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1. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József; Clemen, Felix Christian; Dumitrescu, Adrian (September 2024, Algorithmica)

   Abstract LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$ ${[0,1]}^k$ where$$k \ge 2$$ $k\ge 2$ . According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$ $x_1,x_2,\dots ,x_n$ through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$ , where$$|x-y|$$ $|x-y|$ is the Euclidean distance betweenxandy, and$$c_k$$ $c_k$ is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$ $x_{n+1}\equiv x_1$ . From the other direction, for every$$k \ge 2$$ $k\ge 2$ and$$n \ge 2$$ $n\ge 2$ , there existnpoints in$$[0,1]^k$$ ${[0,1]}^k$ , such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$ . For the plane, the best constant is$$c_2=2$$ $c_2=2$ and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$ for every$$k \ge 3$$ $k\ge 3$ and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$ $c_k=2^{1/k}·\sqrt{k}$ , for every$$k \ge 2$$ $k\ge 2$ . Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$ or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ $c_k=2.91\sqrt{k}(1+o_k\left(1\right))$ . Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$ $c_3\ge 2^{7/6}$ , which disproves the conjecture for$$k=3$$ $k=3$ . Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. 

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2. Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

   https://doi.org/10.1007/s00493-023-00015-w  

   Scott, Alex; Seymour, Paul; Spirkl, Sophie (October 2023, Combinatorica)

   Abstract A graphGisH-freeif it has no induced subgraph isomorphic toH. We prove that a$$P_5$$ $P_5$ -free graph with clique number$$\omega \ge 3$$ $\omega \ge 3$ has chromatic number at most$$\omega ^{\log _2(\omega )}$$ ${\omega }^{{log}_2\left(\omega \right)}$ . The best previous result was an exponential upper bound$$(5/27)3^{\omega }$$ $(5/27)3^{\omega }$ , due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for$$P_5$$ $P_5$ , which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for$$P_5$$ $P_5$ -free graphs, and our result is an attempt to approach that. 

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3. Ramanujan’s partition generating functions modulo $$\ell $$

   https://doi.org/10.1007/s11139-025-01241-0  

   Bringmann, Kathrin; Craig, William; Ono, Ken (November 2025, The Ramanujan Journal)

   Abstract For the partition functionp(n), Ramanujan proved the striking identities$$\begin{aligned} \begin{aligned} \mathcal {P}_5(q):=\sum _{n\ge 0} p(5n+4)q^n&=5\prod _{n\ge 1} \frac{\left( q^5;q^5\right) _{\infty }^5}{(q;q)_{\infty }^6},\\ \mathcal {P}_{7}(q):=\sum _{n\ge 0} p(7n+5)q^n&=7\prod _{n\ge 1}\frac{\left( q^7;q^7\right) _{\infty }^3}{(q;q)_{\infty }^4}+49q \prod _{n\ge 1}\frac{\left( q^7;q^7\right) _{\infty }^7}{(q;q)_{\infty }^8}, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{cc}{P}_5\left(q\right):=\sum _{n\ge 0}p(5n+4)q^n& =5\prod _{n\ge 1}\frac{{\left(q^5,;,q^5\right)}_{\infty }^5}{{(q;q)}_{\infty }^6},\\ P_7\left(q\right):=\sum _{n\ge 0}p(7n+5)q^n& =7\prod _{n\ge 1}\frac{{\left(q^7,;,q^7\right)}_{\infty }^3}{{(q;q)}_{\infty }^4}+49q\prod _{n\ge 1}\frac{{\left(q^7,;,q^7\right)}_{\infty }^7}{{(q;q)}_{\infty }^8},\end{array}\end{array}$ where$$(q;q)_{\infty }:=\prod _{n\ge 1}(1-q^n).$$ ${(q;q)}_{\infty }:={\prod }_{n\ge 1}(1-q^n).$ As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes$$\ell \ge 5,$$ $\ell \ge 5,$ closed form expressions of the power series$$ \mathcal {P}_{\ell }(q):=\sum _{n\ge 0} p(\ell n-\delta _{\ell })q^n\pmod {\ell }, $$ $P_{\ell }\left(q\right):=\sum _{n\ge 0}p(\ell n-{\delta }_{\ell })q^n(mod\ell ),$ where$$\delta _{\ell }:=\frac{\ell ^2-1}{24}.$$ ${\delta }_{\ell }:=\frac{{\ell }^2-1}{24}.$ In this paper, we prove that$$ \mathcal {P}_{\ell }(q)\equiv c_{\ell } \dfrac{\mathcal {T}_{\ell }(q)}{\left( q^\ell ; q^\ell \right) _\infty } \pmod {\ell }, $$ $P_{\ell }\left(q\right)\equiv c_{\ell }\frac{T_{\ell }\left(q\right)}{{\left(q^{\ell },;,q^{\ell }\right)}_{\infty }}(mod\ell ),$ where$$c_{\ell }\in \mathbb Z$$ $c_{\ell }\in Z$ is explicit and$${\mathcal {T}}_{\ell }(q)$$ $T_{\ell }\left(q\right)$ is the generating function for the Hecke traces of$$\ell $$ $\ell$ -ramified values of special Dirichlet series for weight$$\ell -1$$ $\ell -1$ cusp forms on$$\textrm{SL}_2(\mathbb Z)$$ ${\text{SL}}_2\left(Z\right)$ . This is a new proof of Ramanujan’s congruences modulo 5, 7, and 11, as there are no nontrivial cusp forms of weight 4, 6, and 10. 

