}\right) - f\left( \mathrm{z}\right) \parallel = \parallel \mathrm{x} - \mathrm{z}\parallel \), for every \( \mathrm{x},\mathrm{z} \in {c}_{0} \), however, the equation \( f\left( \mathrm{x}\right) = \mathrm{x} \) is satisfied only if \( \mathrm{x} = \left( {1,1,\ldots }\right) \) which is not in \( {c}_{0} \). | 17895_Variational Methods in Nonlinear Analysis_ With Applications in Optimization and Partial Differentia | 5 | 33 | Analysis | Operator theory |
Proposition 13.4.18. For \( n \geq 1 \) an element \( x \in {G}_{n} \) is thin if and only if \( {\Phi x} = 0 \) . | 10462_Nonabelian Algebraic Topology_ filtered spaces_ crossed complexes_ cubical higher homotopy groupoids | 2 | 11 | Algebra | Group theory and generalizations |
Corollary 46. (The Divergence Theorem) If \( X \) is a vector field on \( \left( {M, g}\right) \) with compact support, then\n\n\[ \n{\int }_{M}\operatorname{div}X \cdot d\operatorname{vol} = 0 \n\] | 18462_Riemannian geometry | 4 | 18 | Geometry and Topology | Differential geometry |
Proposition 4. Let \( A \) be a simple algebra over \( K \) . Then every automorphism \( \alpha \) of \( A \) over \( K \) is of the form \( x \rightarrow {a}^{-1}{xa} \) with \( a \in {A}^{ \times } \) . | 7839_Basic Number Theory | 2 | 4 | Algebra | General algebraic systems |
Exercise 9.1 Solve the following differential equations.\n\n1. \( \frac{\mathrm{d}y}{\mathrm{\;d}x} = \frac{x + y}{x - y} \) | 22067_Algebraic and Differential Methods for Nonlinear Control Theory_ Elements of Commutative Algebra and | 6 | 35 | Differential Equations and Dynamical Systems | Ordinary differential equations |
Corollary 3.2. Let the notation be as above. If \( \gcd \left( {m, n}\right) = 1 \) then \( \# A\left( {\mathbb{F}}_{{q}^{mn}}\right) = \) \( \# B\left( {\mathbb{F}}_{{q}^{m}}\right) \) . If \( n \mid m \) then \( \# B\left( {\mathbb{F}}_{{q}^{m}}\right) = {\left( \# A\left( {\mathbb{F}}_{{q}^{m/n}}\right) \right) }^{n} \) . | 24975_Public-Key Cryptography and Computational Number Theory_ Proceedings of the International Conference | 2 | 5 | Algebra | Field theory and polynomials |
Lemma 5.3.4. Assume that \( \mathrm{G} = \mathrm{T} \) is a torus. Then the theorem holds. | 22751_Geometry of Moduli Spaces and Representation Theory | 4 | 16 | Geometry and Topology | Geometry |
Theorem 2. For any element \( g \in {\mathbb{F}}_{p} \) of multiplicative order \( q \leq {p}^{1 - \varepsilon } \), any fixed \( \gamma > 0 \) and any function \( H \) with\n\n\[ U \leq \frac{{\left| \mathcal{M}\right| }^{2}{q}^{3 - \gamma }}{{p}^{3}} \]\n\ngiven an oracle \( {\mathcal{O}}_{\ell }^{NR} \) with \( \ell = \left\lceil {{\log }^{1/2}q}\right\rceil + \left\lceil {\log \log q}\right\rceil \), there exists a probabilistic polynomial time algorithm to recover the signer’s NR secret key \( \alpha \), from \( O\left( {{\log }^{1/2}q}\right) \) signatures \( \left( {r\left( {k,\mu }\right), v\left( {k,\mu }\right) }\right) \) with \( k \in \left\lbrack {0, q - 1}\right\rbrack \) and \( \mu \in \mathcal{M} \) selected independently and uniformly at random. The probability of success is at least \( 1 - {2}^{-{\left( \log q\right) }^{1/2}\log \log q} \) . | 28100_Cryptography and Lattices_ International Conference_ CaLC 2001 Providence_ RI_ USA_ March 29_30_ 200 | 3 | 13 | Number Theory | Number theory |
Theorem 4.16 (Weiss). (Weiss 1985) If \( \pi \) : \( \left( {X, X,\mu, T}\right) \rightarrow \left( {Y,\mathcal{Y}, v, S}\right) \) is a factor map with \( \left( {X, X,\mu, T}\right) \) ergodic and \( \left( {\widehat{Y},{\mathcal{B}}_{\widehat{Y}},\widehat{v},\widehat{S}}\right) \) is a uniquely ergodic model for \( \left( {Y,\mathcal{Y}, v, T}\right) \), then there is a uniquely ergodic model \( \left( {\widehat{X},{\mathcal{B}}_{\widehat{X}},\widehat{\mu },\widehat{T}}\right) \; \) for \( \left( {X, X,\mu, T}\right) \) and a factor map \( \widehat{\pi } : \widehat{X} \rightarrow \widehat{Y} \) which is a model for \( \pi : X \rightarrow Y \) . | 14139_Ergodic Theory _Encyclopedia of Complexity and Systems Science Series_ | 6 | 37 | Differential Equations and Dynamical Systems | Dynamical systems and ergodic theory |
Lemma 58.5. A skeletal category \( \mathcal{C} \) is equivalent to the category Set if and only if it satisfies the following properties:\n\n(1) \( \mathcal{C} \) is locally small;\n\n(2) \( \mathcal{C} \) is balanced;\n\n(3) \( \mathcal{C} \) has equalizers;\n\n(4) \( \mathcal{C} \) has arbitrary coproducts;\n\n(5) \( \mathcal{C} \) has a terminal object 1 ;\n\n(6) 1 is a \( \mathcal{C} \) -generator;\n\n(7) \( \mathcal{C} \) is element-separating. | 8796_Classical Set Theory_ Theory of Sets and Classes | 0 | 1 | Foundations and Logic | Category theory |
