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Laszlo Lovasz - One of the best experts on this subject based on the ideXlab platform.
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a tight bound for green s arithmetic triangle removal lemma in vector spaces
Advances in Mathematics, 2017Co-Authors: Jacob Fox, Laszlo LovaszAbstract:Abstract Let p be a fixed prime. A triangle in F p n is an Ordered Triple ( x , y , z ) of points satisfying x + y + z = 0 . Let N = p n = | F p n | . Green proved an arithmetic triangle removal lemma which says that for every ϵ > 0 and prime p , there is a δ > 0 such that if X , Y , Z ⊂ F p n and the number of triangles in X × Y × Z is at most δ N 2 , then we can delete ϵN elements from X , Y , and Z and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that 1 / δ can be taken to be an exponential tower of twos of height logarithmic in 1 / ϵ . We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in F p n . We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is an explicit number C p such that we may take δ = ( ϵ / 3 ) C p , and we must have δ ≤ ϵ C p − o ( 1 ) . In particular, C 2 = 1 + 1 / ( 5 / 3 − log 2 3 ) ≈ 13.239 , and C 3 = 1 + 1 / c 3 with c 3 = 1 − log b log 3 , b = a − 2 / 3 + a 1 / 3 + a 4 / 3 , and a = 33 − 1 8 , which gives C 3 ≈ 13.901 . The proof uses the essentially sharp bound on multicolored sum-free sets due to work of Kleinberg–Sawin–Speyer, Norin, and Pebody, which builds on the recent breakthrough on the cap set problem by Croot–Lev–Pach, and the subsequent work by Ellenberg–Gijswijt, Blasiak–Church–Cohn–Grochow–Naslund–Sawin–Umans, and Alon.
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a tight bound for green s arithmetic triangle removal lemma in vector spaces
Symposium on Discrete Algorithms, 2017Co-Authors: Jacob Fox, Laszlo LovaszAbstract:Let p be a fixed prime. A triangle in 𝔽np is an Ordered Triple (x, y, z) of points satisfying x + y + z = 0. Let N = pn = |𝔽np|. Green proved an arithmetic triangle removal lemma which says that for every ϵ > 0 and prime p, there is a Δ > 0 such that if X, Y, Z ⊂ 𝔽np and the number of triangles in X × Y × Z is at most ΔN2, then we can delete ϵN elements from X, Y, and Z and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that 1/Δ can be taken to be an exponential tower of twos of height logarithmic in 1/ϵ. We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in 𝔽np. We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is a computable number Cp such that we may take Δ = (ϵ/3)Cp, and we must have Δ ≤ ϵCp − 0(1). In particular, C2 = 1 + 1/(5/3 − log2 3) ≈ 13.239, and C3 = 1 + 1/c3 with [EQUATION], and [EQUATION], which gives C3 ≈ 13.901. The proof uses Kleinberg, Sawin, and Speyer's essentially sharp bound on multicolored sum-free sets, which builds on the recent breakthrough on the cap set problem by Croot-Lev-Pach, and the subsequent work by Ellenberg-Gijswijt, Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, and Alon.
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a tight bound for green s arithmetic triangle removal lemma in vector spaces
arXiv: Combinatorics, 2016Co-Authors: Jacob Fox, Laszlo LovaszAbstract:Let $p$ be a fixed prime. A triangle in $\mathbb{F}_p^n$ is an Ordered Triple $(x,y,z)$ of points satisfying $x+y+z=0$. Let $N=p^n=|\mathbb{F}_p^n|$. Green proved an arithmetic triangle removal lemma which says that for every $\epsilon>0$ and prime $p$, there is a $\delta>0$ such that if $X,Y,Z \subset \mathbb{F}_p^n$ and the number of triangles in $X \times Y \times Z$ is at most $\delta N^2$, then we can delete $\epsilon N$ elements from $X$, $Y$, and $Z$ and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that $1/\delta$ can be taken to be an exponential tower of twos of height logarithmic in $1/\epsilon$. We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in $\mathbb{F}_p^n$. We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is a computable number $C_p$ such that we may take $\delta = (\epsilon/3)^{C_p}$, and we must have $\delta \leq \epsilon^{C_p-o(1)}$. In particular, $C_2=1+1/(5/3 - \log_2 3) \approx 13.239$, and $C_3=1+1/c_3$ with $c_3=1-\frac{\log b}{\log 3}$, $b=a^{-2/3}+a^{1/3}+a^{4/3}$, and $a=\frac{\sqrt{33}-1}{8}$, which gives $C_3 \approx 13.901$. The proof uses Kleinberg, Sawin, and Speyer's essentially sharp bound on multicolored sum-free sets, which builds on the recent breakthrough on the cap set problem by Croot-Lev-Pach, and the subsequent work by Ellenberg-Gijswijt, Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, and Alon.