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4. Modular apparatus for nuclear reactions spectroscopy (MARS): characterization and first application to determine $$^{12}$$C($$^6$$Li,$$^4$$He)$$^{14}\hbox {N}^{g.s}$$ nuclear reaction cross sections

   https://doi.org/10.1140/epjp/s13360-025-06305-0  

   Garrido-Gómez, L; Vegas-Díaz, A; Alvarez, M_A G; Fernández-García, J P; Fernández, B; Ferrer, F J; Lopez-Aires, D (May 2025, The European Physical Journal Plus)

   Abstract This work presents MARS (Modular apparatus for nuclear reactions spectroscopy) and its characterization prior to its first application to measure$$^6$$ $_{}^6$ Li+$$^{12}$$ $_{}^{12}$ C nuclear reactions. Measurements were performed at the 3 MV tandem accelerator of the CNA (National Accelerator Center), in Seville, Spain. The$$^{6}$$ $_{}^6$ Li projectiles were accelerated at energies around the$$^6$$ $_{}^6$ Li+$$^{12}$$ $_{}^{12}$ C Coulomb barrier ($$V^{\text {cm}}_{B}\sim 3.0$$ $V_B^{\text{cm}}\sim 3.0$ MeV - center of mass and$$V^{\text {lab}}_{B}\sim 4.5$$ $V_B^{\text{lab}}\sim 4.5$ MeV - laboratory frame). Using a$$^{6}\hbox {Li}^{2+}$$ $_{}^6{\text{Li}}^{2+}$ beam, we measured at 13 laboratory energies from 4.00 to 7.75 MeV. Thus, we present the excitation function of$$^{12}$$ $_{}^{12}$ C($$^6$$ $_{}^6$ Li,$$^4$$ $_{}^4$ He)$$^{14}\hbox {N}^{g.s.}$$ $_{}^{14}{\text{N}}^{g.s.}$ reaction, at 2 backward angles ($$110.0^\circ $$ $110.0^{\circ }$ and$$140.0^\circ $$ $140.0^{\circ }$ ). The projectile dissociation, leading to this reaction, increases with the bombarding energies around the Coulomb barrier. This dissociation is favored at an optimum energy$$E_{b}^{\text {op}}$$ $E_b^{\text{op}}$ $$\ge $$ $\ge$ $$V_{B}$$ $V_B$ +$$|Q_{bu}|$$ $|Q_{\mathrm{bu}}|$ , where$$V_{B}$$ $V_B$ is the Coulomb barrier of the system, and$$|Q_{bu}|$$ $|Q_{\mathrm{bu}}|$ is the module ofQ-value for the$$^6$$ $_{}^6$ Li dissociation into$$^4$$ $_{}^4$ He+$$^2$$ $_{}^2$ H. This result corroborates a systematic analysis of weakly bound projectiles reacting on several targets [1]. 

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5. Universality and phase transitions in low moments of secular coefficients of critical holomorphic multiplicative chaos

   https://doi.org/10.1007/s00440-025-01443-z  

   Gu, Haotian; Zhang, Zhenyuan (November 2025, Probability Theory and Related Fields)

   Abstract We investigate the low moments$$\mathbb {E}[|A_N|^{2q}],\, 0 $E\left[\right|A_N{|}^{2q}],0<q⩽1$ of secular coefficients$$A_N$$ $A_N$ of the critical non-Gaussian holomorphic multiplicative chaos, i.e. coefficients of$$z^N$$ $z^N$ in the power series expansion of$$\exp (\sum _{k=1}^\infty X_kz^k/\sqrt{k})$$ $exp({\sum }_{k=1}^{\infty }{X}_k{z}^k/\sqrt{k})$ , where$$\{X_k\}_{k\geqslant 1}$$ ${\left\{X_k\right\}}_{k⩾1}$ are i.i.d. rotationally invariant unit variance complex random variables. Inspired by Harper’s remarkable result on random multiplicative functions, Soundararajan and Zaman recently showed that if each$$X_k$$ $X_k$ is standard complex Gaussian,$$A_N$$ $A_N$ features better-than-square-root cancellation:$$\mathbb {E}[|A_N|^2]=1$$ $E\left[\right|A_N{|}^2]=1$ and$$\mathbb {E}[|A_N|^{2q}]\asymp (\log N)^{-q/2}$$ $E\left[\right|A_N{|}^{2q}]\asymp {(logN)}^{-q/2}$ for fixed$$q\in (0,1)$$ $q\in (0,1)$ as$$N\rightarrow \infty $$ $N\to \infty$ . We show that this asymptotics holds universally if$$\mathbb {E}[e^{\gamma |X_k|}]<\infty $$ $E\left[e^{\gamma |X_k|}\right]<\infty$ for some$$\gamma >2q$$ $\gamma >2q$ . As a consequence, we establish the universality for the tightness of the normalized secular coefficients$$A_N(\log (1+N))^{1/4}$$ $A_N{(log(1+N))}^{1/4}$ , generalizing a result of Najnudel, Paquette, and Simm. Another corollary is the almost sure regularity of some critical non-Gaussian holomorphic chaos in appropriate Sobolev spaces. Moreover, we characterize the asymptotics of$$\mathbb {E}[|A_N|^{2q}]$$ $E\left[\right|A_N{|}^{2q}]$ for$$|X_k|$$ $|X_k|$ following a stretched exponential distribution with an arbitrary scale parameter, which exhibits a completely different behavior and underlying mechanism from the Gaussian universality regime. As a result, we unveil a double-layer phase transition around the critical case of exponential tails. Our proofs combine Harper’s robust approach with a careful analysis of the (possibly random) leading terms in the monomial decomposition of$$A_N$$ $A_N$ . 

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