Lemma 11.2.3 Let \( \left\{ {s}_{n}\right\} \) be a kth-order linearly recursive sequence in \( {\mathbb{F}}_{2} \) with primitive characteristic polynomial \( f\left( x\right) \) . For convenience of notation, we write \( s\left( n\right) = {s}_{n}, n \geq 0 \) . Let \( {r}_{s} = {2}^{k} - 1 \) denote the (maximal) period of \( s \) . Fix \( b, c \geq 0 \) with \( \gcd \left( {c,{r}_{s}}\right) = 1 \), and let \( s\left( {b + {ac}}\right), a \geq 0 \), denote the subsequence of \( s \) . Then the period of \( s\left( {b + {ac}}\right) \) is \( {r}_{s} \) . | 9776_Cryptography for Secure Encryption | 3 | 13 | Number Theory | Number theory |
Proposition 6. Let \( \\Omega \) be a nonempty bounded subset of \( X \) such that \( \\alpha \\left( \\Omega \\right) > 0 \) and \( C \) a closed convex subset of \( X \) . Suppose that \( T : C \\rightarrow C \) is a nonexpansive mapping such that \( I - T \) is \( \\psi \) -expansive. Then for every \( \\varepsilon > 0,0 < c < \\alpha \\left( \\Omega \\right) + \\varepsilon \), and all \( \\delta ,{\\delta }^{\\prime } > 0 \) with \( 0 < \\delta + {\\delta }^{\\prime } < \\psi \\left( c\\right) \), we have\n\n\[ \n\\left\\lbrack {{F}_{\\delta }\\left( {T,\\Omega }\\right) \\times {F}_{{\\delta }^{\\prime }}\\left( {T,\\Omega }\\right) }\\right\\rbrack \\cap {N}_{\\varepsilon }^{c}\\left( \\Omega \\right) = \\varnothing .\n\] | 13979_Applied Mathematics in Tunisia_ International Conference on Advances in Applied Mathematics _ICAAM__ | 2 | 7 | Algebra | Linear and multilinear algebra; matrix theory |
Theorem 12.15 (Polya) If \( d = 1,2 \), then the \( d \) -dimensional simple random walk comes back to \( \mathbf{0} \) infinitely often. If \( d \geq 3 \), eventually the random walk stops coming back to 0 . In other words, if \( d = 1,2 \), the chain is recurrent and if \( d \geq 3 \) the chain is transient. | 10912_Probability and Stochastic Processes | 9 | 44 | Probability and Statistics | Probability theory and stochastic processes |
Theorem 2.1 (Maximum principle for non-zero-sum games).\n\n(i) Let \( \\left( {\\widehat{\\pi },{\\widehat{\\theta }}}\\right) \\in {\\mathcal{A}}_{\\Pi } \\times {\\mathcal{A}}_{\\Theta } \) be a Nash equilibrium with corresponding state process\n\n\( \\widehat{X}\\left( t\\right) = {X}^{\\left( \\widehat{\\pi },{\\widehat{\\theta }}\\right) }\\left( t\\right) \), i.e.,\n\n\[ \n{J}_{1}\\left( {\\pi ,{\\widehat{\\theta }}}\\right) \\leq {J}_{1}\\left( {\\widehat{\\pi },{\\widehat{\\theta }}}\\right) ,\\;\\text{ for all }\\pi \\in {\\mathcal{A}}_{\\Pi },\n\]\n\n\[ \n{J}_{2}\\left( {\\widehat{\\pi },\\theta }\\right) \\leq {J}_{2}\\left( {\\widehat{\\pi },{\\widehat{\\theta }}}\\right) ,\\;\\text{ for all }\\theta \\in {\\mathcal{A}}_{\\Theta }.\n\]\n\nAssume that the random variables \( \\frac{\\partial {f}_{i}}{\\partial x} \) and \( {F}_{i}\\left( {t, s}\\right), i = 1,2 \), belong to \( {\\mathbb{D}}_{1,2} \) . Then\n\n\[ \n{\\left. {\\mathbb{E}}^{x}\\left\\lbrack {\\nabla }_{\\pi }{\\widehat{H}}_{1}\\left( t,{X}^{\\left( \\pi ,{\\widehat{\\theta }}\\right) }\\left( t\\right) ,\\pi ,{\\widehat{\\theta }},\\omega \\right) {\\left. \\right| }_{\\pi = \\widehat{\\pi }}{\\mathcal{E}}_{t}\\right) \\right| }_{\\pi = \\widehat{\\pi }} = 0, \\tag{22.22}\n\]\n\n\[ \n{\\left. {\\mathbb{E}}^{x}\\left\\lbrack {\\nabla }_{\\theta }{\\widehat{H}}_{2}\\left( t,{X}^{\\left( \\widehat{\\pi },\\theta \\right) }\\left( t\\right) ,{\\widehat{\\pi }},\\theta ,\\omega \\right) \\left. \\right| {}_{\\theta = \\widehat{\\theta }}\\;\\right) {\\mathcal{E}}_{t}\\right| }_{\\theta = \\widehat{\\theta }} = 0, \\tag{22.23}\n\]\n\nfor a.a. \( t,\\omega \) .\n\n(ii) Conversely, suppose that there exists \( \\left( {\\widehat{\\pi },{\\widehat{\\theta }}}\\right) \\in {\\mathcal{A}}_{\\Pi } \\times {\\mathcal{A}}_{\\Theta } \) such that Eqs. (22.22) and (22.23) hold. Then\n\n\[ \n{\\left. \\frac{\\partial }{\\partial y}{J}_{1}\\left( \\widehat{\\pi } + y\\beta ,{\\widehat{\\theta }}\\right) \\right| }_{y = 0} = 0\\;\\text{ for all }\\beta ,\n\]\n\n\[ \n{\\left. \\frac{\\partial }{\\partial v}{J}_{2}\\left( \\widehat{\\pi },{\\widehat{\\theta } + v\\eta }\\right) \\right| }_{v = 0} = 0\\;\\text{ for all }\\eta \n\]\n\nIn particular, if\n\n\[ \n\\pi \\rightarrow {J}_{1}\\left( {\\pi ,{\\widehat{\\theta }}}\\right) \\;\\text{ and }\\;\\theta \\rightarrow {J}_{2}\\left( {\\widehat{\\pi },\\theta }\\right) , \\tag{22.24}\n\]\n\nare concave, then \( \\left( {\\widehat{\\pi },{\\widehat{\\theta }}}\\right) \) is a Nash equilibrium. | 2914_Malliavin Calculus and Stochastic Analysis A Festschrift in Honor of David Nualart | 12 | 58 | Social and Behavioral Sciences | Game theory, economics, social and behavioral sciences |
