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Lemma 3.5. The order of any element in
\[
{\left. \operatorname{Ker}\left( {\psi }^{p + 1} - 1\right) \right| }_{B{P}_{q{p}^{j + 1} - 2}\left( {P \land P}\right) }
\]
divides \( {p}^{j + 2} \) .
Proof of Theorem 3.1. Suppose \( {\theta }_{j} \) exists and lifts to \( {\widetilde{\theta }}_{j} \in {\pi }_{q{p}^{j +... | 2304_Topology and Its Applications 2000-03-03_ Vol 101 Iss 3 | 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 \( {p}^{j + 2} \) . | Proof of Theorem 3.1. Suppose \( {\theta }_{j} \) exists and lifts to \( {\widetilde{\theta }}_{j} \in {\pi }_{q{p}^{j + 1} - 2}^{s}\left( {P \land P}\right) \) ; let \( {\mathbf{\Theta }}_{j} \) be the \( {BP} \) -Hurewicz image of \( {\widetilde{\theta }}_{j} \) and \( {n}_{j} \) be the order of \( {\mathbf{\Theta }}... | 2 | Unknown | Algebra | Unknown |
Theorem 17.1 Under assumption 1 and 2, the channel capacity of the genetic code is 2.92 Shannon bits.
Proof.
\[
\text{Capacity} = H\left( Y\right) = H\left( \frac{2,6,3,1,4,6,4,4,4,2,2,2,2,2,2,2,2,3,1,6,4}{64}\right)
\]
\[
= {2.92}
\]
0
This result is not surprising due to the fact that there is much redundancy ... | 9381_Cryptography_ information theory_ and error-correction_ a handbook for the 21st century | Theorem 17.1 Under assumption 1 and 2, the channel capacity of the genetic code is 2.92 Shannon bits. | \[ \text{Capacity} = H\left( Y\right) = H\left( \frac{2,6,3,1,4,6,4,4,4,2,2,2,2,2,2,2,2,3,1,6,4}{64}\right) \] \[ = {2.92} \] | 10 | 47 | Computer Science and Engineering | Information and communication, circuits |
Corollary 1.1.9 If \( L, M \) are finite separable extensions of \( k \), their composi-tum is separable as well.
Proof: By definition of \( {LM} \) there exist finitely many separable elements \( {\alpha }_{1},\ldots ,{\alpha }_{m} \) of \( L \) such that \( {LM} = M\left( {{\alpha }_{1},\ldots ,{\alpha }_{m}}\righ... | 1509_Galois groups and fundamental groups | Corollary 1.1.9 If \( L, M \) are finite separable extensions of \( k \), their compositum is separable as well. | Proof: By definition of \( {LM} \) there exist finitely many separable elements \( {\alpha }_{1},\ldots ,{\alpha }_{m} \) of \( L \) such that \( {LM} = M\left( {{\alpha }_{1},\ldots ,{\alpha }_{m}}\right) \). As the \( {\alpha }_{i} \) are separable over \( k \), they are separable over \( M \), and so the extension \... | 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
\[
{\widetilde{f}}_... | 22824_Partial Differential Equations_ Foundations and Integral Representations Part 1 | 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}... | Proof: We choose \( 0 \leq r < R \), and calculate with the aid of (3) as follows:\n\n\[ \n{\widetilde{f}}_{kl}\left( r\right) = {\int }_{\left| \xi \right| = 1}{\Delta u}\left( {r\xi }\right) {H}_{kl}\left( \xi \right) {d\sigma }\left( \xi \right) \n\]\n\n\[ \n= {\int }_{\left| \xi \right| = 1}\left\{ {\left( {\frac{{... | 5 | 25 | Analysis | Potential theory |
Proposition 6.9. For an equilibrium point \( \left( {{x}_{e},{y}_{e}}\right) \) as defined by Prop. 6.2 if the following equations hold:
\[
\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... | 12899_Analytical and Stochastic Modeling Techniques and Applications_ 16th International Conference_ ASMTA | 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... | ## Proof (Proof of Prop. 6.9)\n\nTo analyse of the equilibrium point \( \left( {{x}_{e},{y}_{e}}\right) \), we take the Jacobian \( J \) of the rate equations (2) and (3) at this point. The partial derivatives \( \frac{\partial g}{\partial x} \) and \( \frac{\partial g}{\partial y} \) at \( \left( {{x}_{e},{y}_{e}}\rig... | 6 | 36 | Differential Equations and Dynamical Systems | Partial differential equations |
Lemma 7.5. Let \( m \) be a nonnegative real number.
