The Experts below are selected from a list of 183 Experts worldwide ranked by ideXlab platform
Hugh C. Williams - One of the best experts on this subject based on the ideXlab platform.
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An Upper Bound on the Least Inert Prime in a Real Quadratic Field
2007Co-Authors: Andrew Granville Mollin, Andrew Granville, Richard Mollin, Hugh C. WilliamsAbstract:It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D ? 3705, there is always at least one prime p ! p D=2 such that the Kronecker Symbol (D=p) = \Gamma1. 1991 Mathematics Subject Classification 11R11, 11Y40 The first author is a Presidential Faculty Fellow. His research is partiallly supported by the NSF. The research of the second two authors is partially supported by NSERC of Canada 1 1 Introduction Let D be the fundamental discriminant of a real quadratic field and let S = f5; 8; 12; 13; 17; 24; 28; 33; 40; 57; 60; 73; 76; 88; 97; 105; 124; 129; 136; 145; 156; 184; 204; 249; 280; 316; 345; 364; 385; 424; 456; 520; 609; 616; 924; 940; 984; 1065; 1596; 2044; 2244; 3705g: At the end of Chapter 6 of [5], the second author made the following conjecture. Conjecture. The values of D for which the least prime p such that the Kronecker Symbol (D=p) = \Gamma1 satisfies p ? p D=2 are precisely those in S. He also veri..
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An upper bound on the least inert prime in a real quadratic field
Canadian Journal of Mathematics, 2000Co-Authors: Andrew Granville, Richard Mollin, Hugh C. WilliamsAbstract:AbstractIt is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D > 3705, there is always at least one prime p < √D/2 such that the Kronecker Symbol (D/p) = −1.
Lanza De Cristoforis, Massimo - One of the best experts on this subject based on the ideXlab platform.
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Weakly Singular and Microscopically Hypersingular Load Perturbation for a Nonlinear Traction Boundary Value Problem: a Functional Analytic Approach
'Springer Science and Business Media LLC', 2011Co-Authors: Dalla Riva M, Lanza De Cristoforis, MassimoAbstract:Let $\Omega^{i}$ and $\Omega^{o}$ be two bounded open subsets of ${\mathbb{R}}^{n}$ containing $0$. Let $G^{i}$ be a (nonlinear) map from $\partial\Omega^{i}\times {\mathbb{R}}^{n}$ to $ {\mathbb{R}}^{n}$. Let $a^{o}$ be a map from $\partial\Omega^{o}$ to the set $M_{n}({\mathbb{R}})$ of $n\times n$ matrices with real entries. Let $g$ be a function from $\partial\Omega^{o}$ to ${\mathbb{R}}^{n}$. Let $\gamma$ be a positive valued function defined on a right neighborhood of $0$ in the real line. Let $T$ be a map from $]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})$ to $M_{n}({\mathbb{R}})$. Then we consider the problem \[ \left\{ \begin{array}{ll} {\mathrm{div}}\, (T(\omega,Du))=0\qquad\qquad\qquad\qquad\qquad\qquad\qquad \quad \qquad {\mathrm{in}} &\Omega^{o}\setminus\epsilon{\mathrm{cl}}\Omega^{i}, \\ -T(\omega,Du(x))\nu_{\epsilon\Omega^{i}}(x)=\frac{1}{\gamma(\epsilon)}G^{i}(x/ \epsilon, \gamma(\epsilon)\epsilon^{-1} (\log \epsilon)^{-\delta_{2,n}} u(x)) & \forall x\in \epsilon\partial\Omega^{i}, \\ T(\omega,Du(x))\nu^{o}(x)=a^{o}(x)u(x)+g(x) & \forall x\in\partial \Omega^{o}, \end{array} \right. \] where $\nu_{\epsilon\Omega^{i}}$ and $\nu^{o}$ denote the outward unit normal to $\epsilon\partial \Omega^{i}$ and $\partial\Omega^{o}$, respectively, and where $\epsilon>0$ is a small parameter. Here $(\omega-1)$ plays the role of ratio between the first and second Lam\'{e} constants and $T(\omega,\cdot)$ plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor. Under the condition that $\lim_{\epsilon\to 0^{+}}{\gamma(\epsilon)}{\epsilon^{-1}(\log\epsilon)^{-\delta_{2,n}}}=0$ and $\lim_{\epsilon\to 0^{+}}{\epsilon^{n-1}}{\gamma(\epsilon)^{-1}}=0$, we prove that under suitable assumptions the above problem has a family of solutions $\{u(\epsilon,\cdot)\}_{\epsilon\in ]0,\epsilon'[}$ for $\epsilon'$ sufficiently small and we analyze the behaviour of such a family as $\epsilon$ approaches $0$ by an approach which is alternative to those of asymptotic analysis. Here $\delta_{2,n}$ denotes the Kronecker Symbol
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Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach
'Informa UK Limited', 2010Co-Authors: Dalla Riva M, Lanza De Cristoforis, MassimoAbstract:Let Omega(i) and Omega(o) be two bounded open subsets of R(n) containing 0. Let G(i) be a (nonlinear) map of partial derivative Omega(i) x R(n) to R(n). Let a(o) be a map of partial derivative Omega(o) to the set M(n)(R) of n x n matrices with real entries. Let g be a function of partial derivative Omega(o) to R(n). Let gamma be a positive valued function defined on a right neighbourhood of 0 on the real line. Let T be a map of] 1 - (2/n), + infinity[xM(n)(R) to M(n)(R). Then we consider the problem {divdT(omega, Du)) = 0 in Omega(o) backslash epsilon c1 Omega(i), -T(omega, Du(x))v(epsilon Omega)(i)(x) = 1/gamma(epsilon) G(i)(x.epsilon, u(x)) for all x is an element of epsilon partial derivative Omega(i), T(omega, Du(x))v(o)(x) = a(o)(x)u(x) + g(x) for all x is an element of partial derivative Omega(o) where nu(epsilon)Omega(i) and nu(o) denote the outward unit normal to epsilon partial derivative Omega(i) and partial derivative Omega(o), respectively, and where epsilon > 0 is a small parameter. Here (omega - 1) plays the role of ratio between the first and second Lame constants and T(omega, .) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor. Under the condition that lim(epsilon -> 0) gamma(-1) (epsilon)epsilon(log epsilon)(delta 2,n) exists in R, we prove that under suitable assumptions the above problem has a family of solutions {u(epsilon, .)}(epsilon epsilon]0,epsilon'[) for epsilon' sufficiently small and we analyse the behaviour of such a family as epsilon approaches 0 by an approach which is alternative to those of asymptotic analysis. Here delta(2, n) denotes the Kronecker Symbol
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Hypersingularly perturbed loads for a nonlinear traction boundary value problem. A functionalanalytic approach
L. N. Gumilev Eurasian National University, 2010Co-Authors: Dalla Riva M, Lanza De Cristoforis, MassimoAbstract:Let $\Omega^{i}$ and $\Omega^{o}$ be two bounded open subsets of ${\mathbb{R}}^{n}$ containing $0$. Let $G^{i}$ be a (nonlinear) map from $\partial\Omega^{i}\times {\mathbb{R}}^{n}$ to $ {\mathbb{R}}^{n}$. Let $a^{o}$ be a map from $\partial\Omega^{o}$ to the set $M_{n}({\mathbb{R}})$ of $n\times n$ matrices with real entries. Let $g$ be a function from $\partial\Omega^{o}$ to ${\mathbb{R}}^{n}$. Let $\gamma$ be a positive valued function defined on a right neighborhood of $0$ in the real line. Let $T$ be a map from $]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})$ to $M_{n}({\mathbb{R}})$. Then we consider the problem \[ \left\{ \begin{array}{ll} {\mathrm{div}}\, (T(\omega,Du))=0 & {\mathrm{in}}\ \Omega^{o}\setminus\epsilon{\mathrm{cl}}\Omega^{i}\,, \\ -T(\omega,Du(x))\nu_{\epsilon\Omega^{i}}(x)=\frac{1}{\gamma(\epsilon)}G^{i}(x/ \epsilon, \gamma(\epsilon)\epsilon^{-1} (\log \epsilon)^{-\delta_{2,n}} u(x)) & \forall x\in \epsilon\partial\Omega^{i}\,, \\ T(\omega,Du(x))\nu^{o}(x)=a^{o}(x)u(x)+g(x) & \forall x\in\partial \Omega^{o}\,, \end{array} \right. \] where $\nu_{\epsilon\Omega^{i}}$ and $\nu^{o}$ denote the outward unit normal to $\epsilon\partial \Omega^{i}$ and $\partial\Omega^{o}$, respectively, and where $\epsilon>0$ is a small parameter. Here $(\omega-1)$ plays the role of ratio between the first and second Lam\'{e} constants, and $T(\omega,\cdot)$ denotes (a constant multiple of) the linearized Piola Kirchhoff stress tensor, and $\delta_{2,n}$ denotes the Kronecker Symbol. Under the condition that $\gamma$ generates a very strong singularity, \textit{i.e.}, the case in which $\lim_{\epsilon\to 0^{+}}\frac{\gamma(\epsilon)}{\epsilon^{n-1}}$ exists in $ [0,+\infty[$, we prove that under suitable assumptions the above problem has a family of solutions $\{u(\epsilon,\cdot)\}_{\epsilon\in ]0,\epsilon'[}$ for $\epsilon'$ sufficiently small and we analyze the behavior of such a family as $\epsilon$ is close to $0$ by an approach which is alternative to those of asymptotic analysis
Dalla Riva M - One of the best experts on this subject based on the ideXlab platform.
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Weakly Singular and Microscopically Hypersingular Load Perturbation for a Nonlinear Traction Boundary Value Problem: a Functional Analytic Approach
'Springer Science and Business Media LLC', 2011Co-Authors: Dalla Riva M, Lanza De Cristoforis, MassimoAbstract:Let $\Omega^{i}$ and $\Omega^{o}$ be two bounded open subsets of ${\mathbb{R}}^{n}$ containing $0$. Let $G^{i}$ be a (nonlinear) map from $\partial\Omega^{i}\times {\mathbb{R}}^{n}$ to $ {\mathbb{R}}^{n}$. Let $a^{o}$ be a map from $\partial\Omega^{o}$ to the set $M_{n}({\mathbb{R}})$ of $n\times n$ matrices with real entries. Let $g$ be a function from $\partial\Omega^{o}$ to ${\mathbb{R}}^{n}$. Let $\gamma$ be a positive valued function defined on a right neighborhood of $0$ in the real line. Let $T$ be a map from $]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})$ to $M_{n}({\mathbb{R}})$. Then we consider the problem \[ \left\{ \begin{array}{ll} {\mathrm{div}}\, (T(\omega,Du))=0\qquad\qquad\qquad\qquad\qquad\qquad\qquad \quad \qquad {\mathrm{in}} &\Omega^{o}\setminus\epsilon{\mathrm{cl}}\Omega^{i}, \\ -T(\omega,Du(x))\nu_{\epsilon\Omega^{i}}(x)=\frac{1}{\gamma(\epsilon)}G^{i}(x/ \epsilon, \gamma(\epsilon)\epsilon^{-1} (\log \epsilon)^{-\delta_{2,n}} u(x)) & \forall x\in \epsilon\partial\Omega^{i}, \\ T(\omega,Du(x))\nu^{o}(x)=a^{o}(x)u(x)+g(x) & \forall x\in\partial \Omega^{o}, \end{array} \right. \] where $\nu_{\epsilon\Omega^{i}}$ and $\nu^{o}$ denote the outward unit normal to $\epsilon\partial \Omega^{i}$ and $\partial\Omega^{o}$, respectively, and where $\epsilon>0$ is a small parameter. Here $(\omega-1)$ plays the role of ratio between the first and second Lam\'{e} constants and $T(\omega,\cdot)$ plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor. Under the condition that $\lim_{\epsilon\to 0^{+}}{\gamma(\epsilon)}{\epsilon^{-1}(\log\epsilon)^{-\delta_{2,n}}}=0$ and $\lim_{\epsilon\to 0^{+}}{\epsilon^{n-1}}{\gamma(\epsilon)^{-1}}=0$, we prove that under suitable assumptions the above problem has a family of solutions $\{u(\epsilon,\cdot)\}_{\epsilon\in ]0,\epsilon'[}$ for $\epsilon'$ sufficiently small and we analyze the behaviour of such a family as $\epsilon$ approaches $0$ by an approach which is alternative to those of asymptotic analysis. Here $\delta_{2,n}$ denotes the Kronecker Symbol