Theorem 12.5.5 Assume that \( d = {12} \) . Then \( {n}_{h} = 1 \), and the following hold.\n\n(i) We have \( {n}_{S} = 2{\left( {n}_{k} + 1\right) }^{2}\left( {{n}_{k}^{4} + {n}_{k}^{2} + 1}\right) \).\n\n(ii) We have \( {m}_{st} = {n}_{k}^{6},{m}_{{\lambda }_{h}} = {n}_{k}^{6},{m}_{{\lambda }_{k}} = 1 \), and\n\n\[ \n{m}_{{\chi }_{1}} = \frac{{n}_{k}}{6}{\left( {n}_{k} + 1\right) }^{2}\left( {{n}_{k}^{2} + {n}_{k} + 1}\right) \n\]\n\n\[ \n{m}_{{\chi }_{2}} = \frac{{n}_{k}}{2}{\left( {n}_{k} + 1\right) }^{2}\left( {{n}_{k}^{2} - {n}_{k} + 1}\right) \n\]\n\n\[ \n{m}_{{\chi }_{3}} = \frac{2{n}_{k}}{3}\left( {{n}_{k}^{4} + {n}_{k}^{2} + 1}\right) \n\]\n\n\[ \n{m}_{{\chi }_{4}} = \frac{{n}_{k}}{2}{\left( {n}_{k} + 1\right) }^{2}\left( {{n}_{k}^{2} - {n}_{k} + 1}\right) \n\]\n\n\[ \n{m}_{{\chi }_{5}} = \frac{{n}_{k}}{6}{\left( {n}_{k} + 1\right) }^{2}\left( {{n}_{k}^{2} + {n}_{k} + 1}\right) . \n\] | 1526_Theory of Association Schemes _Springer Monographs in Mathematics_ | 2 | 5 | Algebra | Field theory and polynomials |
Corollary 7 From (115) the Moments associated with the Dirichlet beta-L series are given by the following relationship, namely\n\n\[ \n{M}_{O}^{\left( k\right) } = \left\{ \begin{array}{l} 2 \cdot {M}_{{L}_{\left( 4,1\right) }}^{\left( k\right) } - {M}_{\widetilde{O}}^{\left( k\right) } \\ 2 \cdot {M}_{{L}_{\left( 4,3\right) }}^{\left( k\right) } + {M}_{\widetilde{O}}^{\left( k\right) } \end{array}\right\} \tag{141} \n\] | 11466_Frontiers in Functional Equations and Analytic Inequalities | 5 | 27 | Analysis | Special functions |
Corollary 9.4. Any element of \( {\left( {L}_{l}\left( \mathbb{Z}\left\lbrack {\mu }_{5}\right\rbrack \left\lbrack \frac{1}{5}\right\rbrack \right) \right) }_{i}^{\diamond } \) for \( i \leq 2 \) is geometric. | 27435_Arithmetic and Geometry Around Galois Theory | 2 | 6 | Algebra | Commutative algebra |
Theorem 6.8.4. A metric space \( X \) is a uniform AR (resp. a uniform ANR) if and only if \( X \) is a uniform \( {AE} \) (resp. a uniform \( {ANE} \) ). | 1774_Geometric Aspects of General Topology | Unknown | Unknown | Unknown | Unknown |
Exercise 230 Let \( n \) be a positive integer and let \( V \) be the subspace of \( \mathbb{R}\left\lbrack X\right\rbrack \) composed of all polynomials of degree at most \( n \) . Let \( \alpha : V \rightarrow V \) be the linear transformation given by \( \alpha : p\left( X\right) \mapsto p\left( {X + 1}\right) - p\left( X\right) \) . Find \( \ker \left( \alpha \right) \) and \( \operatorname{im}\left( \alpha \right) \) . | 32020_The Linear Algebra a Beginning Graduate Student Ought to Know_ Second Edition | 2 | Unknown | Algebra | Unknown |
Exercise 9.5.14. Let \( \lambda \) be Lebesgue measure on \( \left\lbrack {0,1}\right\rbrack \), and let \( f\left( {x, y}\right) = \) \( {8xy}\left( {{x}^{2} - {y}^{2}}\right) {\left( {x}^{2} + {y}^{2}\right) }^{-3} \) for \( \left( {x, y}\right) \neq \left( {0,0}\right) \), with \( f\left( {0,0}\right) = 0 \) . (a) Compute \( {\int }_{0}^{1}\left( {{\int }_{0}^{1}f\left( {x, y}\right) \lambda \left( {dy}\right) }\right) \lambda \left( {dx}\right) \) . [Hint: Make the substitution \( \left. {u = {x}^{2} + {y}^{2}, v = x\text{, so }{du} = {2y}\mathrm{\;d}y\text{,}{dv} = {dx}\text{, and }{x}^{2} - {y}^{2} = 2{v}^{2} - u\text{. }}\right\rbrack | 21494_A First Look at Rigorous Probability Theory | Unknown | Unknown | Unknown | Unknown |
Example 10.8.7. The Ornstein-Uhlenbeck process \( {X}_{t} \) in Example 10.8.2 is a diffusion process with the diffusion coefficient and drift\n\n\[ Q\left( {t, x}\right) = I,\;\rho \left( {t, x}\right) = - x, \] | 6611_Introduction to Stochastic Integration | Unknown | Unknown | Unknown | Unknown |
Problem 7. Find the \( {17}^{\text{th }} \) term of the AP with first term 5 and common difference 2. | 6704_Discrete Mathematics and Structures | 2 | 7 | Algebra | Linear and multilinear algebra; matrix theory |
Theorem 1. For a given generator (2), group transformations (1) can be found by solving the Lie equations\n\n\\[ \n\\frac{d{\\bar{x}}^{i}}{da} = {\\xi }^{i}\\left( {\\bar{x},\\bar{u}}\\right) ,{\\left. \\;{\\bar{x}}^{i}\\right| }_{a = 0} = {x}^{i},\\;i = 1,\\ldots, n \n\\] \n\n\\[ \n\\frac{d{\\bar{u}}^{\\mu }}{da} = {\\eta }^{\\mu }\\left( {\\bar{x},\\bar{u}}\\right) ,{\\left. \\;{\\bar{u}}^{\\mu }\\right| }_{a = 0} = {u}^{\\mu },\\;\\mu = 1,\\ldots, m.\n\\] | 20333_Handbook of Fractional Calculus with Applications_ Volume 2_ Fractional Differential Equations | 6 | 35 | Differential Equations and Dynamical Systems | Ordinary differential equations |
Theorem 4.3.4 (Closed Graph Theorem). Let \( V \) and \( W \) be Banach spaces and let \( T \) be a linear transformation from \( V \) to \( W \) . If the graph of \( T \) is closed in \( V \times W \), then \( T \) is bounded. | 10882_Fundamentals of Mathematical Analysis | 5 | 33 | Analysis | Operator theory |