(a) The following equality holds for every positive real number \( c \) :
\[
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}_... | 11657_Fourier_series_in_control | 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... | Proof.\n\n(a) Using the power series expansion (7.4), we see that \( y \mathrel{\text{:=}} {J}_{m}\left( x\right) \) satisfies the differential equation\n\n\[ {x}^{2}{y}^{\prime \prime } + x{y}^{\prime } + \left( {{x}^{2} - {m}^{2}}\right) y = 0\;\text{ in }\;\left( {0,\infty }\right) . \tag{7.6} \]\n\nMultiplying this... | 6 | 35 | Differential Equations and Dynamical Systems | Ordinary differential equations |
Lemma 11.3. The operation \( \odot \) on \( {\mathbb{Z}}_{n} \) is associative and commutative and has [1] as an identity element.
PROOF. Make the obvious changes in the proof of Theorem 11.1.
Let \( {\mathbb{Z}}_{n}^{\# } \) denote the set \( \{ \left\lbrack 1\right\rbrack ,\left\lbrack 2\right\rbrack ,\ldots ,\le... | 175_Modern Algebra_ An Introduction_ Sixth Edition | Lemma 11.3. The operation \( \odot \) on \( {\mathbb{Z}}_{n} \) is associative and commutative and has [1] as an identity element. | PROOF. Make the obvious changes in the proof of Theorem 11.1. | 2 | 8 | Algebra | Associative rings and algebras |
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 | 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. | Null | 4 | 19 | Geometry and Topology | General topology |
Problem 12.3. Let the process \( X\left( t\right), t \in \left\lbrack {0, T}\right\rbrack \), be the geometric Lévy pro-
cess
\[
{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... | 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 | 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}... | Null | 6 | 35 | Differential Equations and Dynamical Systems | Ordinary differential equations |
Corollary 7.1 (Odd prime \( p \) ) For idempotents \( e \in {Z}_{p - 1} \) as naturals \( e < p : {e}^{p} \equiv \) \( e{\;\operatorname{mod}\;{p}^{3}} \Rightarrow e = 1 \) .
For \( q = p - 1 \) and some carry \( 0 \leq c < q \) : idempotent \( {e}^{2} \equiv e{\;\operatorname{mod}\;q} \) implies
\[
{e}^{2} = {cq} ... | 22305_Associative Digital Network Theory_ An Associative Algebra Approach to Logic_ Arithmetic and State M | Corollary 7.1 (Odd prime \( p \) ) For idempotents \( e \in {Z}_{p - 1} \) as naturals \( e < p : {e}^{p} \equiv \) \( e{\;\operatorname{mod}\;{p}^{3}} \Rightarrow e = 1 \) . | For \( q = p - 1 \) and some carry \( 0 \leq c < q \) : idempotent \( {e}^{2} \equiv e{\;\operatorname{mod}\;q} \) implies\n\n\[ \n{e}^{2} = {cq} + e = c\left( {p - 1}\right) + e < {q}^{2}. \tag{#} \n\]\n\nNotice that carry \( c = 0 \) resp. \( c > 0 \) yield:\n\n\[ \nc = 0 \Leftrightarrow e = 1,\;\text{ and }\;e > 1 \... | 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
\[
\mathbf{A}\left( x\right) = {g}^{-1}\mathbf{A}\left( {gx}\right) \;\forall g \in O\left( 3\right)
\]
where \( O\left( 3\right) \) 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 | 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{... | Null | 5 | Unknown | Analysis | Unknown |
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 \( \mathrm{X} \) ... | 18074_The Pi Calculus | 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... | Let us reuse the notations \( {V}_{j},{M}_{\mathrm{A}} \), etc. from Example 20.1.1, and write \( {\mathrm{A}}^{\prime \prime } \) for the class obtained from A by applying the transformation. So,\n\n\[ \n{\mathcal{O}}_{\mathrm{A}} \equiv \left( {{\nu g}, p}\right) \left( {{V}_{j} \mid \left( {\nu \widetilde{g},\wideti... | 10 | 46 | Computer Science and Engineering | Computer science |
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.
