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1. Divergence beneath the Brillouin sphere and the phenomenology of prediction error in spherical harmonic series approximations of the gravitational field

   https://doi.org/10.1088/1361-6633/ad44d5  

   Bevis, M; Ogle, C; Costin, O; Jekeli, C; Costin, R D; Guo, J; Fowler, J; Dunne, G V; Shum, C K; Snow, K (June 2024, Reports on Progress in Physics)

   The Brillouin sphere is defined as the smallest sphere, centered at the origin of the geocentric coordinate system, that incorporates all the condensed matter composing the planet. The Brillouin sphere touches the Earth at a single point, and the radial line that begins at the origin and passes through that point is called the singular radial line. For about 60 years there has been a persistent anxiety about whether or not a spherical harmonic (SH) expansion of the external gravitational potential,V, will converge beneath the Brillouin sphere. Recently, it was proven that the probability of such convergence is zero. One of these proofs provided an asymptotic relation, called Costin’s formula, for the upper bound,E_N, on the absolute value of the prediction error,e_N, of a SH series model, $V_N(\theta ,\lambda ,r)$, truncated at some maximum degree, $N=n_{max}$. When the SH series is restricted to (or projected onto) a particular radial line, it reduces to a Taylor series (TS) in $1/r$. Costin’s formula is $E_N\simeq B{N}^{-b}(R/r{)}^N$, whereRis the radius of the Brillouin sphere. This formula depends on two positive parameters:b, which controls the decay of error amplitude as a function ofNwhenris fixed, and a scale factorB. We show here that Costin’s formula derives from a similar asymptotic relation for the upper bound,A_non the absolute value of the TS coefficients,a_n, for the same radial line. This formula, $A_n\simeq K{n}^{-k}$, depends on degree,n, and two positive parameters,kandK, that are analogous tobandB. We use synthetic planets, for which we can compute the potential,V, and also the radial component of gravitational acceleration, $g_r=\partial V/\partial r$, to hundreds of significant digits, to validate both of these asymptotic formulas. Let superscriptVrefer to asymptotic parameters associated with the coefficients and prediction errors for gravitational potential, and superscriptgto the coefficients and predictions errors associated withg_r. For polyhedral planets of uniform density we show that $b^V=k^V=7/2$and $b^g=k^g=5/2$almost everywhere. We show that the frequency of oscillation (around zero) of the TS coefficients and the series prediction errors, for a given radial line, is controlled by the geocentric angle,α, between that radial line and the singular radial line. We also derive useful identities connecting $K^V,B^V,K^g$, andB^g. These identities are expressed in terms of quotients of the various scale factors. The only other quantities involved in these identities areαandR. The phenomenology of ‘series divergence’ and prediction error (whenr < R) can be described as a function of the truncation degree,N, or the depth,d, beneath the Brillouin sphere. For a fixed $r\text{⩽}R$, asNincreases from very low values, the upper error boundE_Nshrinks until it reaches its minimum (best) value whenNreaches some particular or optimum value, $N_{\mathrm{opt}}$. When $N>N_{\mathrm{opt}}$, prediction error grows asNcontinues to increase. Eventually, when $N\gg N_{\mathrm{opt}}$, prediction errors increase exponentially with risingN. If we fix the value ofNand allow $R/r$to vary, then we find that prediction error in free space beneath the Brillouin sphere increases exponentially with depth,d, beneath the Brillouin sphere. Because $b^g=b^V-1$everywhere, divergence driven prediction error intensifies more rapidly forg_rthan forV, both in terms of its dependence onNandd. If we fix bothNandd, and focus on the ‘lateral’ variations in prediction error, we observe that divergence and prediction error tend to increase (as doesB) as we approach high-amplitude topography. 

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2. Search for CP violation in $$ {D}_{(s)}^{+}\to {K}_S^0{K}^{-}{\pi}^{+}{\pi}^{+} $$ decays using triple and quadruple products

   https://doi.org/10.1007/JHEP04(2025)036  

   Aggarwal, L; Ahmed, H; Aihara, H; Akopov, N; Aloisio, A; Althubiti, N; Anh_Ky, N; Asner, D M; Atmacan, H; Aushev, V; et al (April 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We perform the first search forCPviolation in$$ {D}_{(s)}^{+}\to {K}_S^0{K}^{-}{\pi}^{+}{\pi}^{+} $$ $D_{\left(s\right)}^{+}\to K_S^0{K}^{-}{\pi }^{+}{\pi }^{+}$decays. We use a combined data set from the Belle and Belle II experiments, which studye⁺e⁻collisions at center-of-mass energies at or near the Υ(4S) resonance. We use 980 fb⁻¹of data from Belle and 428 fb⁻¹of data from Belle II. We measure sixCP-violating asymmetries that are based on triple products and quadruple products of the momenta of final-state particles, and also the particles’ helicity angles. We obtain a precision at the level of 0.5% for$$ {D}^{+}\to {K}_S^0{K}^{-}{\pi}^{+}{\pi}^{+} $$ $D^{+}\to K_S^0{K}^{-}{\pi }^{+}{\pi }^{+}$decays, and better than 0.3% for$$ {D}_s^{+}\to {K}_S^0{K}^{-}{\pi}^{+}{\pi}^{+} $$ $D_s^{+}\to K_S^0{K}^{-}{\pi }^{+}{\pi }^{+}$decays. No evidence ofCPviolation is found. Our results for the triple-product asymmetries are the most precise to date for singly-Cabibbo-suppressedD⁺decays. Our results for the other asymmetries are the first such measurements performed for charm decays. 

