?9=qJ]Ga,n/Oyz,G_r-RoQNdPh(*?JDm
October 6, 2026 at 12:10 PM
?9=qJ]Ga,n/Oyz,G_r-RoQNdPh(*?JDm
Some states deliver meals to Medicaid patients that address their dietary needs. Evidence shows it aids recovery & saves healthcare costs, but fed'l budget cuts may lead some states to cancel it.
NPR: apple.news/Al-g_R-XkSMa...
😱doing this to people is so cruel, no compassion at all in the regime
NPR: apple.news/Al-g_R-XkSMa...
😱doing this to people is so cruel, no compassion at all in the regime
Meal deliveries can save money and improve health. Will they survive Medicaid cuts? — NPR
Some states deliver meals to Medicaid patients that address their dietary needs. Evidence shows it aids recovery and saves healthcare costs, but federal budget cuts may lead some states to cancel it.
apple.news
September 27, 2026 at 6:26 AM
Some states deliver meals to Medicaid patients that address their dietary needs. Evidence shows it aids recovery & saves healthcare costs, but fed'l budget cuts may lead some states to cancel it.
NPR: apple.news/Al-g_R-XkSMa...
😱doing this to people is so cruel, no compassion at all in the regime
NPR: apple.news/Al-g_R-XkSMa...
😱doing this to people is so cruel, no compassion at all in the regime
2602.03716
$S=\langle d_1,\dots,d_m\rangle$ を数値半群とし、$k[S]$ をその半群環とする。$k[S]$のヒルベルト数式は、シジジー次数における交互の冪和を符号化する正規化された交互シジジー冪和$K_p(S)$を決定する。フェルは、すべての $p\ge 0$ に対して、ギャップ冪和 $G_r(S)=\sum_{g\notin S} g^r$ と、生成元に...
$S=\langle d_1,\dots,d_m\rangle$ を数値半群とし、$k[S]$ をその半群環とする。$k[S]$のヒルベルト数式は、シジジー次数における交互の冪和を符号化する正規化された交互シジジー冪和$K_p(S)$を決定する。フェルは、すべての $p\ge 0$ に対して、ギャップ冪和 $G_r(S)=\sum_{g\notin S} g^r$ と、生成元に...
![$S=\langle d_1,\dots,d_m\rangle$ を数値半群とし、$k[S]$ をその半群環とする。
$k[S]$のヒルベルト数式は、シジジー次数における交互の冪和を符号化する正規化された交互シジジー冪和$K_p(S)$を決定する。
フェルは、すべての $p\ge 0$ に対して、ギャップ冪和 $G_r(S)=\sum_{g\notin S} g^r$ と、生成元による冪和 $σ_k=\sum_i d_i^k$ （および $δ_k=(σ_k-1)/2^k$）で評価される普遍対称多項式 $T_n$ を用いて、$K_p(S)$ の明示的な公式を推測した。
指数生成関数と係数抽出を用いてフェルの予想を証明し、導出に必要な$T_n$の普遍恒等式を導出する。
この議論はLean/Mathlibで完全に形式化されており、自然言語で記述された予想からAxiomProverによって自動的に生成されたものである。](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:z5xxhxqv6elnjzulyf7t22wk/bafkreigsiwtf5lc4gl2wkecayi2yuysfv6xu7ygullsypscbu2lale5hha)
February 8, 2026 at 12:07 AM
2602.03716
$S=\langle d_1,\dots,d_m\rangle$ を数値半群とし、$k[S]$ をその半群環とする。$k[S]$のヒルベルト数式は、シジジー次数における交互の冪和を符号化する正規化された交互シジジー冪和$K_p(S)$を決定する。フェルは、すべての $p\ge 0$ に対して、ギャップ冪和 $G_r(S)=\sum_{g\notin S} g^r$ と、生成元に...
$S=\langle d_1,\dots,d_m\rangle$ を数値半群とし、$k[S]$ をその半群環とする。$k[S]$のヒルベルト数式は、シジジー次数における交互の冪和を符号化する正規化された交互シジジー冪和$K_p(S)$を決定する。フェルは、すべての $p\ge 0$ に対して、ギャップ冪和 $G_r(S)=\sum_{g\notin S} g^r$ と、生成元に...
[16/30] 198 Likes, 28 Comments, 2 Posts
2602.03716, math․CO | math․AC | math․NT, 03 Feb 2026
🆕Fel's Conjecture on Syzygies of Numerical Semigroups
Evan Chen, Chris Cummins, GSM, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Aram Marko...