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Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach
'Informa UK Limited', 2010Co-Authors: Dalla Riva M, Lanza De Cristoforis, MassimoAbstract:Let Omega(i) and Omega(o) be two bounded open subsets of R(n) containing 0. Let G(i) be a (nonlinear) map of partial derivative Omega(i) x R(n) to R(n). Let a(o) be a map of partial derivative Omega(o) to the set M(n)(R) of n x n matrices with real entries. Let g be a function of partial derivative Omega(o) to R(n). Let gamma be a positive valued function defined on a right neighbourhood of 0 on the real line. Let T be a map of] 1 - (2/n), + infinity[xM(n)(R) to M(n)(R). Then we consider the problem {divdT(omega, Du)) = 0 in Omega(o) backslash epsilon c1 Omega(i), -T(omega, Du(x))v(epsilon Omega)(i)(x) = 1/gamma(epsilon) G(i)(x.epsilon, u(x)) for all x is an element of epsilon partial derivative Omega(i), T(omega, Du(x))v(o)(x) = a(o)(x)u(x) + g(x) for all x is an element of partial derivative Omega(o) where nu(epsilon)Omega(i) and nu(o) denote the outward unit normal to epsilon partial derivative Omega(i) and partial derivative Omega(o), respectively, and where epsilon > 0 is a small parameter. Here (omega - 1) plays the role of ratio between the first and second Lame constants and T(omega, .) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor. Under the condition that lim(epsilon -> 0) gamma(-1) (epsilon)epsilon(log epsilon)(delta 2,n) exists in R, we prove that under suitable assumptions the above problem has a family of solutions {u(epsilon, .)}(epsilon epsilon]0,epsilon'[) for epsilon' sufficiently small and we analyse the behaviour of such a family as epsilon approaches 0 by an approach which is alternative to those of asymptotic analysis. Here delta(2, n) denotes the Kronecker Symbol
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Hypersingularly perturbed loads for a nonlinear traction boundary value problem. A functionalanalytic approach
L. N. Gumilev Eurasian National University, 2010Co-Authors: Dalla Riva M, Lanza De Cristoforis, MassimoAbstract:Let $\Omega^{i}$ and $\Omega^{o}$ be two bounded open subsets of ${\mathbb{R}}^{n}$ containing $0$. Let $G^{i}$ be a (nonlinear) map from $\partial\Omega^{i}\times {\mathbb{R}}^{n}$ to $ {\mathbb{R}}^{n}$. Let $a^{o}$ be a map from $\partial\Omega^{o}$ to the set $M_{n}({\mathbb{R}})$ of $n\times n$ matrices with real entries. Let $g$ be a function from $\partial\Omega^{o}$ to ${\mathbb{R}}^{n}$. Let $\gamma$ be a positive valued function defined on a right neighborhood of $0$ in the real line. Let $T$ be a map from $]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})$ to $M_{n}({\mathbb{R}})$. Then we consider the problem \[ \left\{ \begin{array}{ll} {\mathrm{div}}\, (T(\omega,Du))=0 & {\mathrm{in}}\ \Omega^{o}\setminus\epsilon{\mathrm{cl}}\Omega^{i}\,, \\ -T(\omega,Du(x))\nu_{\epsilon\Omega^{i}}(x)=\frac{1}{\gamma(\epsilon)}G^{i}(x/ \epsilon, \gamma(\epsilon)\epsilon^{-1} (\log \epsilon)^{-\delta_{2,n}} u(x)) & \forall x\in \epsilon\partial\Omega^{i}\,, \\ T(\omega,Du(x))\nu^{o}(x)=a^{o}(x)u(x)+g(x) & \forall