Theorem 6.2.4 A metric space \( \left( {M, d}\right) \) is compact iff \( {C}_{b}\left( M\right) = C\left( M\right) \) . | 27579_An Introduction to Mathematical Analysis for Economic Theory and Econometrics | 4 | 19 | Geometry and Topology | General topology |
Lemma 2. Let \( h\left( \widehat{z}\right) \) be an holomorphic function near the origin. Then there exists a unique formal series expansion \( \psi \left( {x,\widehat{z}}\right) = \mathop{\sum }\limits_{{n \geq 0}}{a}_{n}\left( \widehat{z}\right) {x}^{n} \) solution of (24) such that the \( {a}_{n}\left( \widehat{z}\right) \) are holomorphic functions near \( \widehat{z} = 0 \), with \( {a}_{0}\left( \widehat{z}\right) = 1 \) and \( {a}_{1}\left( \widehat{z}\right) = h\left( \widehat{z}\right) \) . In this case, one has\n\n\[ \n{a}_{n}\left( \widehat{z}\right) = \frac{1}{n - 1}{\int }_{0}^{1}{u}^{n - 1}\left( {{a}_{n - 1}^{\prime \prime }\left( {{u}^{2}\widehat{z}}\right) + {b}_{n - 2}\left( {{u}^{2}\widehat{z}}\right) }\right) {du},\;\text{ for }n \geq 2, \tag{28} \n\] \n\nwhere the \( {b}_{n} \) ’s are defined by (26). Furthermore, if \( h\left( z\right) \) is even, then every \( {a}_{n}\left( z\right) \) is even. | 702_Algebraic Analysis of Differential Equations_ from Microlocal Analysis to Exponential Asymptotics | 5 | 24 | Analysis | Functions of a complex variable |
Exercise 6.11.20 Under the conditions of Theorem 6.11.19, show that if \( K \) is a compact subset of \( M \), then there exists an \( N \) such that for all \( n \geq N, K \subset {K}_{N} \) . | 27579_An Introduction to Mathematical Analysis for Economic Theory and Econometrics | 4 | 21 | Geometry and Topology | Manifolds and cell complexes |
Lemma 5.1.3. Let \( A \) be regular, \( I \) an ideal of \( A \), and \( x \in A \) . Then \( \widehat{x} \) belongs locally to \( I \) at each point of \( h{\left( x\right) }^{0} \), the interior of \( h\left( x\right) \), and at each point of \( \Delta \left( A\right) \smallsetminus h\left( I\right) \) . | 25969_A Course in Commutative Banach Algebras _Graduate Texts in Mathematics_ 246_ | 2 | 6 | Algebra | Commutative algebra |
Theorem 5.13 (Riemann’s Inequality) Let \( {g}_{X} \) denote the genus of the non singular projective curve \( X \), defined over the algebraically closed field \( k \) . Then for any divisor \( D \) on \( X \) ,\n\n\[ \n\ell \left( D\right) \geq \deg \left( D\right) + 1 - {g}_{X} \n\] | 16380_A royal road to algebraic geometry | 4 | 14 | Geometry and Topology | Algebraic geometry |
Theorem 1.2.18 (Long tables). Let \( \Gamma \) be a simplicial complex and fix \( {r}_{2},\ldots ,{r}_{m} \) . There exists a number \( b\left( {\Gamma ,{r}_{2},\ldots ,{r}_{m}}\right) < \infty \) such that the 1-norms of the elements of any minimal Markov basis for \( \Gamma \) on \( s \times {r}_{2} \times \cdots \times {r}_{m} \) tables are less than or equal to \( b\left( {\Gamma ,{r}_{2},\ldots ,{r}_{m}}\right) \) . This bound is independent of \( s \), which can grow large. | 23092_Lectures on Algebraic Statistics | 1 | 2 | Combinatorics | Combinatorics |
Lemma 10.2.2. Let \( f \in C\left\lbrack t\right\rbrack \) . Let \( l \) be a positive integer such that \( l \equiv 0{\;\operatorname{mod}\;r} \) . Then\n\n\[ \frac{{f}^{{p}^{l}} - f}{{t}^{{p}^{l}} - t} \in C\left\lbrack t\right\rbrack \] | 26724_Hilbert_s Tenth Problem_ Diophantine Classes and Extensions to Global Fields _New Mathematical Monog | 2 | 5 | Algebra | Field theory and polynomials |
Corollary 9.1.4.\n\n\\[ \n\\operatorname{Pf}\\left( {A\\left( D\\right) }\\right) = s\\left( {D, M}\\right) \\mathcal{P}\\left( {D, M}\\right) \n\\] | 12046_Discrete Mathematics in Statistical Physics_ Introductory Lectures | 0 | 0 | Foundations and Logic | Mathematical logic and foundations |
Theorem 4.1.1. Let \( \{ \overrightarrow{\xi }\left( m\right), m \in \mathbb{Z}\} \) be a stochastic sequence which defines the stationary GM increment sequence \( {\chi }_{\bar{\mu },\bar{s}}^{\left( d\right) }\left( {\overrightarrow{\xi }\left( m\right) }\right) = {\left\{ {\chi }_{\bar{\mu },\bar{s}}^{\left( d\right) }\left( {\xi }_{p}\left( m\right) \right) \right\} }_{p = 1}^{T} \) with the absolutely continuous spectral function \( F\left( \lambda \right) \) which has spectral density \( f\left( \lambda \right) \) . Let \( \{ \overrightarrow{\eta }\left( m\right), m \in \mathbb{Z}\} \) be an uncorrelated with the sequence \( \overrightarrow{\xi }\left( m\right) \) stationary stochastic sequence with an absolutely continuous spectral function \( G\left( \lambda \right) \) which has spectral density \( g\left( \lambda \right) \) . Let minimality condition (4.2) be satisfied. The optimal linear estimate \( {\widehat{A}}_{N}\overrightarrow{\xi } \) of the functional \( {A}_{N}\overrightarrow{\xi } \) which depends on the unknown values of elements \( \overrightarrow{\xi }\left( k\right), k = 0,1,2,\ldots, N \), from observations of the sequence \( \overrightarrow{\xi }\left( m\right) + \overrightarrow{\eta }\left( m\right) \) at points of the set \( Z \smallsetminus \{ 0,1,2,\ldots, N\} \) is calculated by formula (4.8), where the spectral characteristic \( {\overrightarrow{h}}_{\bar{\mu }, N}\left( \lambda \right) \) is calculated by the formula | 12806_Non-Stationary Stochastic Processes Estimation_ Vector Stationary Increments_ Periodically Stationar | 9 | 44 | Probability and Statistics | Probability theory and stochastic processes |