2602.03716, math․CO | math․AC | math․NT, 03 Feb 2026
🆕Fel's Conjecture on Syzygies of Numerical Semigroups
Evan Chen, Chris Cummins, GSM, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Aram Marko...
![Let $S=\langle d_1,\dots,d_m\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring.
The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees.
Fel conjectured an explicit formula for $K_p(S)$, for all $p\ge 0$, in terms of the gap power sums $G_r(S)=\sum_{g\notin S} g^r$ and universal symmetric polynomials $T_n$ evaluated at the generator power sums $σ_k=\sum_i d_i^k$ (and $δ_k=(σ_k-1)/2^k$).
We prove Fel's conjecture via exponential generating functions and coefficient extraction, solating the universal identities for $T_n$ needed for the derivation.
The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the conjecture.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:z5xxhxqv6elnjzulyf7t22wk/bafkreihf2phudvszu5nfvcpt2jgotqufulv6xezb7snsbkb5attqdvc3oe)
February 8, 2026 at 12:07 AM
[16/30] 198 Likes, 28 Comments, 2 Posts
2602.03716, math․CO | math․AC | math․NT, 03 Feb 2026
🆕Fel's Conjecture on Syzygies of Numerical Semigroups
Evan Chen, Chris Cummins, GSM, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Aram Marko...
2602.03716, math․CO | math․AC | math․NT, 03 Feb 2026
🆕Fel's Conjecture on Syzygies of Numerical Semigroups
Evan Chen, Chris Cummins, GSM, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Aram Marko...
its translations by $g_1,...,g_r$ sum up to the constant-1 function. Our main result shows, informally speaking, that fractional divisibility is impossible if at least $r/2$ rotations are ``generic". [2/2 of https://arxiv.org/abs/2501.13522v1]
January 24, 2025 at 6:13 AM
its translations by $g_1,...,g_r$ sum up to the constant-1 function. Our main result shows, informally speaking, that fractional divisibility is impossible if at least $r/2$ rotations are ``generic". [2/2 of https://arxiv.org/abs/2501.13522v1]
arXiv:2501.13522v1 Announce Type: new
Abstract: We say that an r-tuple $(g_1,...,g_r)$ of special orthogonal $d\times d$ matrices fractionally divides the $(d-1)$-dimensional sphere $S$ if there is a non-constant function in $L^2(S)$ such that [1/2 of https://arxiv.org/abs/2501.13522v1]
Abstract: We say that an r-tuple $(g_1,...,g_r)$ of special orthogonal $d\times d$ matrices fractionally divides the $(d-1)$-dimensional sphere $S$ if there is a non-constant function in $L^2(S)$ such that [1/2 of https://arxiv.org/abs/2501.13522v1]
January 24, 2025 at 6:13 AM
arXiv:2501.13522v1 Announce Type: new
Abstract: We say that an r-tuple $(g_1,...,g_r)$ of special orthogonal $d\times d$ matrices fractionally divides the $(d-1)$-dimensional sphere $S$ if there is a non-constant function in $L^2(S)$ such that [1/2 of https://arxiv.org/abs/2501.13522v1]
Abstract: We say that an r-tuple $(g_1,...,g_r)$ of special orthogonal $d\times d$ matrices fractionally divides the $(d-1)$-dimensional sphere $S$ if there is a non-constant function in $L^2(S)$ such that [1/2 of https://arxiv.org/abs/2501.13522v1]
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June 5, 2026 at 2:35 AM
e)C&G_R?tr-9:9(w>8$c]DQapH~5L@Dw
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April 30, 2026 at 10:25 AM
-Z`(%`+eD3J(xL<H/ZNuLab;G_R<zHJ:
monochromatic copy of $G_i$, that is, a color $i \in[r]$ such that $i \in \chi(e)$ for every $e \in E(G_i)$. If $G_1=G_2=\cdots=G_r=G$, then we write $\mathrm{R}_{r,s}(G)$ for short. In 2022, Le asked to find lower and upper bounds for $\mathrm{R}_{s, [2/4 of https://arxiv.org/abs/2505.20652v1]
May 28, 2025 at 6:02 AM
monochromatic copy of $G_i$, that is, a color $i \in[r]$ such that $i \in \chi(e)$ for every $e \in E(G_i)$. If $G_1=G_2=\cdots=G_r=G$, then we write $\mathrm{R}_{r,s}(G)$ for short. In 2022, Le asked to find lower and upper bounds for $\mathrm{R}_{s, [2/4 of https://arxiv.org/abs/2505.20652v1]