x\in\partial \Omega^{o}\,, \end{array} \right. \] where $\nu_{\epsilon\Omega^{i}}$ and $\nu^{o}$ denote the outward unit normal to $\epsilon\partial \Omega^{i}$ and $\partial\Omega^{o}$, respectively, and where $\epsilon>0$ is a small parameter. Here $(\omega-1)$ plays the role of ratio between the first and second Lam\'{e} constants, and $T(\omega,\cdot)$ denotes (a constant multiple of) the linearized Piola Kirchhoff stress tensor, and $\delta_{2,n}$ denotes the Kronecker Symbol. Under the condition that $\gamma$ generates a very strong singularity, \textit{i.e.}, the case in which $\lim_{\epsilon\to 0^{+}}\frac{\gamma(\epsilon)}{\epsilon^{n-1}}$ exists in $ [0,+\infty[$, we prove that under suitable assumptions the above problem has a family of solutions $\{u(\epsilon,\cdot)\}_{\epsilon\in ]0,\epsilon'[}$ for $\epsilon'$ sufficiently small and we analyze the behavior of such a family as $\epsilon$ is close to $0$ by an approach which is alternative to those of asymptotic analysis
Andrew Granville - One of the best experts on this subject based on the ideXlab platform.
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An Upper Bound on the Least Inert Prime in a Real Quadratic Field
2007Co-Authors: Andrew Granville Mollin, Andrew Granville, Richard Mollin, Hugh C. WilliamsAbstract:It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D ? 3705, there is always at least one prime p ! p D=2 such that the Kronecker Symbol (D=p) = \Gamma1. 1991 Mathematics Subject Classification 11R11, 11Y40 The first author is a Presidential Faculty Fellow. His research is partiallly supported by the NSF. The research of the second two authors is partially supported by NSERC of Canada 1 1 Introduction Let D be the fundamental discriminant of a real quadratic field and let S = f5; 8; 12; 13; 17; 24; 28; 33; 40; 57; 60; 73; 76; 88; 97; 105; 124; 129; 136; 145; 156; 184; 204; 249; 280; 316; 345; 364; 385; 424; 456; 520; 609; 616; 924; 940; 984; 1065; 1596; 2044; 2244; 3705g: At the end of Chapter 6 of [5], the second author made the following conjecture. Conjecture. The values of D for which the least prime p such that the Kronecker Symbol (D=p) = \Gamma1 satisfies p ? p D=2 are precisely those in S. He also veri..
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An upper bound on the least inert prime in a real quadratic field
Canadian Journal of Mathematics, 2000Co-Authors: Andrew Granville, Richard Mollin, Hugh C. WilliamsAbstract:AbstractIt is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D > 3705, there is always at least one prime p < √D/2 such that the Kronecker Symbol (D/p) = −1.
Peng Gao - One of the best experts on this subject based on the ideXlab platform.
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The mean square of real character sums
arXiv: Number Theory, 2019Co-Authors: Peng GaoAbstract:In this paper, we evaluate a smoothed sum of the form $\displaystyle \sideset{}{^*}\sum_{d \leq X}\left(\sum_{n \leq Y} \left(\frac{8d}{n} \right ) \right)^2$, where $\left ( \frac{8d}{\cdot} \right )$ is the Kronecker Symbol and $\sideset{}{^*}\sum$ denotes a sum over positive odd square-free integers.
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Mean values of divisors twisted by quadratic characters
arXiv: Number Theory, 2019Co-Authors: Peng GaoAbstract:In this paper, we evaluate the sum $\sum_{m,n}\leg {m}{n}d(n)$, where $\leg {m}{n}$ is the Kronecker Symbol and $d(n)$ is the divisor function.