Proof. The Equation (7.26) shows the gradedness ... | 8322_Krichever-Novikov Type Algebras Theory and Applications | 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. | Proof. The Equation (7.26) shows the gradedness for \( \overline{gl}\left( \infty \right) \) . For \( \overset{⏜}{gl}\left( \infty \right) \) we only have to recall that the cocycle vanishes if \( r \neq - s \), see (7.19). Hence, only for \( r + s = 0 \) will there be a term coming with the central element \( t \), wh... | 4 | 15 | Geometry and Topology | Topological groups, Lie groups |
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]).
For those reasons, throug... | 23563_Topology and Its Applications 2001-06-29_ Vol 113 Iss 1-3 | 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]). | Null | 4 | 20 | Geometry and Topology | Algebraic topology |
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.
With any invariant point selection \( \mathbf{X} \), there... | 23554_Topology and its Applications 2004-02-28_ Vol 137 Iss 1-3 | 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. | Null | 0 | 1 | Foundations and Logic | Category theory |
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
\[
\bar{u} = \overline{{u}^{\prime }} \Leftrightarrow u,{u}^{\prime }\text{ are conjugate in }F\left( {a, b}\... | 27350_Markov_s theorem and 100 years of the uniqueness conjecture_ a mathematical journey from irrational | 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... | Proof. If \( u = w{u}^{\prime }{w}^{-1} \), then\n\n\[ \n\overline{\mathbf{u}} = {\mathbf{u}}^{\mathbf{\pi }} = {\mathbf{w}}^{\mathbf{\pi }}{\mathbf{u}}^{\prime \mathbf{\pi }}{\left( {\mathbf{w}}^{\mathbf{\pi }}\right) }^{-1} = {\mathbf{u}}^{\prime \mathbf{\pi }} = \overline{{\mathbf{u}}^{\prime }}.\n\]\n\nAssume, conv... | 2 | 11 | Algebra | Group theory and generalizations |
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
\[
\operatorname{... | 27353_Selected Aspects of Fractional Brownian Motion | 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\operatornam... | Proof. By bilinearity and approximation, it is enough to show (4.7) for \( \phi \left( x\right) = \) \( {e}^{i\langle t, x{\rangle }_{{\mathbb{R}}^{d}}} \) and \( \psi \left( x\right) = {e}^{i\langle s, x{\rangle }_{{\mathbb{R}}^{d}}} \) when \( s, t \in {\mathbb{R}}^{d} \) are given (and fixed once for all). Set\n\n![... | 9 | Unknown | Probability and Statistics | Unknown |
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 show... | 16173_Reasoning with probabilistic and deterministic graphical models_ exact algorithms | 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 ... | Null | 10 | 47 | Computer Science and Engineering | Information and communication, circuits |
Theorem 9 [35,36] Suppose \( c \in {\mathbb{R}}^{\ell }\langle \langle X\rangle \rangle \) has coefficients which satisfy
\[
\left| \left( {c,\eta }\right) \right| \leq {K}_{c}{M}_{c}^{\left| \eta \right| },\;\forall \eta \in {X}^{ * }.