Theorem 1. Assume that \( \left( {H}_{1}\right) - \left( {H}_{4}\right) \) hold. Moreover, if there exists a constant \( M > 0 \), such that\n\n\[ M\left\lbrack {q\left( M\right) \left( {\frac{\psi \left( b\right) - \psi \left( a\right) }{{2\Gamma }\left( {\alpha - 1}\right) }{\int }_{a}^{b}{\left( \psi \left( b\right) - \psi \left( s\right) \right) }^{\alpha - 2}p\left( s\right) {\psi }^{\prime }\left( s\right) {ds}}\right. }\right.\n\n\[ {\left. \left. +\frac{2}{\Gamma \left( \alpha \right) }{\left( \psi \left( b\right) - \psi \left( a\right) \right) }^{\alpha - 1}{\int }_{a}^{b}p\left( s\right) {\psi }^{\prime }\left( s\right) ds\right) \right\rbrack }^{-1} > 1. \tag{23} \]\n\nThen (4) has at least one solution on \( \left\lbrack {a, b}\right\rbrack \) . | 28491_Fractional Differential Equations_ Inclusions and Inequalities with Applications | 6 | Unknown | Differential Equations and Dynamical Systems | Unknown |
Theorem 4. The Hilbert-Schmidt operator \( \mathbb{K} : \mathcal{H} \rightarrow \mathcal{H} \) from (6) with the integral kernel (5) represents a bounded linear operator on the Hilbert space \( \mathcal{H} = {L}^{2}\left( {Q,\mathbb{C}}\right) \), and we have the estimate\n\n\[ \n\parallel \mathbb{K}\parallel \leq \parallel K\parallel \n\] | 35105_偏微分方程_第2卷_英文 | 5 | 33 | Analysis | Operator theory |
Theorem 2.1.16 (Iskovskikh (1979a)). Let \( X \) be a nonsingular three-dimensional Fano variety of index \( r \) and genus \( g \) . Assume that the anticanon-ical linear system \( \left| {-{K}_{X}}\right| \) determines a morphism \( \varphi : X \rightarrow {X}^{\prime } \subset {\mathbb{P}}^{g + 1} \) which is not an embedding. Then \( \varphi : X \rightarrow {X}^{\prime } \) is a double cover with a smooth ramification divisor \( D \subset {X}^{\prime } \) . The variety \( X \) is completely determined by the pair \( \left( {{X}^{\prime }, D}\right) \), and for this pair only the following cases are possible:\n\n(i) \( {X}^{\prime } \subset {\mathbb{P}}^{6} \) is a cone over the Veronese surface, and \( D \subset {X}^{\prime } \) is cut out by a cubic hypersurface; in this case \( X \) is isomorphic to the variety from 3.1.6 (i);\n\n(ii) \( {X}^{\prime } = {\mathbb{P}}^{3} \), and \( D \subset {\mathbb{P}}^{3} \) is a surface of degree 6; in this case \( r = 1, g\left( X\right) = 2 \) , and \( \operatorname{Pic}\left( X\right) = \mathbb{Z} \) ;\n\n(iii) \( {X}^{\prime } = Q \subset {\mathbb{P}}^{4} \) is a smooth quadric, and \( D \subset Q \) is cut out by a quadric in \( {\mathbb{P}}^{4} \) ; in this case \( r = 1, g\left( X\right) = 3 \), and \( \operatorname{Pic}\left( X\right) = \mathbb{Z} \) ;\n\n(iv) \( {X}^{\prime } = {\mathbb{P}}^{1} \times {\mathbb{P}}^{2} \subset {\mathbb{P}}^{5} \) embedded by Segre, and \( D \) is a divisor of bidegree \( \left( {2,4}\right) \) ; in this case \( r = 1, g\left( X\right) = 4 \), and \( \operatorname{Pic}\left( X\right) = \mathbb{Z} \oplus \mathbb{Z} \) ;\n\n(v) \( {X}^{\prime } = {\mathbb{P}}_{{\mathbb{P}}^{1}}\left( \mathcal{E}\right) \), where \( \mathcal{E} = {\mathcal{O}}_{{\mathbb{P}}^{1}}\left( 2\right) \oplus {\mathcal{O}}_{{\mathbb{P}}^{1}}\left( 1\right) \oplus {\mathcal{O}}_{{\mathbb{P}}^{1}}\left( 1\right) \), and \( {X}^{\prime } \) is embedded in \( {\mathbb{P}}^{6} \) by the linear system \( \left| {{\mathcal{O}}_{\mathbb{P}\left( \mathcal{E}\right) }\left( 1\right) }\right| \), and \( D \in \left| {{\mathcal{O}}_{\mathbb{P}\left( \mathcal{E}\right) }\left( 4\right) }\right| \) ; in this case \( r = 1 \) , \( g\left( X\right) = 5 \), and \( \operatorname{Pic}\left( X\right) = \mathbb{Z} \oplus \mathbb{Z} \) ; the variety \( X \) can also be realized as a blow-up of a Fano variety \( {Y}_{2} \) of index 2 along a nonsingular elliptic curve \( {H}_{1} \cap {H}_{2} \), where \( {H}_{1},{H}_{2} \in \frac{1}{2}\left| {-{K}_{{Y}_{2}}}\right| \) ;\n\n(vi) \( {X}^{\prime } = {\mathbb{P}}^{1} \times {\mathbb{P}}^{2} \subset {\mathbb{P}}^{8}\; \) embedded by the linear system \( \left| {{p}_{1}^{ * }{\mathcal{O}}_{{\mathbb{P}}^{1}}\left( 2\right) \otimes {p}_{2}^{ * }{\mathcal{O}}_{{\mathbb{P}}^{2}}\left( 1\right) }\right| , \) and \( D \in \left| {{p}_{2}^{ * }{\mathcal{O}}_{{\mathbb{P}}^{2}}\left( 4\right) }\right| \) ; in this case \( r = 1, g\left( X\right) = 7 \), and \( \operatorname{Pic}\left( X\right) = {\mathbb{Z}}^{9} \) ; the variety \( X \) is isomorphic to \( {\mathbb{P}}^{1} \times F \), where \( F \) is a del Pezzo surface of degree 2. | 38847_代数几何 Fano簇 | 4 | Unknown | Geometry and Topology | Unknown |
