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This paper studies AdamW-style Shampoo, an effective variant of the classical Shampoo that won the external tuning track of the AlgoPerf neural network training competition. Our analysis unifies one-sided and two-sided preconditioning. When the exponents of the two preconditioners sum to $1/2$, we establish the convergence rate $\frac{1}{K}\sum_{k=1}^KE\left[|| abla f(X_k)||_*\right]\leq O(\frac{\sqrt{m+n}C}{K^{1/4}})$, where $K$ represents the number of iterations, $(m,n)$ denotes the dimensions of the matrix-valued parameters, and $C$ matches the constant appearing in the optimal convergence rate of SGD. Theoretically, the nuclear norm and Frobenius norm satisfy $|| abla f(X)||_F\leq || abla f(X)||_*\leq \sqrt{\min\{m,n\}}|| abla f(X)||_F$, which suggests that our convergence rate is analogous to the optimal $\frac{1}{K}\sum_{k=1}^KE\left[|| abla f(X_k)||_F\right]\leq O(\frac{C}{K^{1/4}})$ convergence rate of SGD in the ideal case where $|| abla f(X)||_*= Θ(\sqrt{\min\{m,n\}})|| abla f(X)||_F$ and $m$ and $n$ are of comparable magnitude. Then, we extend our analysis to settings where the preconditioning exponents do not sum to 1/2, and establish convergence with an explicit but m

We advertise intersection theory for generalised hypergeometric functions as a means of evaluating Mellin-Barnes representations. As an example, we study two-parameter representations of the off-shell one- and two-loop box graphs in exactly four-dimensional configuration space. Closing the integration contours for the MB parameters we transform these into double sums. Polygamma functions in the MB representation of the double box and the occurrence of higher poles are taken into account by parametric differentiation. Summing over any one of the counters results into a $_{p+1}F_p$ that we replace by its Euler integral representation. The process can be repeated a second time and results in a two- or four-parameter Euler integral, respectively. We use intersection theory to derive Pfaffian systems of equations on related sets of master integrals and solve for the box and double box integrals reproducing the known expressions. Finally, we use a trick to re-derive the double box from a two-parameter Euler integral. This second computation requires only very little computing resources.

A childhood observation of Thakur Anukulchandra that "one and one can only be two ones, not simply two" motivates a precise inquiry: what, exactly, is asserted when we pass from two concrete individuals to the numeral "2"? This paper does not challenge the arithmetic theorem 1+1=2, but rather analyzes what this equation means when applied to physical objects. We answer with two complementary, rigorous treatments. Mathematician's proof. We model aggregation by the free commutative monoid of multisets M(U) over a universe of individuals U, so that delta_a + delta_b literally encodes two ones with individuality preserved. Numerals arise only after a declared classification q:U->T (coarse-graining) via the pushforward q*:M(U)->M(T) and the unique counting homomorphism to N. The non-injectivity of q* isolates the exact locus of information loss. Physicist's proof. We represent physical systems by worldtubes, states, and observables, and define a composite operation that preserves labelled constituents. We prove the Non-Identity Addition Theorem: A+B = X+X iff A=B=X; hence a pair of distinct objects cannot equal a doubled copy. The numeral "2" appears only as the readout of a typed

In this work, we systematically study the differential systems governing loop-level wavefunction coefficients of conformally-coupled scalar field theory within a general power-law FRW cosmology. By utilizing the twisted cohomology, hyperplane arrangements, and IBP techniques, we derive the canonical differential equations for two-site one-loop bubble and tadpole systems, revealing distinct structural differences. We present new insights into the one-loop tadpole system, uncovering that its integral family can include multiple parent functions due to distinct pairs of relative hyperplane associated with each function, unlike the single parent function appearing in the one-loop bubble case. Moreover, we demonstrate that the tadpole correlator selectively probes only a subset of the cohomology space, despite the hyperplane arrangement suggesting a higher-dimensional structure. Another novel contribution of this work is the extension of kinematic flow framework to the loop-level scenarios for the first time. Using a graphical approach based on family trees generated by marked tubing graphs, which encode singularity structures, we efficiently construct the differential equations and unc