\]
Then there exists a real number \( \widehat{R} > 0 \) such that for each \( ... | 1529_Discrete Mechanics_ Geometric Integration and Lie_Butcher Series_ DMGILBS_ Madrid_ May 2015 | 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 \(... | Proof Fix \( N \geq 1 \) . From the assumed coefficient bound and Lemma 3, it follows that\n\n\[ \left| {{\widehat{F}}_{c}\left( \widehat{u}\right) \left( N\right) }\right| \leq \mathop{\sum }\limits_{{j = 0}}^{\infty }\mathop{\sum }\limits_{{\eta \in {X}^{j}}}\left| \left( {c,\eta }\right) \right| \left| {{S}_{\eta }\... | 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. Iffor each \( k \neq l \in X \)
\[
{\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) {... | 6842_Advances in Applied Mathematical Analysis and Applications | 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) {... | Proof. Let us consider \( s \in X \) be any arbitrary point in \( \mathrm{X} \) . Now construct a sequence \( \left\{ {s}_{n}\right\} \) in \( \mathrm{X} \) such that \( f{s}_{n} = {s}_{n + 1} \) .\n\nStep - 1: Claim \( \mathop{\lim }\limits_{{n \rightarrow \infty }}d\left( {{s}_{n},{s}_{n + 1}}\right) = 0 \)\n\nFor al... | 5 | 23 | Analysis | Measure and integration |
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... | 6066_An Introduction to Number 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) \... | Null | 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 th... | 23684_A Natural Introduction to Probability Theory_ Second Edition | 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... | If \( X \) has a normal distribution with parameters \( \mu \) and \( {\sigma }^{2} \), we use the fact that \[ Y = \frac{X - \mu }{\sigma } \] has a standard normal distribution. Since \( X = {\sigma Y} + \mu \), this gives, using Exercise 5.8.10, that \( \operatorname{var}\left( X\right) = {\sigma }^{2}\operatorname{... | 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 \( {S}^{-1}R \) .
(ii) If \( \mathcal{D}\mathcal{U}\left( R\right) \neq \varnothing \), then \( \mathcal{D}\mathcal{U}\left( {{S}^{-1}R}\right) \neq \... | 30537_Non-Associative and Non-Commutative Algebra and Operator Theory_ NANCAOT_ Dakar_ Senegal_ May 23_25_ | 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 \( {S}^{-1}R \) . | Proof. (i) Suppose that \( R = \mathop{\bigcup }\limits_{{\alpha \in A}}{R}_{\alpha } \) is a directed union of Artinian subrings and \( {S}_{\alpha } = S \cap {R}_{\alpha } \) be a multiplicative closed subset of \( {R}_{\alpha } \). It is not difficult to show that \( {S}^{-1}R = \mathop{\bigcup }\limits_{{\alpha \in... | 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
\[
\mathop{\lim }\limits_{{s \rightarrow 0}}{sF}\left( s\right) = \mathop{\lim }\limits_{{... | 21999_Fundamentals of Partial Differential Equations | 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_{{... | Proof Since\n\n\[ {sF}\left( s\right) - f\left( 0\right) = \mathcal{L}\left\lbrack {{f}^{\prime }\left( t\right) }\right\rbrack = {\int }_{0}^{\infty }{e}^{-{st}}{f}^{\prime }\left( t\right) \mathrm{d}t \]\n\nwe have\n\n\[ \mathop{\lim }\limits_{{s \rightarrow 0}}{sF}\left( s\right) - f\left( 0\right) = {\int }_{0}^{\i... | 6 | 35 | Differential Equations and Dynamical Systems | Ordinary differential equations |
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 \( \l... | 20333_Handbook of Fractional Calculus with Applications_ Volume 2_ Fractional Differential Equations | 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 \( \lef... | Null | 6 | 36 | Differential Equations and Dynamical Systems | Partial differential equations |
Example 3.20 For the random variable \( \widetilde{X} \), the range set and probability mass function are given as
\[
{R}_{\widetilde{X}} = \{ a, b, c\}
\]
\[
p\left( {x = a}\right) = \frac{1}{4}\;p\left( {x = b}\right) = \frac{2}{4}\;p\left( {x = c}\right) = \frac{1}{4}.