Jacob Fox - One of the best experts on this subject based on the ideXlab platform.
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a tight bound for green s arithmetic triangle removal lemma in vector spaces
Advances in Mathematics, 2017Co-Authors: Jacob Fox, Laszlo LovaszAbstract:Abstract Let p be a fixed prime. A triangle in F p n is an Ordered Triple ( x , y , z ) of points satisfying x + y + z = 0 . Let N = p n = | F p n | . Green proved an arithmetic triangle removal lemma which says that for every ϵ > 0 and prime p , there is a δ > 0 such that if X , Y , Z ⊂ F p n and the number of triangles in X × Y × Z is at most δ N 2 , then we can delete ϵN elements from X , Y , and Z and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that 1 / δ can be taken to be an exponential tower of twos of height logarithmic in 1 / ϵ . We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in F p n . We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is an explicit number C p such that we may take δ = ( ϵ / 3 ) C p , and we must have δ ≤ ϵ C p − o ( 1 ) . In particular, C 2 = 1 + 1 / ( 5 / 3 − log 2 3 ) ≈ 13.239 , and C 3 = 1 + 1 / c 3 with c 3 = 1 − log b log 3 , b = a − 2 / 3 + a 1 / 3 + a 4 / 3 , and a = 33 − 1 8 , which gives C 3 ≈ 13.901 . The proof uses the essentially sharp bound on multicolored sum-free sets due to work of Kleinberg–Sawin–Speyer, Norin, and Pebody, which builds on the recent breakthrough on the cap set problem by Croot–Lev–Pach, and the subsequent work by Ellenberg–Gijswijt, Blasiak–Church–Cohn–Grochow–Naslund–Sawin–Umans, and Alon.
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a tight bound for green s arithmetic triangle removal lemma in vector spaces
Symposium on Discrete Algorithms, 2017Co-Authors: Jacob Fox, Laszlo LovaszAbstract:Let p be a fixed prime. A triangle in 𝔽np is an Ordered Triple (x, y, z) of points satisfying x + y + z = 0. Let N = pn = |𝔽np|. Green proved an arithmetic triangle removal lemma which says that for every ϵ > 0 and prime p, there is a Δ > 0 such that if X, Y, Z ⊂ 𝔽np and the number of triangles in X × Y × Z is at most ΔN2, then we can delete ϵN elements from X, Y, and Z and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that 1/Δ can be taken to be an exponential tower of twos of height logarithmic in 1/ϵ. We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in 𝔽np. We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is a computable number Cp such that we may take Δ = (ϵ/3)Cp, and we must have Δ ≤ ϵCp − 0(1). In particular, C2 = 1 + 1/(5/3 − log2 3) ≈ 13.239, and C3 = 1 + 1/c3 with [EQUATION], and [EQUATION], which gives C3 ≈ 13.901. The proof uses Kleinberg, Sawin, and Speyer's essentially sharp bound on multicolored sum-free sets, which builds on the recent breakthrough on the cap set problem by Croot-Lev-Pach, and the subsequent work by Ellenberg-Gijswijt, Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, and Alon.
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a tight bound for green s arithmetic triangle removal lemma in vector spaces
arXiv: Combinatorics, 2016Co-Authors: Jacob Fox, Laszlo LovaszAbstract:Let $p$ be a fixed prime. A triangle in $\mathbb{F}_p^n$ is an Ordered Triple $(x,y,z)$ of points satisfying $x+y+z=0$. Let $N=p^n=|\mathbb{F}_p^n|$. Green proved an arithmetic triangle removal lemma which says that for every $\epsilon>0$ and prime $p$, there is a $\delta>0$ such that if $X,Y,Z \subset \mathbb{F}_p^n$ and the number of triangles in $X \times Y \times Z$ is at most $\delta N^2$, then we can delete $\epsilon N$ elements from $X$, $Y$, and $Z$ and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that $1/\delta$ can be taken to be an exponential tower of twos of height logarithmic in $1/\epsilon$. We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in $\mathbb{F}_p^n$. We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is a computable number $C_p$ such that we may take $\delta = (\epsilon/3)^{C_p}$, and we must have $\delta \leq \epsilon^{C_p-o(1)}$. In particular, $C_2=1+1/(5/3 - \log_2 3) \approx 13.239$, and $C_3=1+1/c_3$ with $c_3=1-\frac{\log b}{\log 3}$, $b=a^{-2/3}+a^{1/3}+a^{4/3}$, and $a=\frac{\sqrt{33}-1}{8}$, which gives $C_3 \approx 13.901$. The proof uses Kleinberg, Sawin, and Speyer's essentially sharp bound on multicolored sum-free sets, which builds on the recent breakthrough on the cap set problem by Croot-Lev-Pach, and the subsequent work by Ellenberg-Gijswijt, Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, and Alon.