Corollary 8.4.5 (How to recognize a Brownian motion).\n\nLet\n\n\\[ d{Y}_{t} = u\\left( {t,\\omega }\\right) {dt} + v\\left( {t,\\omega }\\right) d{B}_{t} \\]\n\nbe an Itô process in \\( {\\mathbf{R}}^{n} \\) . Then \\( {Y}_{t} \\) is a Brownian motion if and only if\n\n\\[ {E}^{x}\\left\\lbrack {u\\left( {t, \\cdot }\\right) \\mid {\\mathcal{N}}_{t}}\\right\\rbrack = 0\\;\\text{ and }\\;v{v}^{T}\\left( {t,\\omega }\\right) = {I}_{n} \\tag{8.4.14} \\]\n\nfor a.a. \\( \\left( {t,\\omega }\\right) \\) . | 24297_Oksendal Stochastic differential equations | 9 | 44 | Probability and Statistics | Probability theory and stochastic processes |
Problem 2. Characterize diagonals \( {\left\{ \left\langle E{e}_{i},{e}_{i}\right\rangle \right\} }_{i \in I} \) of a self-adjoint operator \( E \), where \( {\left\{ {e}_{i}\right\} }_{i \in I} \) is any orthonormal basis of \( \mathcal{H} \) . | 9449_Excursions in Harmonic Analysis_ Volume 4_ The February Fourier Talks at the Norbert Wiener Center | 5 | 33 | Analysis | Operator theory |
Corollary 1. Let at least one of the price sequences, for example, \( {\left\{ {p}_{i}^{k}\right\} }_{k = 0}^{\infty } \), grow unlimitedly. Then all the other price sequences also grow unlimitedly under the Assumption 1. | 24032_Mathematical Optimization Theory and Operations Research_ 19th International Conference_ MOTOR 2020_ | 8 | 42 | Optimization and Control | Calculus of variations and optimal control; optimization |
Theorem 18.8 (Slutsky’s Theorem). Let \( {\left( {X}_{n}\right) }_{n \geq 1} \) and \( {\left( {Y}_{n}\right) }_{n \geq 1} \) be two sequences of \( {\mathbf{R}}^{d} \) valued random variables, with \( {X}_{n}\overset{\mathcal{D}}{ \rightarrow }X \) and \( \begin{Vmatrix}{{X}_{n} - {Y}_{n}}\end{Vmatrix} \rightarrow 0 \) in probability. Then \( {Y}_{n}\overset{\mathcal{D}}{ \rightarrow }X \) . | 15222_Probability essentials | 9 | 44 | Probability and Statistics | Probability theory and stochastic processes |
Theorem 1.1 Let \( \mathcal{D} \) be the normalized in \( {L}_{p},2 \leq p < \infty \), real \( d \) -variate trigonometric system. Then for any \( {f}_{0} \in {L}_{p} \) the WCGA with weakness parameter \( t \) gives\n\n\[{\begin{Vmatrix}{f}_{C\left( {t, p, d}\right) m\ln \left( {m + 1}\right) }\end{Vmatrix}}_{p} \leq C{\sigma }_{m}{\left( {f}_{0},\mathcal{D}\right) }_{p}. \tag{1.3}\] | 31382_Mathematical Analysis_ Probability and Applications _ Plenary Lectures_ ISAAC 2015_ Macau_ China | 5 | 22 | Analysis | Real functions |
Proposition 11.5.13 ([841, Proposition 2.4]) Let \( \mathcal{C},\mathcal{D} \subseteq {\mathbb{F}}_{q}^{n \times m} \) be rank-metric codes. If \( \mathcal{C} \sim \mathcal{D} \), then\n\n\[ \n{d}_{i}\left( \mathcal{C}\right) = {d}_{i}\left( \mathcal{D}\right) \text{ for }i = 1,2,\ldots ,\dim \left( \mathcal{C}\right) .\n\] | 32121_Concise Encyclopedia Of Coding Theory | 1 | 2 | Combinatorics | Combinatorics |
Theorem 13.5 Assume that in two dimensions at \( {x}_{0} \in {\Phi }_{t}^{-1}{M}_{t} \) the normal \( {n}_{\mathrm{M}}\left( {x}_{0}\right) \neq 0 \) so that the pre-Maxwell set does not have a generalized cusp at \( {x}_{0} \) . Then, the Maxwell set can only have a cusp at \( {\Phi }_{t}\left( {x}_{0}\right) \) if \( {\Phi }_{t}\left( {x}_{0}\right) \in {C}_{t} \) . Moreover, if:\n\n\[ x = {\Phi }_{t}\left( {x}_{0}\right) \in {\Phi }_{t}\left\{ {{\Phi }_{t}^{-1}{C}_{t} \cap {\Phi }_{t}^{-1}{M}_{t}}\right\} \]\n\nthe Maxwell set will have a generalized cusp at \( x \) . | 22349_Analysis and Stochastics of Growth Processes and Interface Models | 4 | 18 | Geometry and Topology | Differential geometry |
Lemma 2.5. Let \( G \) be an \( {\aleph }_{0} \) -bounded \( P \) -group. Then every continuous homomorphic image \( K \) of \( G \) with \( \psi \left( K\right) \leq \omega \) is countable. | 1088_Topology and its Applications 2004-01-28_ Vol 136 Iss 1-3 | 2 | 11 | Algebra | Group theory and generalizations |
Proposition 20 (Proposition 2.30, [8]) If \( G \) is a connected Lie group and Ad denotes the adjoint action of \( G \) on its Lie algebra, then \( \Delta \left( g\right) = \det \left( {\operatorname{Ad}\left( {g}^{-1}\right) }\right) \) . | 23387_Harmonic and Applied Analysis_ From Radon Transforms to Machine Learning | 4 | 15 | Geometry and Topology | Topological groups, Lie groups |
Exercise 6.9 Show that \( {g}^{ij} = \left( {\mathrm{d}{x}^{i} \mid \mathrm{d}{x}^{j}}\right) \) . (Cf. Exercise 6.7.) | 12530_Differentiable Manifolds - A Theoretical Physics Approach | 4 | Unknown | Geometry and Topology | Unknown |