\]
(a) Let \( {\widetilde{X}}_{1}^{N} \sim... | 3010_Information Theory for Electrical Engineers | 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... | ## Solution 3.20\n\n(a) In a strongly typical sequence \( {x}_{1}^{20} \), the symbol ’ \( a \) ’ appears \( {20} \times \frac{1}{4} = 5 \), the symbol ’ \( b \) ’ appears \( {20} \times \frac{2}{4} = {10} \) times, and the symbol ’ \( c \) ’ appears \( {20} \times \frac{1}{4} = 5 \) times. Then, we can form a strongly... | Unknown | Unknown | Unknown | Unknown |
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
... | 13050_Fourier Analysis and Partial Differential Equations | 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\... | Proof. The conditions on \( w \) imply that \( {\partial }_{t}^{2}w\left( t\right) = {c}^{2}{\partial }_{x}^{2}w\left( t\right) \in C(\left\lbrack {0, T}\right\rbrack \) ; \( \left. {H}_{\text{per }}^{s - 2}\right) \) . A simple computation leads to\n\n\[ \n{\partial }_{t}{E}_{s}\left( t\right) = \mathop{\lim }\limits_... | 6 | 36 | Differential Equations and Dynamical Systems | Partial differential equations |
Theorem 6.2 If \( \lambda \in \Lambda \), then any solution of \( J{u}^{\prime } + {qu} = {\lambda wu} \) is identically equal to 0 .
Proof Since \( \lambda \in \Lambda \) there is a point \( {x}_{0} \) for which \( {B}_{ \pm }\left( {{x}_{0},\lambda }\right) \) are not invertible. Of course \( {x}_{0} + \omega \) i... | 25784_From Complex Analysis to Operator Theory_ A Panorama_ In Memory of Sergey Naboko _Operator Theory_ A | Theorem 6.2 If \( \lambda \in \Lambda \), then any solution of \( J{u}^{\prime } + {qu} = {\lambda wu} \) is identically equal to 0 . | Proof Since \( \lambda \in \Lambda \) there is a point \( {x}_{0} \) for which \( {B}_{ \pm }\left( {{x}_{0},\lambda }\right) \) are not invertible. Of course \( {x}_{0} + \omega \) is then also such a point. First we show that \( \operatorname{ran}{B}_{ + }\left( {{x}_{0},\lambda }\right) \) and \( \operatorname{ran}{... | 6 | 35 | Differential Equations and Dynamical Systems | Ordinary differential equations |
Lemma 9.5.29. The following statements are true.
(b1) Equation (9.5.89) has solutions of prime period 2 if and only if \( \alpha = 1 \) .
(b2) Suppose \( \alpha = 1 \) . Let \( \{ x\left( k\right) {\} }_{k = - 1}^{\infty } \) solve (9.5.89). Then \( \{ x\left( k\right) {\} }_{k = - 1}^{\infty } \) is periodic with ... | 26736_Discrete oscillation theory | 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 \( \alpha = 1 \) . | Null | 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
\[
\left( {Mf}\right) \left( x\right) = \mathop{\sup }\limits_{n}\left| {\left( {{\mu }_{n}f}\right) \left( x\right) }\right|, x \i... | 18335_Harmonic Analysis and Partial Differential Equations_ Essays in Honor of Alberto P_ Calderon _Chicag | 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\... | Proof. We start with \( f \\in {\\ell }_{ + }^{1} \) and \( \\lambda > 0 \) . Apply the Calderón-Zygmund decomposition to \( f \) and \( \\rho = \\frac{\\lambda }{4} \) . We get dyadic intervals \( \\left( {Q}_{i}\\right) \), which are pairwise disjoint, such that\n\n\[ \n\\rho < \\frac{1}{\\left| {Q}_{i}\\right| }{\\i... | 5 | 23 | Analysis | Measure and integration |
Theorem 6.14. Euler’s Theorem. If \( m \) is a positive integer and \( a \) is an integer with \( \left( {a, m}\right) = 1 \), then \( {a}^{\phi \left( m\right) } \equiv 1\left( {\;\operatorname{mod}\;m}\right) \) .