Xiang Ming Chen - One of the best experts on this subject based on the ideXlab platform.
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Hybrid improper ferroelectricity in A-site cation Ordered Li2La2Ti3O10 ceramic with Triple-layer Ruddlesden–Popper structure
Applied Physics Letters, 2021Co-Authors: Bi Hui Zhang, B. H. Chen, Xiao Qiang Liu, James Hester, Xiang Ming ChenAbstract:Hybrid improper ferroelectricity has been extensively studied in double-layer Ruddlesden–Popper oxides in recent years. Although the hybrid improper ferroelectricity could be created among Triple-layer Ruddlesden–Popper oxides with an Ordered A-site cation predicted by the first-principles calculations, no experimental result has been reported yet. In the present work, the room-temperature ferroelectricity has been observed in Li2La2Ti3O10 ceramics with an A-site cation Ordered Triple-layer Ruddlesden–Popper structure. The polar phase P21ab has been determined by combining the first-principles calculation and the powder diffraction analysis at room temperature. The hybrid improper ferroelectricity was induced by the Triple-coupled irreps including the A-site cation ordering. The variable temperature differential scanning calorimetry measurements and dielectric responses indicate no evidence of phase transition over the temperature range of 200–1080 K. The present work sheds light on designing the hybrid improper ferroelectrics in A-site Ordered Triple-layer Ruddlesden–Popper compounds.
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hybrid improper ferroelectricity in a site cation Ordered li2la2ti3o10 ceramic with Triple layer ruddlesden popper structure
Applied Physics Letters, 2021Co-Authors: Bi Hui Zhang, B. H. Chen, Xiao Qiang Liu, James Hester, Xiang Ming ChenAbstract:Hybrid improper ferroelectricity has been extensively studied in double-layer Ruddlesden–Popper oxides in recent years. Although the hybrid improper ferroelectricity could be created among Triple-layer Ruddlesden–Popper oxides with an Ordered A-site cation predicted by the first-principles calculations, no experimental result has been reported yet. In the present work, the room-temperature ferroelectricity has been observed in Li2La2Ti3O10 ceramics with an A-site cation Ordered Triple-layer Ruddlesden–Popper structure. The polar phase P21ab has been determined by combining the first-principles calculation and the powder diffraction analysis at room temperature. The hybrid improper ferroelectricity was induced by the Triple-coupled irreps including the A-site cation ordering. The variable temperature differential scanning calorimetry measurements and dielectric responses indicate no evidence of phase transition over the temperature range of 200–1080 K. The present work sheds light on designing the hybrid improper ferroelectrics in A-site Ordered Triple-layer Ruddlesden–Popper compounds.
Bi Hui Zhang - One of the best experts on this subject based on the ideXlab platform.
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Hybrid improper ferroelectricity in A-site cation Ordered Li2La2Ti3O10 ceramic with Triple-layer Ruddlesden–Popper structure
Applied Physics Letters, 2021Co-Authors: Bi Hui Zhang, B. H. Chen, Xiao Qiang Liu, James Hester, Xiang Ming ChenAbstract:Hybrid improper ferroelectricity has been extensively studied in double-layer Ruddlesden–Popper oxides in recent years. Although the hybrid improper ferroelectricity could be created among Triple-layer Ruddlesden–Popper oxides with an Ordered A-site cation predicted by the first-principles calculations, no experimental result has been reported yet. In the present work, the room-temperature ferroelectricity has been observed in Li2La2Ti3O10 ceramics with an A-site cation Ordered Triple-layer Ruddlesden–Popper structure. The polar phase P21ab has been determined by combining the first-principles calculation and the powder diffraction analysis at room temperature. The hybrid improper ferroelectricity was induced by the Triple-coupled irreps including the A-site cation ordering. The variable temperature differential scanning calorimetry measurements and dielectric responses indicate no evidence of phase transition over the temperature range of 200–1080 K. The present work sheds light on designing the hybrid improper ferroelectrics in A-site Ordered Triple-layer Ruddlesden–Popper compounds.