arXiv:2505.20652v1 Announce Type: new
Abstract: The \textit{set-coloring Ramsey number} $\mathrm{R}_{r, s}(G_1,G_2,...,G_r)$ is the least $n \in \mathbb{N}$ such that every coloring $\chi: E\left(K_n\right) \rightarrow\binom{[r]}{s}$ contains a [1/4 of https://arxiv.org/abs/2505.20652v1]
Abstract: The \textit{set-coloring Ramsey number} $\mathrm{R}_{r, s}(G_1,G_2,...,G_r)$ is the least $n \in \mathbb{N}$ such that every coloring $\chi: E\left(K_n\right) \rightarrow\binom{[r]}{s}$ contains a [1/4 of https://arxiv.org/abs/2505.20652v1]
May 28, 2025 at 6:02 AM
arXiv:2505.20652v1 Announce Type: new
Abstract: The \textit{set-coloring Ramsey number} $\mathrm{R}_{r, s}(G_1,G_2,...,G_r)$ is the least $n \in \mathbb{N}$ such that every coloring $\chi: E\left(K_n\right) \rightarrow\binom{[r]}{s}$ contains a [1/4 of https://arxiv.org/abs/2505.20652v1]
Abstract: The \textit{set-coloring Ramsey number} $\mathrm{R}_{r, s}(G_1,G_2,...,G_r)$ is the least $n \in \mathbb{N}$ such that every coloring $\chi: E\left(K_n\right) \rightarrow\binom{[r]}{s}$ contains a [1/4 of https://arxiv.org/abs/2505.20652v1]
$G_r$ of all right translations of $G$ is normal in $Aut(\Gamma)$. At first, we reduce the case of an arbitrary abelian group to the case of an abelian $p$-group. Further, we obtain several results on CI-property of normal [3/4 of https://arxiv.org/abs/2503.00859v1]
March 4, 2025 at 6:06 AM
$G_r$ of all right translations of $G$ is normal in $Aut(\Gamma)$. At first, we reduce the case of an arbitrary abelian group to the case of an abelian $p$-group. Further, we obtain several results on CI-property of normal [3/4 of https://arxiv.org/abs/2503.00859v1]
condition for the periodicity of the Cayley graph $G_R$. We also completely determine the rings $R$ for which $G_R$ exhibits perfect state transfer. The quadratic unitary Cayley graph $\mathcal{G}_R$ has vertex set $R$, [3/7 of https://arxiv.org/abs/2502.10217v1]
February 17, 2025 at 6:11 AM
condition for the periodicity of the Cayley graph $G_R$. We also completely determine the rings $R$ for which $G_R$ exhibits perfect state transfer. The quadratic unitary Cayley graph $\mathcal{G}_R$ has vertex set $R$, [3/7 of https://arxiv.org/abs/2502.10217v1]
unitary Cayley graphs. Let $R$ be a finite commutative ring. The unitary Cayley graph $G_R$ has vertex set $R$, where two vertices $u$ and $v$ are adjacent if $u-v$ is a unit in $R$. We provide a necessary and sufficient [2/7 of https://arxiv.org/abs/2502.10217v1]
February 17, 2025 at 6:11 AM
unitary Cayley graphs. Let $R$ be a finite commutative ring. The unitary Cayley graph $G_R$ has vertex set $R$, where two vertices $u$ and $v$ are adjacent if $u-v$ is a unit in $R$. We provide a necessary and sufficient [2/7 of https://arxiv.org/abs/2502.10217v1]
= 6$ for the $E_1$ space and $C_\Gamma = 4$ for the $E_2$ space, $g_R$ is the characteristic length scale of the quantum system, and $L$ is the side physical length of the fundamental cubic region. Using the three dimensional Dirac potential as a toy [5/8 of https://arxiv.org/abs/2505.08603v1]
May 14, 2025 at 6:20 AM
= 6$ for the $E_1$ space and $C_\Gamma = 4$ for the $E_2$ space, $g_R$ is the characteristic length scale of the quantum system, and $L$ is the side physical length of the fundamental cubic region. Using the three dimensional Dirac potential as a toy [5/8 of https://arxiv.org/abs/2505.08603v1]
g_R$), to leading order, the energy eigenvalues for both the $E_1$ and $E_2$ spaces can be written in the unified form $E\simeq -\frac{\hbar^2}{2mg_R^2}(1 + C_\Gamma\,\frac{2g_R}{L}\,e^{-L/g_R})$, where the topology dependent coefficient is $C_\Gamma [4/8 of https://arxiv.org/abs/2505.08603v1]
May 14, 2025 at 6:20 AM
g_R$), to leading order, the energy eigenvalues for both the $E_1$ and $E_2$ spaces can be written in the unified form $E\simeq -\frac{\hbar^2}{2mg_R^2}(1 + C_\Gamma\,\frac{2g_R}{L}\,e^{-L/g_R})$, where the topology dependent coefficient is $C_\Gamma [4/8 of https://arxiv.org/abs/2505.08603v1]