Before we prove Euler's theorem, we illustrate the idea behind the proof with an example. | 31750_Elementary Number Theory and Its Applications_Fourth Edition | Theorem 6.14. Euler’s Theorem. If \( m \) is a positive integer and \( a \) is an integer with \( \left( {a, m}\right) = 1 \), then \( {a}^{\phi \left( m\right) } \equiv 1\left( {\;\operatorname{mod}\;m}\right) \) . | Null | 3 | 13 | Number Theory | Number theory |
Example 6.3.3. The balance equations for the \( M/M/2/3 \) queueing system with \( \lambda = {2\mu } \)
are:
state \( j \) departure rate from \( j = \) arrival rate to \( j \)
\[
{2\mu }{\pi }_{0} = \mu {\pi }_{1}
\]
\[
1\;\left( {{2\mu } + \mu }\right) {\pi }_{1} = {2\mu }{\pi }_{0} + \left( {2 \times \mu }\rig... | 8535_Basic Probability Theory with Applications | 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 \tim... | It is a simple matter to solve this system of linear equations. The equations for states 2 and 3 yield that \( {\pi }_{1} = {\pi }_{2} = {\pi }_{3} \) . Then, making use of the equation for state 0, we can write that\n\n\[ \n{\pi }_{0} + 2{\pi }_{0} + 2{\pi }_{0} + 2{\pi }_{0} = 1\; \Rightarrow \;{\pi }_{0} = \frac{1}{... | 8 | 43 | Optimization and Control | Operations research, mathematical programming |
Problem 472. Describe the sample space, if two distinguishable coins are rolled simultaneously.
Solution. Since the pair \( \{ H, T\} \) and \( \{ T, H\} \) are indistinguishable, the sample space consists of three elements \( \{ H, H\} ,\{ T, T\} ,\{ H, T\} \), where the first two can be identified with \( \{ H\} \... | 24820_Discrete Mathematics with Cryptographic Applications_ A Self-Teaching Introduction | Problem 472. Describe the sample space, if two distinguishable coins are rolled simultaneously. | Solution. Since the pair \( \{ H, T\} \) and \( \{ T, H\} \) are indistinguishable, the sample space consists of three elements \( \{ H, H\} ,\{ T, T\} ,\{ H, T\} \), where the first two can be identified with \( \{ H\} \) and \( \{ T\} \), respectively. However, if the coins are distinguishable, then the sets \( \left... | 1 | 2 | Combinatorics | Combinatorics |
Example
7.3.8
---
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}... | 7422_Probability and Statistics 4 | 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 pos... | \[{\mu }_{1} = \frac{{0.25} \times 2 + {46} \times 1 \times {1.329}}{{0.25} + {46} \times 1} = {1.333},\]\[{v}_{1}^{2} = \frac{{0.25} \times 1}{{0.25} + {46} \times 1} = {0.0054}\]\n\nThe mean \( {\mu }_{1} \) of the posterior distribution of \( \theta \), as given in Eq. (7.3.1), can be rewritten as follows:\n\n\[{\mu... | 9 | 45 | Probability and Statistics | Statistics |
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 \( {S}_{h} = B \) .
Proof By the theorem, there exists \( g \... | 30436_A Course in Mathematical Analysis_ Volume III_ Complex Analysis_ Measure and Integration | 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 \( {S}_{h} = B \) . | Proof By the theorem, there exists \( g \) with \( {S}_{g} = A \) such that \( f - g \) has removable singularities at the points of \( A \) . Remove them, and set \( h = f - g \) . | 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 | 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. | Null | 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 }\limit... | 17895_Variational Methods in Nonlinear Analysis_ With Applications in Optimization and Partial Differentia | 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_... | Null | 5 | 32 | Analysis | Functional analysis |
Proposition 13.4.18. For \( n \geq 1 \) an element \( x \in {G}_{n} \) is thin if and only if \( {\Phi x} = 0 \) .