Theorem 1.2.15. Let \( \mu \in \mathcal{F}\mathcal{P}\left( G\right) \) and let \( a = \land \{ \eta \left( e\right) \mid \mu \subseteq \eta ,\eta \in \mathcal{F}\left( G\right) \} \) . Then \[ \langle \mu \rangle = {e}_{a} \cup \left( {{ \cup }_{n = 1}^{\infty }{\left( \mu \cup {\mu }^{-1}\right) }^{n}}\right) = { \cup }_{n = 1}^{\infty }{\left( {e}_{a} \cup \mu \cup {\mu }^{-1}\right) }^{n}. \] | 14520_Fuzzy Group Theory | 2 | 11 | Algebra | Group theory and generalizations |
Proposition 8.1. (see [BF],[M1]) The set \( {B}_{2, K}^{k} \) can be given a natural scheme structure. Its expected dimension is\n\n\[ \n{\rho }_{2, K}^{k} = \dim U\left( {2, K}\right) - \left( \begin{matrix} k + 1 \\ 2 \end{matrix}\right) .\n\]\n\nThe tangent space to \( {B}_{2, K}^{k} \) at a point \( E \) is naturally identified to the orthogonal to the image of the symmetric Petri map\n\n\[ \n{S}^{2}\left( {{H}^{0}\left( {C, E}\right) }\right) \rightarrow {H}^{0}\left( {C,{S}^{2}\left( E\right) }\right) .\n\] | 15980_Moduli Spaces and Vector Bundles | 4 | 14 | Geometry and Topology | Algebraic geometry |
Proposition 8.2.4 The following formulas are valid in PLTL, for any formulas \( \alpha \) and \( \beta \) :\n\n1. \( \sim \sim \alpha \leftrightarrow \alpha \) ,\n\n2. \( \sim \left( {\alpha \land \beta }\right) \leftrightarrow \sim \alpha \vee \sim \beta \) ,\n\n3. \( \sim \left( {\alpha \vee \beta }\right) \leftrightarrow \sim \alpha \land \sim \beta \) ,\n\n\[ \text{4.} \sim \left( {\alpha \rightarrow \beta }\right) \leftrightarrow \alpha \land \sim \beta \text{,} \]\n\n\[ \text{5.} \sim \neg \alpha \leftrightarrow \neg \sim \alpha \text{,} \]\n\n\[ \text{6.} \sim \mathrm{X}\alpha \leftrightarrow \mathrm{X} \sim \alpha \text{,} \]\n\n\[ \text{7.} \sim \mathrm{F}\alpha \leftrightarrow \mathrm{G} \sim \alpha \text{,} \]\n\n\[ \text{8.} \sim \mathrm{G}\alpha \leftrightarrow \mathrm{F} \sim \alpha \text{.} \] | 13166_Proof Theory of N4-Paraconsistent Logics | 0 | 0 | Foundations and Logic | Mathematical logic and foundations |
Theorem 8.2 (The Gauss Formula). Suppose \( \left( {M, g}\right) \) is an embedded Riemannian submanifold of a Riemannian or pseudo-Riemannian manifold \( \left( {\widetilde{M},\widetilde{g}}\right) \) . If \( X, Y \in \) \( \mathfrak{X}\left( M\right) \) are extended arbitrarily to smooth vector fields on a neighborhood of \( M \) in \( \widetilde{M} \), the following formula holds along \( M \) :\n\n\[ \n{\widetilde{\nabla }}_{X}Y = {\nabla }_{X}Y + \coprod \left( {X, Y}\right) \n\] | 31699_Introduction to Riemannian manifolds _Corrected version of second edition_ | 4 | 18 | Geometry and Topology | Differential geometry |
Theorem 9.8. [46] The central fiber of this map is the polarized toric variety for the torus \( {T}_{N} \) given by the polyhedron \( {\bar{\Xi }}_{f} \) and the generic fiber is a compactification of the cluster variety \( \mathcal{A} \) . | 7930_Representation Theory _ Current Trends and Perspectives_Henning Krause_Peter Littelmann_Gunter Malle | 4 | Unknown | Geometry and Topology | Unknown |
Lemma 6.41. Let \( {H}_{1} \) and \( {H}_{2} \) be subgroups of \( G \), and let \( g \in G \) . Then\n\n\[ \n\# {H}_{1}g{H}_{2} = \frac{\# {H}_{1} \cdot \# {H}_{2}}{\# \left( {{g}^{-1}{H}_{1}g \cap {H}_{2}}\right) }. \tag{6.16} \n\] | 11112_Abstract Algebra_ An Integrated Approach | 2 | 11 | Algebra | Group theory and generalizations |
Theorem 9.3.4. Let \( p \in \left\lbrack {1, + \infty }\right\rbrack \) . Let \( F \) be as in (9.3.2) satisfying (9.2.5), (9.2.9),(9.3.3) ÷ (9.3.6),(9.2.2),(9.3.7),(9.3.8),(9.2.3), and let \( {f}_{F} \) be given by (9.1.6). Then \( {f}_{F} \) is convex and lower semicontinuous, and (9.3.1) holds.\n\nConversely, given \( f : {\mathbf{R}}^{n} \rightarrow \left\lbrack {0, + \infty }\right\rbrack \) convex and lower semicontinuous, and defined \( F \) by (9.3.1) with \( {f}_{F} = f \), it turns out that conditions (9.2.5), (9.2.9), (9.3.3)÷(9.3.6), (9.2.2), (9.3.7), (9.3.8), (9.2.3) are satisfied by \( F \) . | 17661_Unbounded Functionals in the Calculus of Variations_ Representation_ Relaxation_ and Homogenization | 5 | 22 | Analysis | Real functions |
Lemma 2. Suppose \( B \) is a \( d \times d \) strictly positive definite matrix. Let\n\n\[ \n{D}_{n} = {\int }_{{\mathbb{R}}^{d}}\exp \left( {-\frac{n}{2}{x}^{T}{Bx}}\right) {dx} \]\n\nand\n\n\[ \n{D}_{n}\left( r\right) = {\int }_{\left| x\right| \leq r}\exp \left( {-\frac{n}{2}{x}^{T}{Bx}}\right) {dx}. \]\n\nThen for \( r > 0 \) ,\n\n\[ \n\mathop{\lim }\limits_{{n \rightarrow \infty }}{D}_{n}\left( r\right) /{D}_{n} = 1 \]\n | 27890_Discrete Geometry_ Combinatorics and Graph Theory_ 7th China-Japan Conference_ CJCDGCGT 2005_ Tianji | 5 | 23 | Analysis | Measure and integration |