We study the problem of maximizing Nash social welfare, which is the geometric mean of agents' utilities, in two well-known models. The first model involves one-sided preferences, where a set of indivisible items is allocated among a group of agents (commonly studied in fair division). The second model deals with two-sided preferences, where a set of workers and firms, each having numerical valuations for the other side, are matched with each other (commonly studied in matching-under-preferences literature). We study these models under capacity constraints, which restrict the number of items (respectively, workers) that an agent (respectively, a firm) can receive. We develop constant-factor approximation algorithms for both problems under a broad class of valuations. Specifically, our main results are the following: (a) For any $ε> 0$, a $(6+ε)$-approximation algorithm for the one-sided problem when agents have submodular valuations, and (b) a $1.33$-approximation algorithm for the two-sided problem when the firms have subadditive valuations. The former result provides the first constant-factor approximation algorithm for Nash welfare in the one-sided problem with submodular val

Existence of the eigenvalues of the discrete-time quantum walks is deeply related to localization. Also, for the study of open quantum systems, non-Hermitian systems have attracted much attention. As mathematical models for such systems, non-unitary quantum walks with the chiral symmetry are essential for the study of the topological insulator. In this paper, we give the whole picture of the eigenvalues of a non-unitary one-dimensional two-state quantum walks with one defect and the chiral symmetry.

We experimentally investigate a superconducting circuit composed of two flux qubits ultrastrongly coupled to a common LC resonator. Owing to the large anharmonicity of the flux qubits, the system can be described well by a generalized Dicke Hamiltonian containing spin spin interaction terms. In the experimentally measured spectrum, we observed two key phenomena. First, an avoided level crossing provides evidence of the exotic interaction that allows the simultaneous excitation of two artificial atoms by absorbing one photon from the resonator. Second, we identified a pronounced spectral asymmetry that is a clear signature of light matter decoupling. This multi atom ultrastrongly coupled system opens the door to studying novel processes for quantum optics and quantum-information tasks on a chip.

Under the spin-position decoupling approximation, a vector with a phase in 3D orientation space endowed with geometric algebra, substitutes the vector-matrix spin model built on the Pauli spin operator. The standard quantum operator-state spin formalism is replaced with vectors transforming by proper and improper rotations in the same 3D space -- isomorphic to the space of Pauli matrices. In the single spin case the novel spin 1/2 representation: (1) is Hermitian; (2) shows handedness; (3) yields all the standard results and its modulus equals the total spin angular momentum S_tot; (4) formalizes irreversibility in measurement; (5) permits adaptive embedding of the 2D spin space in 3D. Maximally entangled spin pairs: (1) are in phase and have opposite handedness; (2) relate by one of the four basic improper rotations in 3D: plane-reflections for triplets and inversion for singlet; (3) yield the standard total angular momentum; (4) all standard expectation values for bipartite and partial observations follow. Depending on whether proper and improper rotors act one or two sided, the formalism appears in two complementary forms, the spinor or the vector form, respectively. The propose

Strategic behavior in two-sided matching markets has been traditionally studied in a "one-sided" manipulation setting where the agent who misreports is also the intended beneficiary. Our work investigates "two-sided" manipulation of the deferred acceptance algorithm where the misreporting agent and the manipulator (or beneficiary) are on different sides. Specifically, we generalize the recently proposed accomplice manipulation model (where a man misreports on behalf of a woman) along two complementary dimensions: (a) the two for one model, with a pair of misreporting agents (man and woman) and a single beneficiary (the misreporting woman), and (b) the one for all model, with one misreporting agent (man) and a coalition of beneficiaries (all women). Our main contribution is to develop polynomial-time algorithms for finding an optimal manipulation in both settings. We obtain these results despite the fact that an optimal one for all strategy fails to be inconspicuous, while it is unclear whether an optimal two for one strategy satisfies the inconspicuousness property. We also study the conditions under which stability of the resulting matching is preserved. Experimentally, we show th

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