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3. On the Distributed Resistor-Constant Phase Element Transmission Line in a Reflective Bounded Domain

   https://doi.org/10.1149/1945-7111/add212  

   Allagui, Anis; Balaguera, Enrique H; Wang, Chunlei (May 2025, Journal of The Electrochemical Society)

   In this work we derive and study the analytical solution of the voltage and current diffusion equation for the case of a finite-length resistor-constant phase element (CPE) transmission line (TL) network that can represent a model for porous electrodes in the absence of any Faradic processes. The energy storage component is considered to be an elemental CPE per unit length of impedancez_c(s) = 1/(c_αs^α) with constant parameters (c_α, α) instead of the ideal capacitor of impedancez(s) = 1/(cs) usually assumed in TL modeling. The problem becomes a time-fractional diffusion equation for the voltage that we solve under galvanostatic charging, and derive from it a reduced impedance function of the form $z_{\alpha }\left(s_n\right)=s_n^{-\alpha /2}\coth \left(s_n^{\alpha /2}\right)$, wheres_n = jω_nis a normalized frequency. We also derive the system’s step response, and the distribution function of relaxation times associated with it. The analysis can be viewed and used as a support for the fractal finite-length Warburg model. 

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4. L-Systems and the Lovász Number

   https://doi.org/10.1007/s00493-025-00136-4  

   Linz, William (April 2025, Combinatorica)

   Abstract Given integers$$n> k > 0$$ $n>k>0$, and a set of integers$$L \subset [0, k-1]$$ $L\subset [0,k-1]$, anL-systemis a family of sets$$\mathcal {F}\subset \left( {\begin{array}{c}[n]\\ k\end{array}}\right) $$ $F\subset \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$such that$$|F \cap F'| \in L$$ $|F\cap F^{\prime }|\in L$for distinct$$F, F'\in \mathcal {F}$$ $F,F^{\prime }\in F$.L-systems correspond to independent sets in a certain generalized Johnson graphG(n, k, L), so that the maximum size of anL-system is equivalent to finding the independence number of the graphG(n, k, L). TheLovász number$$\vartheta (G)$$ $\vartheta \left(G\right)$is a semidefinite programming approximation of the independence number$$\alpha $$ $\alpha$of a graphG. In this paper, we determine the leading order term of$$\vartheta (G(n, k, L))$$ $\vartheta \left(G\right(n,k,L\left)\right)$of any generalized Johnson graph withkandLfixed and$$n\rightarrow \infty $$ $n\to \infty$. As an application of this theorem, we give an explicit construction of a graphGonnvertices with a large gap between the Lovász number and the Shannon capacityc(G). Specifically, we prove that for any$$\epsilon > 0$$ $ϵ>0$, for infinitely manynthere is a generalized Johnson graphGonnvertices which has ratio$$\vartheta (G)/c(G) = \Omega (n^{1-\epsilon })$$ $\vartheta \left(G\right)/c\left(G\right)=\Omega \left(n^{1-ϵ}\right)$, which improves on all known constructions. The graphGa fortiorialso has ratio$$\vartheta (G)/\alpha (G) = \Omega (n^{1-\epsilon })$$ $\vartheta \left(G\right)/\alpha \left(G\right)=\Omega \left(n^{1-ϵ}\right)$, which greatly improves on the best known explicit construction. 

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5. More on codes for combinatorial composite DNA

   https://doi.org/10.1007/s10623-025-01634-8  

   Ye, Zuo; Sabary, Omer; Gabrys, Ryan; Yaakobi, Eitan; Elishco, Ohad (May 2025, Designs, Codes and Cryptography)

   Abstract In this paper, we focus on constructing unique-decodable and list-decodable codes for the recently studied (t, e)-composite-asymmetric error-correcting codes ((t, e)-CAECCs). Let$$\mathcal {X}$$ $X$be an$$m \times n$$ $m\times n$binary matrix in which each row has Hamming weightw. If at mosttrows of$$\mathcal {X}$$ $X$contain errors, and in each erroneous row, there are at mosteoccurrences of$$1 \rightarrow 0$$ $1\to 0$errors, we say that a (t, e)-composite-asymmetric error occurs in$$\mathcal {X}$$ $X$. For general values ofm, n, w, t, ande, we propose new constructions of (t, e)-CAECCs with redundancy at most$$(t-1)\log (m) + O(1)$$ $(t-1)log\left(m\right)+O\left(1\right)$, whereO(1) is independent of the code lengthm. In particular, this yields a class of (2, e)-CAECCs that are optimal in terms of redundancy. Whenmis a prime power, the redundancy can be further reduced to$$(t-1)\log (m) - O(\log (m))$$ $(t-1)log\left(m\right)-O(log(m\left)\right)$. To further increase the code size, we introduce a combinatorial object called a weak$$B_e$$ $B_e$-set. When$$e = w$$ $e=w$, we present an efficient encoding and decoding method for our codes. Finally, we explore potential improvements by relaxing the requirement of unique decoding to list-decoding. We show that when the list size ist! or an exponential function oft, there exist list-decodable (t, e)-CAECCs with constant redundancy. When the list size is two, we construct list-decodable (3, 2)-CAECCs with redundancy$$\log (m) + O(1)$$ $log\left(m\right)+O\left(1\right)$. 

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