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1. Improved Bounds on a Generalization of Tuza's Conjecture

   https://doi.org/10.37236/10829  

   Basit, Abdul; McGinnis, Daniel; Simmons, Henry; Sinnwell, Matt; Zerbib, Shira (October 2022, The Electronic Journal of Combinatorics)

   For an $$r$$-uniform hypergraph $$H$$, let $$\nu^{(m)}(H)$$ denote the maximum size of a set $$M$$ of edges in $$H$$ such that every two edges in $$M$$ intersect in less than $$m$$ vertices, and let $$\tau^{(m)}(H)$$ denote the minimum size of a collection $$C$$ of $$m$$-sets of vertices such that every edge in $$H$$ contains an element of $$C$$. The fractional analogues of these parameters are denoted by $$\nu^{*(m)}(H)$$ and $$\tau^{*(m)}(H)$$, respectively. Generalizing a famous conjecture of Tuza on covering triangles in a graph, Aharoni and Zerbib conjectured that for every $$r$$-uniform hypergraph $$H$$, $$\tau^{(r-1)}(H)/\nu^{(r-1)}(H) \leq \lceil{\frac{r+1}{2}}\rceil$$. In this paper we prove bounds on the ratio between the parameters $$\tau^{(m)}$$ and $$\nu^{(m)}$$, and their fractional analogues. Our main result is that, for every $$r$$-uniform hypergraph~$$H$$,\[ \tau^{*(r-1)}(H)/\nu^{(r-1)}(H) \le \begin{cases} \frac{3}{4}r - \frac{r}{4(r+1)} &\text{for }r\text{ even,}\\\frac{3}{4}r - \frac{r}{4(r+2)} &\text{for }r\text{ odd.} \\\end{cases} \]This improves the known bound of $$r-1$.We also prove that, for every $$r$$-uniform hypergraph $$H$$, $$\tau^{(m)}(H)/\nu^{*(m)}(H) \le \operatorname{ex}_m(r, m+1)$$, where the Turán number $$\operatorname{ex}_r(n, k)$$ is the maximum number of edges in an $$r$$-uniform hypergraph on $$n$$ vertices that does not contain a copy of the complete $$r$$-uniform hypergraph on $$k$$ vertices. Finally, we prove further bounds in the special cases $(r,m)=(4,2)$ and $(r,m)=(4,3)$. 

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2. Advancing replicable and reproducible GIScience: an approach with KNIME

   https://doi.org/10.1080/15230406.2024.2446556  

   Fu, Xiaokang; Liu, Lingbo; Guan, Weihe Wendy; Kalra, Yogya; Bao, Shuming; Kötter, Tobias; Sturm, Kevin (February 2025, Cartography and Geographic Information Science)

   The reproducibility and replicability (R&R) crisis poses a significant challenge across disciplines, particularly in spatiotemporal studies. This paper focuses on the unique challenges within spatiotemporal research in the context of R&R, including data availability, methodological conception transparency, interdisciplinary collaboration complexities, the balance between R&R and innovation, and R&R education. Recognizing the potential of Scientific Workflow Management Systems (SWMS) to enhance R&R, we introduce a pioneering SWMS-based integrated spatiotemporal research approach (SISRA) utilizing KNIME, an open-source SWMS, to tackle these R&R challenges. First, we developed a set of KNIME extensions, including Geospatial and Dataverse extensions, to enhance spatiotemporal software availability in SWMS. Then we created spatial data virtual laboratory architecture to support multidisciplinary collaboration. Finally, we suggested a geographical research lifecycle that integrates SWMS-based methods to improve practices, efficiency, and innovation in R&R research and education. Our approach exemplifies how executable workflows can not only alleviate the R&R burden on researchers but also strengthen R&R education in geographical research, illustrating the benefits of our approach in training, teaching, and multidisciplinary collaboration. 

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3. Asymptotics of singular values for quantum derivatives

   https://doi.org/10.1090/tran/8827  

   Frank, Rupert; Sukochev, Fedor; Zanin, Dmitriy (March 2023, Transactions of the American Mathematical Society)

   We obtain Weyl type asymptotics for the quantised derivative \dj \mkern 1muf of a function f f from the homgeneous Sobolev space W ˙ d 1 ( R d ) \dot {W}^1_d(\mathbb {R}^d) on R d . \mathbb {R}^d. The asymptotic coefficient ‖ ∇ f ‖ L d ( R d ) \|\nabla f\|_{L_d(\mathbb R^d)} is equivalent to the norm of \dj \mkern 1muf in the principal ideal L d , ∞ , \mathcal {L}_{d,\infty }, thus, providing a non-asymptotic, uniform bound on the spectrum of \dj \mkern 1muf. Our methods are based on the C ∗ C^{\ast } -algebraic notion of the principal symbol mapping on R d \mathbb {R}^d , as developed recently by the last two authors and collaborators. 

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4. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie; Malinnikova, Eugenia (June 2021, Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

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5. On realizations of the subalgebra 𝒜^{ℝ}(1) of the ℝ-motivic Steenrod algebra

   https://doi.org/10.1090/btran/114  

   Bhattacharya, P.; Guillou, B.; Li, A. (July 2022, Transactions of the American Mathematical Society, Series B)

   In this paper, we show that the finite subalgebra A R ( 1 ) \mathcal {A}^\mathbb {R}(1) , generated by S q 1 \mathrm {Sq}^1 and S q 2 \mathrm {Sq}^2 , of the R \mathbb {R} -motivic Steenrod algebra A R \mathcal {A}^\mathbb {R} can be given 128 different A R \mathcal {A}^\mathbb {R} -module structures. We also show that all of these A \mathcal {A} -modules can be realized as the cohomology of a 2 2 -local finite R \mathbb {R} -motivic spectrum. The realization results are obtained using an R \mathbb {R} -motivic analogue of the Toda realization theorem. We notice that each realization of A R ( 1 ) \mathcal {A}^\mathbb {R}(1) can be expressed as a cofiber of an R \mathbb {R} -motivic v 1 v_1 -self-map. The C 2 {\mathrm {C}_2} -equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the R O ( C 2 ) \mathrm {RO}({\mathrm {C}_2}) -graded Steenrod operations on a C 2 {\mathrm {C}_2} -equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. This technique is then used to identify the geometric fixed-point spectra of the C 2 {\mathrm {C}_2} -equivariant realizations of A C 2 ( 1 ) \mathcal {A}^{\mathrm {C}_2}(1) . We find another application of the R \mathbb {R} -motivic Toda realization theorem: we produce an R \mathbb {R} -motivic, and consequently a C 2 {\mathrm {C}_2} -equivariant, analogue of the Bhattacharya-Egger spectrum Z \mathcal {Z} , which could be of independent interest. 

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