“The paper identifies universal polar dual pairs of spherical codes C and D such that for a large class of potential functions h the minima of the discrete h-potential of C on the sphere occur at the points of D and vice versa.”

“The paper characterizes the general form of bijective linear maps that preserve Lie products and operator products equal to fixed elements.”

“The paper determines the robust optimal growth rate, constructs a worst-case admissible model, and characterizes the robust growth-optimal strategy via a solution to a certain partial differential equation (PDE).”

“Acyclic production systems are always viable.”

“An acyclic mcp graph is amenable if and only if a.e. component has at most two nonvanishing ends, while it is nowhere amenable exactly when a.e. component has a nonempty perfect (closed) set of nonvanishing ends.”

“Temperature, chemical potential, and pressure arise as conjugate variables of a single information-theoretic functional.”

“For fixed $m$ and $n$, the paper characterizes the pairs of parameters $k_1,k_2$ for which $ζ_{G(p,m,n,k_1)}(s)=ζ_{G(p,m,n,k_2)}(s)$.”

“A skew-gentle algebra is τ-tilting finite if and only if it is representation-finite.”

“The paper demonstrates that a previously proposed condition for consistency does not hold and provides new characterizations and conditions.”

“The paper introduces the term Hoffman-London to refer to graphs that are minimal in this sense (minimizing H-colorings with paths).”

“Our results mathematically characterize the background magnetic field exerts the stabilizing effect, and bridge the gap left by previous work with respect to the asymptotic behavior in time.”

“The paper uncovers an isospin cascade sequence of phase transitions that gives rise to a rich variety of ordered states, including electronic crystal phases with non-zero Chern numbers.”

“The paper completely characterizes the information-theoretic threshold for exact recovery in the GSBM.”

“The study reveals a transition from a single axisymmetric vortex ring to azimuthally unstable, multi-vortex structures as film thickness decreases.”

“The paper finds an information critical phase where the coherent information saturates to a fractional value, indicating that a finite fraction of logical information is still preserved.”

“The paper establishes a theoretical upper bound on excess risk characterized by a distinct phase transition. In the initial optimization phase, the excess risk decays exponentially relative to the computational cost. However, once a specific resource allocation threshold is crossed, the system enters a statistical phase, where the generalization error follows a power-law decay of Θ(C−1/6).”

“The paper completely characterizes a class of permutation polynomials of the form $L(X)+γTr_q^{q^3}(c_1X+c_2X^2+c_3X^3+c_4X^{q+2})$ over $\mathbb{F}_{q^3}$.”

“The study characterizes quantum synchronization.”

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