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hybrid improper ferroelectricity in a site cation Ordered li2la2ti3o10 ceramic with Triple layer ruddlesden popper structure
Applied Physics Letters, 2021Co-Authors: Bi Hui Zhang, B. H. Chen, Xiao Qiang Liu, James Hester, Xiang Ming ChenAbstract:Hybrid improper ferroelectricity has been extensively studied in double-layer Ruddlesden–Popper oxides in recent years. Although the hybrid improper ferroelectricity could be created among Triple-layer Ruddlesden–Popper oxides with an Ordered A-site cation predicted by the first-principles calculations, no experimental result has been reported yet. In the present work, the room-temperature ferroelectricity has been observed in Li2La2Ti3O10 ceramics with an A-site cation Ordered Triple-layer Ruddlesden–Popper structure. The polar phase P21ab has been determined by combining the first-principles calculation and the powder diffraction analysis at room temperature. The hybrid improper ferroelectricity was induced by the Triple-coupled irreps including the A-site cation ordering. The variable temperature differential scanning calorimetry measurements and dielectric responses indicate no evidence of phase transition over the temperature range of 200–1080 K. The present work sheds light on designing the hybrid improper ferroelectrics in A-site Ordered Triple-layer Ruddlesden–Popper compounds.
Yuichi Shimakawa - One of the best experts on this subject based on the ideXlab platform.
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Spin orders in the charge disproportionated phases of A-site layer Ordered Triple perovskite LaCa2Fe3O9
Physical Review B: Condensed Matter and Materials Physics, 2018Co-Authors: Angel Arevalo-lopez, Yoshiteru Hosaka, Haichuan Guo, Fabio Romero, Takashi Saito, J. Paul Attfield, Yuichi ShimakawaAbstract:The coupling between spins and charge disproportionation states has been investigated in the LaCa 2 Fe 3 O 9 oxide with neutron powder diffraction. This A-site layer Ordered Triple perovskite LaCa 2 Fe 3 O 9 undergoes charge disproportionation on cooling and shows two different charge ordering patterns. At 230 K, Fe 3.67+ disproportionates into a 2:1 ratio of Fe 3+ :Fe 5+ which order in a layered manner along the direction of the pseudocubic unit cell. At lower temperatures (T < 170 K), the charge ordering pattern changes to a layered arrangement along the direction. Neutron powder diffraction data shows that in the intermediate temperature range (170 K < T < 230 K) the spins order into a cycloidal structure on the ac plane for the Fe 3+ cations while the Fe 5+ cations remain paramagnetic. For the lowest temperature range (2 K < T < 190 K), the spin structure follows the charge ordering and evolves to a layered magnetic structure. Introduction.
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spin order in the charge disproportionated phases of the a site layer Ordered Triple perovskite lac a 2 f e 3 o 9
Physical Review B, 2018Co-Authors: Angel M Arevalolopez, Yoshiteru Hosaka, Haichuan Guo, Takashi Saito, Fabio Denis Romero, Paul J Attfield, Yuichi ShimakawaAbstract:The coupling between spins and charge disproportionation states has been investigated in the $\mathrm{LaC}{\mathrm{a}}_{2}\mathrm{F}{\mathrm{e}}_{3}{\mathrm{O}}_{9}$ oxide with neutron powder diffraction. This $A$-site layer Ordered Triple perovskite $\mathrm{LaC}{\mathrm{a}}_{2}\mathrm{F}{\mathrm{e}}_{3}{\mathrm{O}}_{9}$ undergoes charge disproportionation on cooling and shows two different charge ordering patterns. At 230 K, $\mathrm{F}{\mathrm{e}}^{3.67+}$ disproportionates into a 2:1 ratio of $\mathrm{F}{\mathrm{e}}^{3+}:\mathrm{F}{\mathrm{e}}^{5+}$, which order in a layered manner along the $\ensuremath{\langle}010\ensuremath{\rangle}$ direction of the pseudocubic unit cell. At lower temperatures $(Tl170\phantom{\rule{0.16em}{0ex}}\mathrm{K})$, the charge ordering pattern changes to a layered arrangement along the $\ensuremath{\langle}111\ensuremath{\rangle}$ direction. Neutron powder diffraction data show that in the intermediate temperature range ($170\phantom{\rule{0.16em}{0ex}}\mathrm{K}lTl230\phantom{\rule{0.16em}{0ex}}\mathrm{K}$) the spins order into a cycloidal structure on the $ac$ plane for the $\mathrm{F}{\mathrm{e}}^{3+}$ cations while the $\mathrm{F}{\mathrm{e}}^{5+}$ cations remain paramagnetic. For the lowest temperature range ($2\phantom{\rule{0.16em}{0ex}}\mathrm{K}lTl190\phantom{\rule{0.16em}{0ex}}\mathrm{K}$), the spin structure follows the charge ordering and evolves to a $\ensuremath{\langle}111\ensuremath{\rangle}$ layered magnetic structure.