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Emili Besalú - One of the best experts on this subject based on the ideXlab platform.
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Stereographic Projection of Density Functions (DF) and the Holographic Electronic Density Theorem (HEDT).
Journal of chemical theory and computation, 2012Co-Authors: Emili Besalú, Ramon Carbó-dorcaAbstract:Mezey's holographic electronic Density Theorem is discussed from the point of view of stereographic projection techniques. Such a mathematical procedure is analyzed in depth from the point of view of first-order Density functions; the procedure is then extended to any relevant quantum chemical function. This endeavor provides the background to construct a Holographic General Function Theorem (HGFT) for multivariate well-behaved functions. Stereographic projections, applied first as a way to obtain pictures of molecular quantum chemical functions, are shown to provide a flexible and original vantage point for visualizing, from any location and any chosen perspective, the form of any well-behaved function, irrespective of the number of variables involved. The pictographic possibilities of the HGFT are explored and exploited in several examples.
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mathematical aspects of the lcao mo first order Density function 4 a discussion on the connection of taylor series expansion of electronic Density tsed function with the holographic electron Density Theorem hedt and the hohenberg kohn Theorem hkt
Journal of Mathematical Chemistry, 2011Co-Authors: Ramon Carbodorca, Emili BesalúAbstract:Taylor series expansion of electronic Density (TSED) functions are set up in order to propose them as an alternative description of the holographic electron Density Theorem (HEDT) functions. The manipulation of the obtained TSED general formulation leads to a connection between TSED, HEDT and Hohenberg-Kohn Theorem (HKT).
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communications on quantum similarity 2 a geometric discussion on holographic electron Density Theorem and confined quantum similarity measures
Journal of Computational Chemistry, 2010Co-Authors: Ramon Carbodorca, Emili BesalúAbstract:The so-called holographic electron Density Theorem (HEDT) is analyzed from an algebraic perspective, and a brief analytical point of view is also given. The connection of the HEDT with quantum similarity measures (QSM) over electronic Density functions (DF) is studied using GTO functions, atomic ASA DF, and promolecular ASA DF. Restricted integration of QSM over a box of finite side length is discussed for all this DF. This work emphasizes the geometric aspects of HEDT, but for the sake of completeness, some analytical insight based on a general Taylor series expansion is also given at the end. works of Munch and Reiss and Mezey is that the previous authors result could not apply for complete molecular densities but only for a finite, bounded domain of space, so it was valid only for artificial molecular models. However, Mezey proposed a general framework applicable to complete molecules, by cir- cumventing the limitations of analytic continuation Theorems for compact sets. Mezey, using the fact that beyond some distance of the origin, electron Density converges uniformly to zero, con- nected this Density function (DF) behavior with the Alexandrov one-point compactification Theorem. Doing the analysis in an abstract four-dimensional space, Mezey could prove the HEDT for real molecules, without artificial boundaries. Moreover, Mezey in this mentioned work also connected the HEDT set up with quantum similarity measures and indices.
Ethan Smith - One of the best experts on this subject based on the ideXlab platform.
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A generalization of the Barban-Davenport-Halberstam Theorem to number fields
2014Co-Authors: Ethan SmithAbstract:Abstract. Let L/K be a Galois extension of number fields. The problem of counting the number of prime ideals p of K with fixed Frobenius class in Gal(L/K) and norm satisfying a congruence condition is considered. We show that the square of the error term arising from the Chebotarëv Density Theorem for this problem is small “on average. ” The result may be viewed as a variation on the classical Barban-Davenport-Halberstam Theorem. 1
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(APPEARED IN PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY)
2014Co-Authors: Ethan SmithAbstract:Abstract. Let K be a fixed number field, and assume that K is Galois over Q. Previously, the author showed that when estimating the number of prime ideals with norm congruent to a modulo q via the Chebotarëv Density Theorem, the mean square error in the approximation is small when averaging over all q ≤ Q and all appropriate a. In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical Barban-Davenport-Halberstam Theorem in the case K = Q. 1
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A generalization of the Barban-Davenport-Halberstam Theorem to number fields
2014Co-Authors: Ethan SmithAbstract:Abstract. For a fixed number field K, we consider the mean square error in estimating the number of primes with norm congruent to a modulo q by the Chebotarëv Density Theorem when averaging over all q ≤ Q and all appropriate a. Using a large sieve inequality, we obtain an upper bound similar to the Barban-Davenport-Halberstam Theorem. 1
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a generalization of the barban davenport halberstam Theorem to number fields
Journal of Number Theory, 2009Co-Authors: Ethan SmithAbstract:For a fixed number field K, we consider the mean square error in estimating the number of primes with norm congruent to a modulo q by the Chebotarev Density Theorem when averaging over all q⩽Q and all appropriate a. Using a large sieve inequality, we obtain an upper bound similar to the Barban–Davenport–Halberstam Theorem.
Ramon Carbodorca - One of the best experts on this subject based on the ideXlab platform.
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mathematical aspects of the lcao mo first order Density function 4 a discussion on the connection of taylor series expansion of electronic Density tsed function with the holographic electron Density Theorem hedt and the hohenberg kohn Theorem hkt
Journal of Mathematical Chemistry, 2011Co-Authors: Ramon Carbodorca, Emili BesalúAbstract:Taylor series expansion of electronic Density (TSED) functions are set up in order to propose them as an alternative description of the holographic electron Density Theorem (HEDT) functions. The manipulation of the obtained TSED general formulation leads to a connection between TSED, HEDT and Hohenberg-Kohn Theorem (HKT).