Lemma 1. For every Spectrum Allocation instance \( \left( {G, D, k}\right) \) and its optimal solution \( I \), there exists a permutation \( \operatorname{Per}\left( D\right) \) of the demands, such that demands from Per (D) are allocated one by one with the lowest available wavelengths will deliver the optimal solution \( I \) . | 682_Mathematical Optimization Theory and Operations Research_ Recent Trends_ 21st International Conference_ MOTOR 2022_ Petrozavodsk_ Russia_ July 2_6_ 2022_ Revised Selected Papers | 1 | 2 | Combinatorics | Combinatorics |
Proposition 3.1 - Let \( A \) be a finitely generated \( k \) -algebra. If \( \mathfrak{m} \in \operatorname{Spm}\left( A\right) \) , then the canonical map \( k \rightarrow A/\mathfrak{m} \) is an isomorphism. | 15589_Introduction to algebraic groups | 2 | 6 | Algebra | Commutative algebra |
Proposition 2.3.15 ([4, Proposition 3.5]). If \( i, j, k \) are not all the same, then(2.3.6) | 25377_2-Kac-Moody Algebras | 1 | 2 | Combinatorics | Combinatorics |
Example 3.23 Find the surface area of rotation of \( y = \sqrt{x} \) about the \( x \)-axis from \( x = 0 \) to \( x = 4 \) . | 17898_Fast Start Advanced Calculus | 5 | 22 | Analysis | Real functions |
Theorem 3.4.1. If a finite set of closed \( d \) -dimensional balls of radius \( \frac{\pi }{2} \) (i.e., of closed hemispheres) in the \( d \) -dimensional spherical space \( {\mathbb{S}}^{d}, d \geq 2 \) is rearranged so that the (spherical) distance between each pair of centers does not increase, then the (spherical) d-dimensional volume of the intersection does not decrease and the (spherical) \( d \) -dimensional volume of the union does not increase. | 22389_Lectures on Sphere Arrangements - the Discrete Geometric Side | 4 | 16 | Geometry and Topology | Geometry |
Lemma 6 (Lemma III.4,[5]). Let \( \sum = \left( {{A}_{1},{A}_{2},{B}_{1},{B}_{2}, C, D}\right) \) be a \( {2D} \) linear system and \( X\left( {{z}_{1},{z}_{2}}\right) \) the corresponding matrix defined in (4). Then \( \sum \) is modally reachable if and only if the matrix \( X\left( {{z}_{1},{z}_{2}}\right) \) is \( \ell {FP} \) . | 1561_Coding Theory and Applications _ 5th International Castle Meeting_ ICMCTA 2017_ Vihula_ Estonia_ Aug | 13 | 59 | Systems Theory and Control | Systems theory; control |
Exercise 7.17 Let \( V = {\mathbb{R}}^{n} \) with the standard inner product. Let \( \mathbf{x} \in {\mathbb{R}}^{n} \) . Use the Cauchy-Schwarz inequality to prove that\n\n\[ \left| {x}_{1}\right| + \left| {x}_{2}\right| + \cdots + \left| {x}_{n}\right| \leq \sqrt{n}\parallel \mathbf{x}\parallel \] | 22051_Elements of Linear Algebra | 5 | 29 | Analysis | Harmonic analysis on Euclidean spaces |
Corollary 9.8 [41] Let the sequence \( \left\{ {x}_{n}\right\} \) be generated by the mapping\n\n\[ \n{x}_{n + 1} = {\alpha }_{n}{\gamma f}\left( {x}_{n}\right) + \left( {I - \mu {\alpha }_{n}F}\right) T{x}_{n},\n\]\n\nwhere \( T \) is nonexpansive, \( {\alpha }_{n} \) is a sequence in \( \left( {0,1}\right) \) satisfying the following conditions:\n\n\[ \n\left\{ \begin{array}{ll} \text{ (i) } & \mathop{\lim }\limits_{{n \rightarrow \infty }}{\alpha }_{n} = 0,\;\sum {\alpha }_{n} = \infty ; \\ \text{ (ii) } & \sum \left| {{\alpha }_{n + 1} - {\alpha }_{n}}\right| < \infty ,\;\sum \left| {{\beta }_{n + 1} - {\beta }_{n}}\right| < \infty ; \\ \text{ (iii) } & 0 \leq \mathop{\max }\limits_{i}{k}_{i} \leq {\beta }_{n} < a < 1,\forall n \geq 0. \end{array}\right. \tag{9.71}\n\]\n\nIt was proved in [41] that \( \left\{ {x}_{n}\right\} \) converged strongly to the common fixed-point \( {x}^{ * } \) of \( T \), which is the solution of variational inequality problem\n\n\[ \n\left\langle {\left( {{\gamma f} - {\mu F}}\right) {x}^{ * }, x - {x}^{ * }}\right\rangle \leq 0,\;\forall x \in \operatorname{Fix}\left( T\right) . \tag{9.72}\n\] | 31124_Mathematical Analysis and Applications_ Selected Topics | 6 | 38 | Differential Equations and Dynamical Systems | Difference and functional equations |
Theorem 8.2.4 If \( 1 \leq k \leq p - 1 \), then\n\n\[ S\left( {p, k}\right) = {kS}\left( {p - 1, k}\right) + S\left( {p - 1, k - 1}\right) . \] | 37834_组合数学 英文版 | 1 | 2 | Combinatorics | Combinatorics |
Corollary 2.5. If \( \Omega \) is a symmetric admissible Tricomi domain, then the reflected Tricomi operator \( R{T}_{AC} : {W}_{A} \subset {W}_{{AC} \cup \sigma }^{1} \rightarrow {L}^{2}\left( \Omega \right) \) admits infinitely many positive and negative eigenvalues \( {\lambda }_{j}^{ \pm } = {\left( {\mu }_{j}^{ \pm }\right) }^{-1} \) with associated eigen-functions \( {\left\{ {e}_{j}^{ \pm }\right\} }_{j \in \mathbb{N}} \) where \( {\lambda }_{j}^{ \pm } \rightarrow \pm \infty \) as \( j \rightarrow + \infty \) . | 10376_Nonlinear Analysis and its Applications to Differential Equations | 6 | 36 | Differential Equations and Dynamical Systems | Partial differential equations |
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