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1. A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality

   https://doi.org/10.1063/5.0091111  

   Carlen, Eric A.; Lieb, Elliott H. (June 2022, Journal of Mathematical Physics)

   The Golden–Thompson trace inequality, which states that Tr  e H+ K ≤ Tr  e H e K , has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or [Formula: see text] and K = potential, Tr  e H+(1− u) K e uK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr  X s Y t X 1− s Y 1− t ≤ Tr  XY for positive operators X, Y and for [Formula: see text], and [Formula: see text]. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the [Formula: see text] range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem. 

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2. The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

   https://doi.org/10.1090/tran/8822  

   Blumberg, Andrew; Mandell, Michael (January 2022, Transactions of the American Mathematical Society)

   Let p ∈ Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum S {\mathbb {S}} admits an “eigensplitting” that generalizes known splittings on K K -theory and T C TC . We identify the summands in the fiber as the covers of Z p {\mathbb {Z}}_{p} -Anderson duals of summands in the K ( 1 ) K(1) -localized algebraic K K -theory of Z {\mathbb {Z}} . Analogous results hold for the ring Z {\mathbb {Z}} where we prove that the K ( 1 ) K(1) -localized fiber sequence is self-dual for Z p {\mathbb {Z}}_{p} -Anderson duality, with the duality permuting the summands by i ↦ p − i i\mapsto p-i (indexed mod p − 1 p-1 ). We explain an intrinsic characterization of the summand we call Z Z in the splitting T C ( Z ) p ∧ ≃ j ∨ Σ j ′ ∨ Z TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z in terms of units in the p p -cyclotomic tower of Q p {\mathbb {Q}}_{p} . 

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3. Measurement of ψ (2S) production as a function of charged-particle pseudorapidity density in pp collisions at $$ \sqrt{s} $$ = 13 TeV and p–Pb collisions at $$ \sqrt{s_{\textrm{NN}}} $$ = 8.16 TeV with ALICE at the LHC

   https://doi.org/10.1007/JHEP06(2023)147  

   Acharya, S.; Adamová, D.; Adler, A.; Aglieri Rinella, G.; Agnello, M.; Agrawal, N.; Ahammed, Z.; Ahmad, S.; Ahn, S. U.; Ahuja, I.; et al (June 2023, Journal of High Energy Physics)

   A bstract Production of inclusive charmonia in pp collisions at center-of-mass energy of $$ \sqrt{s} $$ s = 13 TeV and p–Pb collisions at center-of-mass energy per nucleon pair of $$ \sqrt{s_{\textrm{NN}}} $$ s NN = 8 . 16 TeV is studied as a function of charged-particle pseudorapidity density with ALICE. Ground and excited charmonium states ( J/ψ , ψ (2S)) are measured from their dimuon decays in the interval of rapidity in the center-of-mass frame 2 . 5 < y cms < 4 . 0 for pp collisions, and 2 . 03 < y cms < 3 . 53 and −4 . 46 < y cms < −2 . 96 for p–Pb collisions. The charged-particle pseudorapidity density is measured around midrapidity (| η | < 1 . 0). In pp collisions, the measured charged-particle multiplicity extends to about six times the average value, while in p-Pb collisions at forward (backward) rapidity a multiplicity corresponding to about three (four) times the average is reached. The ψ (2S) yield increases with the charged-particle pseudorapidity density. The ratio of ψ (2S) over J/ψ yield does not show a significant multiplicity dependence in either colliding system, suggesting a similar behavior of J/ψ and ψ (2S) yields with respect to charged-particle pseudorapidity density. Results for the ψ (2S) yield and its ratio with respect to J/ψ agree with available model calculations. 

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4. Heegaard Floer homology and cosmetic surgeries in $S^3$

   https://doi.org/10.4171/JEMS/1218  

   Hanselman, Jonathan (March 2022, Journal of the European Mathematical Society)

   If a knot K in S^3 admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either ±2 or ±1/q for some value of q that is explicitly determined by the knot Floer homology of K. Moreover, in the former case the genus of K must be 2, and in the latter case there is a bound relating q to the genus and the Heegaard Floer thickness of K. As a consequence, we show that the cosmetic crossing conjecture holds for alternating knots (or more generally, Heegaard Floer thin knots) with genus not equal to 2. We also show that the conjecture holds for any knot K for which each prime summand of K has at most 16 crossings; our techniques rule out cosmetic surgeries in this setting except for slopes ±1 and ±2 on a small number of knots, and these remaining examples can be checked by comparing hyperbolic invariants. These results make use of the surgery formula for Heegaard Floer homology, which has already proved to be a powerful tool for obstructing cosmetic surgeries; we get stronger obstructions than previously known by considering the full graded theory. We make use of a new graphical interpretation of knot Floer homology and the surgery formula in terms of immersed curves, which makes the grading information we need easier to access. 

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5. Query-Efficient Algorithm to Find all Nash Equilibria in a Two-Player Zero-Sum Matrix Game

   https://doi.org/10.1145/3732787  

   Maiti, Arnab; Boczar, Ross; Jamieson, Kevin; Ratliff, Lillian (April 2025, ACM Transactions on Economics and Computation)

   We study the query complexity of finding the set of all Nash equilibria\(\mathcal {X}_\ast \times \mathcal {Y}_\ast \)in two-player zero-sum matrix games. Fearnley and Savani [18] showed that for any randomized algorithm, there exists ann×ninput matrix where it needs to queryΩ(n²) entries in expectation to compute asingleNash equilibrium. On the other hand, Bienstock et al. [5] showed that there is a special class of matrices for which one can queryO(n) entries and compute its set of all Nash equilibria. However, these results do not fully characterize the query complexity of finding the set of all Nash equilibria in two-player zero-sum matrix games. In this work, we characterize the query complexity of finding the set of all Nash equilibria\(\mathcal {X}_\ast \times \mathcal {Y}_\ast \)in terms of the number of rowsnof the input matrix\(A \in \mathbb {R}^{n \times n} \), row support size\(k_1 := |\bigcup \limits _{x \in \mathcal {X}_\ast } \text{supp}(x)| \), and column support size\(k_2 := |\bigcup \limits _{y \in \mathcal {Y}_\ast } \text{supp}(y)| \). We design a simple yet non-trivial randomized algorithm that returns the set of all Nash equilibria\(\mathcal {X}_\ast \times \mathcal {Y}_\ast \)by querying at mostO(nk⁵· polylog(n)) entries of the input matrix\(A \in \mathbb {R}^{n \times n} \)in expectation, wherek≔ max{k₁,k₂}. This upper bound is tight up to a factor of poly(k), as we show that for any randomized algorithm, there exists ann×ninput matrix with min {k₁,k₂} = 1, for which it needs to queryΩ(nk) entries in expectation in order to find the set of all Nash equilibria\(\mathcal {X}_\ast \times \mathcal {Y}_\ast \). 

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