arXiv:2503.16155v1 Announce Type: new
Abstract: Very little is known on the Hilbert series of graded algebras $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_r)$, $r>n$, $g_i$ generic form of degree $e_i$, in general. One instance when the series is known, [1/7 of https://arxiv.org/abs/2503.16155v1]
Abstract: Very little is known on the Hilbert series of graded algebras $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_r)$, $r>n$, $g_i$ generic form of degree $e_i$, in general. One instance when the series is known, [1/7 of https://arxiv.org/abs/2503.16155v1]
March 21, 2025 at 6:01 AM
arXiv:2503.16155v1 Announce Type: new
Abstract: Very little is known on the Hilbert series of graded algebras $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_r)$, $r>n$, $g_i$ generic form of degree $e_i$, in general. One instance when the series is known, [1/7 of https://arxiv.org/abs/2503.16155v1]
Abstract: Very little is known on the Hilbert series of graded algebras $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_r)$, $r>n$, $g_i$ generic form of degree $e_i$, in general. One instance when the series is known, [1/7 of https://arxiv.org/abs/2503.16155v1]
the simple $G_rT$-modules of $p^r$-restricted highest weight. Further connections between the representations of $\text{Dist}(G_r)^T$ and $G_rT$ are shown, demonstrating the potential usefulness of this algebra. [3/3 of https://arxiv.org/abs/2504.03121v1]
April 7, 2025 at 6:06 AM
the simple $G_rT$-modules of $p^r$-restricted highest weight. Further connections between the representations of $\text{Dist}(G_r)^T$ and $G_rT$ are shown, demonstrating the potential usefulness of this algebra. [3/3 of https://arxiv.org/abs/2504.03121v1]
the subalgebra of $\text{Dist}(G_r)$ consisting of fixed points for the adjoint action of a maximal torus $T$ of $G$. We analyze the structure of this algebra, and classify its simple modules, which essentially are just the non-zero weight spaces of [2/3 of https://arxiv.org/abs/2504.03121v1]
April 7, 2025 at 6:06 AM
the subalgebra of $\text{Dist}(G_r)$ consisting of fixed points for the adjoint action of a maximal torus $T$ of $G$. We analyze the structure of this algebra, and classify its simple modules, which essentially are just the non-zero weight spaces of [2/3 of https://arxiv.org/abs/2504.03121v1]
arXiv:2504.03121v1 Announce Type: new
Abstract: Let $G$ be a simple and simply connected algebraic group over a field of characteristic $p>0$, and $G_r$ its $r$-th Frobenius kernel. In this paper, we initiate a general study of $\text{Dist}(G_r)^T$, [1/3 of https://arxiv.org/abs/2504.03121v1]
Abstract: Let $G$ be a simple and simply connected algebraic group over a field of characteristic $p>0$, and $G_r$ its $r$-th Frobenius kernel. In this paper, we initiate a general study of $\text{Dist}(G_r)^T$, [1/3 of https://arxiv.org/abs/2504.03121v1]
April 7, 2025 at 6:06 AM
arXiv:2504.03121v1 Announce Type: new
Abstract: Let $G$ be a simple and simply connected algebraic group over a field of characteristic $p>0$, and $G_r$ its $r$-th Frobenius kernel. In this paper, we initiate a general study of $\text{Dist}(G_r)^T$, [1/3 of https://arxiv.org/abs/2504.03121v1]
Abstract: Let $G$ be a simple and simply connected algebraic group over a field of characteristic $p>0$, and $G_r$ its $r$-th Frobenius kernel. In this paper, we initiate a general study of $\text{Dist}(G_r)^T$, [1/3 of https://arxiv.org/abs/2504.03121v1]
Paul Sobaje: The Torus Centralizing Subalgebra of $\text{Dist}(G_r)$ https://arxiv.org/abs/2504.03121 https://arxiv.org/pdf/2504.03121 https://arxiv.org/html/2504.03121
April 7, 2025 at 6:06 AM
Paul Sobaje: The Torus Centralizing Subalgebra of $\text{Dist}(G_r)$ https://arxiv.org/abs/2504.03121 https://arxiv.org/pdf/2504.03121 https://arxiv.org/html/2504.03121
August 25, 2024 at 12:44 AM