Proof. We have shown that \( \Phi {\varepsilon }_{j}y = 0,\Phi {\Gamma }_{j}y = 0 \) for all \( y \in {G}_{n - 1} \) (see Proposition 13.4.11). It follows from Proposition 13.4.14 that \( {\Phi x} = 0 \... | 10462_Nonabelian Algebraic Topology_ filtered spaces_ crossed complexes_ cubical higher homotopy groupoids | Proposition 13.4.18. For \( n \geq 1 \) an element \( x \in {G}_{n} \) is thin if and only if \( {\Phi x} = 0 \) . | Proof. We have shown that \( \Phi {\varepsilon }_{j}y = 0,\Phi {\Gamma }_{j}y = 0 \) for all \( y \in {G}_{n - 1} \) (see Proposition 13.4.11). It follows from Proposition 13.4.14 that \( {\Phi x} = 0 \) whenever \( x \) is thin. To see the converse, we recall the definition\n\n\[ \n{\Phi }_{j}x = {\left\lbrack -{\vare... | 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
\[
{\int }_{M}\operatorname{div}X \cdot d\operatorname{vol} = 0
\]
Proof. Just observe
\[
\operatorname{div}X \cdot d\mathrm{{vol}} = {L}_{X}d\mathrm{{vol}}
\]
\[
= {i}_{X}d\left( {d\mathrm... | 18462_Riemannian geometry | Corollary 46. (The Divergence Theorem) If \( X \) is a vector field on \( \left( {M, g}\right) \) with compact support, then\n\n\[ \n{\int }_{M}\operatorname{div}X \cdot d\operatorname{vol} = 0 \n\] | Proof. Just observe\n\n\[ \n\operatorname{div}X \cdot d\mathrm{{vol}} = {L}_{X}d\mathrm{{vol}} \n\]\n\n\[ \n= {i}_{X}d\left( {d\mathrm{{vol}}}\right) + d\left( {{i}_{X}d\mathrm{{vol}}}\right) \n\]\n\n\[ \n= d\left( {{i}_{X}d\mathrm{{vol}}}\right) \n\]\n\nand use Stokes' theorem. | 6 | 36 | Differential Equations and Dynamical Systems | Partial differential equations |
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 \( a \in {A}^{ \times } \) .
Take a basis \( \left\{ {{a}_{1},\ldots ,{a}_{N}}\right\} \) of \( A \) over \( K \) . Then every element of \( ... | 7839_Basic Number Theory | 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 \( a \in {A}^{ \times } \) . | Take a basis \( \left\{ {{a}_{1},\ldots ,{a}_{N}}\right\} \) of \( A \) over \( K \) . Then every element of \( A \otimes {A}^{0} \) can be written in one and only one way as \( \sum {a}_{i} \otimes {b}_{i} \), with \( {b}_{i} \in {A}^{0} \) for \( 1 \leq i \leq N \) . By prop. 3, \( \alpha \) can therefore be written ... | 2 | 9 | Algebra | Nonassociative rings and algebras |
Exercise 9.1 Solve the following differential equations.
1. \( \frac{\mathrm{d}y}{\mathrm{\;d}x} = \frac{x + y}{x - y} \)
2. \( \sqrt{{x}^{2} + {y}^{2}}\mathrm{\;d}x = y\mathrm{\;d}y \)
3. \( \left( {{x}^{2} + {y}^{2}}\right) \mathrm{d}{xxy}\mathrm{\;d}y = 0 \)
4. \( \frac{\mathrm{d}y}{\mathrm{\;d}x} = \frac{1 + ... | 22067_Algebraic and Differential Methods for Nonlinear Control Theory_ Elements of Commutative Algebra and | Exercise 9.1 Solve the following differential equations.\n\n1. \( \frac{\mathrm{d}y}{\mathrm{\;d}x} = \frac{x + y}{x - y} \) | Null | 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) }^... | 24975_Public-Key Cryptography and Computational Number Theory_ Proceedings of the International Conference | 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... | Proof. Let \( {\zeta }_{n} = \exp \left( {{2\pi }\sqrt{-1}/n}\right) \) and let \( {P}_{A}\left( T\right) = \mathop{\prod }\limits_{{i = 1}}^{{2g}}\left( {1 - {\alpha }_{i}}\right) \) . Then \( {P}_{B}\left( T\right) = \) \( \mathop{\prod }\limits_{i}\mathop{\prod }\limits_{{j = 1}}^{n}\left( {1 - {\zeta }_{n}^{j}{\alp... | 2 | 5 | Algebra | Field theory and polynomials |
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