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communications on quantum similarity 2 a geometric discussion on holographic electron Density Theorem and confined quantum similarity measures
Journal of Computational Chemistry, 2010Co-Authors: Ramon Carbodorca, Emili BesalúAbstract:The so-called holographic electron Density Theorem (HEDT) is analyzed from an algebraic perspective, and a brief analytical point of view is also given. The connection of the HEDT with quantum similarity measures (QSM) over electronic Density functions (DF) is studied using GTO functions, atomic ASA DF, and promolecular ASA DF. Restricted integration of QSM over a box of finite side length is discussed for all this DF. This work emphasizes the geometric aspects of HEDT, but for the sake of completeness, some analytical insight based on a general Taylor series expansion is also given at the end. works of Munch and Reiss and Mezey is that the previous authors result could not apply for complete molecular densities but only for a finite, bounded domain of space, so it was valid only for artificial molecular models. However, Mezey proposed a general framework applicable to complete molecules, by cir- cumventing the limitations of analytic continuation Theorems for compact sets. Mezey, using the fact that beyond some distance of the origin, electron Density converges uniformly to zero, con- nected this Density function (DF) behavior with the Alexandrov one-point compactification Theorem. Doing the analysis in an abstract four-dimensional space, Mezey could prove the HEDT for real molecules, without artificial boundaries. Moreover, Mezey in this mentioned work also connected the HEDT set up with quantum similarity measures and indices.
Yahui Wang - One of the best experts on this subject based on the ideXlab platform.
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the Density Theorem of a class of dilation and modulation systems on the half real line
Results in Mathematics, 2019Co-Authors: Yahui WangAbstract:In practice, the time variable cannot be negative. The space $$L^2({\mathbb {R}}_+)$$ of square integrable functions defined on the right half real line $${\mathbb {R}}_+$$ models the causal signal space. This paper focuses on a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$. The Density Theorem for Gabor systems in $$L^2(\mathbb {R})$$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $$L^2({\mathbb {R}})$$ in terms of the index set alone-independently of window functions. The space $$L^2({\mathbb {R}}_+)$$ admits no nontrivial Gabor system since $${\mathbb {R}}_+$$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ and the notion of $$\Theta $$-transform matrix. Using the $$\Theta $$-transform matrix method we obtain the Density Theorem of the dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ under the condition that $$\log _ba$$ is a positive rational number, where a and b are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in $$L^2({\mathbb {R}}_+)$$ is that $$\log _ba \le 1$$. Simultaneously, we obtain a $$\Theta $$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $$L^2(\mathbb {R}_+)$$.
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the Density Theorem of a class of dilation and modulation systems on the half real line
arXiv: Functional Analysis, 2017Co-Authors: Yahui WangAbstract:In the practice, time variable cannot be negative. The space $L^2(\Bbb R_+)$ of square integrable functions defined on the right half real line $\Bbb R_+$ models causal signal space. This paper focuses on a class of dilation-and-modulation systems in $L^2(\Bbb R_+)$. The Density Theorem for Gabor systems in $L^2(\Bbb R)$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $L^2(\Bbb R)$ in terms of the index set alone-independently of window functions. The space $L^2(\Bbb R_+)$ admits no nontrivial Gabor system since $\Bbb R_+$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $L^2(\Bbb R_+)$ and the notion of $\Theta$-transform matrix. Using $\Theta$-transform matrix method we obtain the Density Theorem of the dilation-and-modulation systems in $L^2(\Bbb R_+)$ under the condition that $\log_ba$ is a positive rational number, where $a$ and $b$ are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in $L^2(\Bbb R_+)$ is that $\log_ba \leq 1$. Simultaneously, we obtain a $\Theta$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $L^2(\Bbb R_+)$.
Smith Ethan - One of the best experts on this subject based on the ideXlab platform.
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A generalization of the Barban-Davenport-Halberstam Theorem to number fields
'Elsevier BV', 2012Co-Authors: Smith EthanAbstract:For a fixed number field $K$, we consider the mean square error in estimating the number of primes with norm congruent to $a$ modulo $q$ by the Chebotar\"ev Density Theorem when averaging over all $q\le Q$ and all appropriate $a$. Using a large sieve inequality, we obtain an upper bound similar to the Barban-Davenport-Halberstam Theorem.Comment: Preprint of an old pape
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A Barban-Davenport-Halberstam asymptotic for number fields
'American Mathematical Society (AMS)', 2012Co-Authors: Smith EthanAbstract:Let $K$ be a fixed number field, and assume that $K$ is Galois over $\qq$. Previously, the author showed that when estimating the number of prime ideals with norm congruent to $a$ modulo $q$ via the Chebotar\"ev Density Theorem, the mean square error in the approximation is small when averaging over all $q\le Q$ and all appropriate $a$. In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical Barban-Davenport-Halberstam Theorem in the case $K=\qq$.Comment: Preprint of an old pape
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A barban-davenport-halberstam asymptotic for number fields
'American Mathematical Society (AMS)', 2010Co-Authors: Smith EthanAbstract:Let K be a fixed number field, and assume that K is Galois over ℚ. Previously, the author showed that when estimating the number of prime ideals with norm congruent to a modulo q via the Chebotarëv Density Theorem, the mean square error in the approximation is small when averaging over all q ≤ Q and all appropriate a. In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical Barban- DavenportHalberstam Theorem in the case K = ℚ. © 2010 American Mathematical Society