As large language models (LLMs) are increasingly applied to real-world scenarios, it becomes crucial to understand their ability to follow multiple instructions simultaneously. To systematically evaluate these capabilities, we introduce two specialized benchmarks for fundamental domains where multiple instructions following is important: Many Instruction-Following Eval (ManyIFEval) for text generation with up to ten instructions, and Style-aware Mostly Basic Programming Problems (StyleMBPP) for code generation with up to six instructions. Our experiments with the created benchmarks across ten LLMs reveal that performance consistently degrades as the number of instructions increases. Furthermore, given the fact that evaluating all the possible combinations of multiple instructions is computationally impractical in actual use cases, we developed three types of regression models that can estimate performance on both unseen instruction combinations and different numbers of instructions which are not used during training. We demonstrate that a logistic regression model using instruction count as an explanatory variable can predict performance of following multiple instructions with approximately 10% error, even for unseen instruction combinations. We show that relatively modest sample sizes (500 for ManyIFEval and 300 for StyleMBPP) are sufficient for performance estimation, enabling efficient evaluation of LLMs under various instruction combinations.

We show that starting with a cosmological constant in a curved space-time, the Einstein-Hilbert term of general relativity is generated through a ghost condensation. We fix Weyl symmetry, or equivalently local scale symmetry by a gauge condition $R = 0$ à la BRST formalism, and see that the condensation of the Faddeev-Popov ghosts, $\langle \bar c c \rangle \neq 0$ leads to a generation of the Einstein-Hilbert action of general relativity. This dynamical mechanism of symmetry breakdown for a global scale symmetry is new in the sense that the reduction of fermionic degrees of freedom effectively leads to a generation of bosonic degrees of freedom. We also discuss this mechanism from the viewpoint of the problem of a bound state, and show that asymptotic fields corresponding to the bound states are ``confined'' to the unphysical Hilbert space.

Invasion by the red imported fire ant, Solenopsis invicta Buren, has destructive effects on native biodiversity, agriculture, and public health. This ant's aggressive foraging behaviour and high reproductive capability have enabled its establishment of wild populations in most regions into which it has been imported. An important aspect of eradication is thorough nest monitoring and destruction during early invasion to prevent range expansion. The question is: How intense must monitoring be on temporal and spatial scales to eradicate the fire ant? Assuming that the ant was introduced into a region and that monitoring was conducted immediately after nest detection in an effort to detect all other potentially established nests, we developed a mathematical model to investigate detection rates. Setting the monitoring limit to three years, the detection rate was maximized when monitoring was conducted shifting bait trap locations and setting them at intervals of 30 m for each monitoring. Monitoring should be conducted in a radius of at least 4 km around the source nest, or wider --depending on how late a nest is found. For ease of application, we also derived equations for finding the minimum bait interval required in an arbitrary ant species for thorough monitoring.

We consider a generic scale invariant scalar quantum field theory and its symmetry breakdown. Based on the dimension counting identity, we give a concise proof that dilaton is exactly massless at the classical level if scale invariance is broken spontaneously. On the other hand, on the basis of the generalized dimension counting identity, we prove that the dilaton becomes massive at the quantum level if scale invariance is explicitly broken by quantum anomaly. It is pointed out that a subtlety occurs when scale invariance is spontaneously broken through a scale invariant regularization method where the renormalization scale is replaced with the dilaton field. In this case, the dilaton remains massless even at the quantum level after spontaneous symmetry breakdown of scale symmetry, but when the massless dilaton couples non-minimally to the Einstein-Hilbert term and is applied for cosmology, it is phenomenologically ruled out by solar system tests unless its coupling to matters is much suppressed compared to the gravitational interaction.

We consider a generic scale invariant scalar quantum field theory and its symmetry breakdown. Based on the dimension counting identity, we give a concise proof that dilaton is exactly massless at the classical level if scale invariance is broken spontaneously. On the other hand, on the basis of the generalized dimension counting identity, we prove that the dilaton becomes massive at the quantum level if scale invariance is explicitly broken by quantum anomaly. It is pointed out that a subtlety occurs when scale invariance is spontaneously broken through a scale invariant regularization method where the renormalization scale is replaced with the dilaton field. In this case, the dilaton remains massless even at the quantum level after spontaneous symmetry breakdown of scale symmetry, but when the massless dilaton couples non-minimally to the Einstein-Hilbert term and is applied for cosmology, it is phenomenologically ruled out by solar system tests unless its coupling to matters is much suppressed compared to the gravitational interaction.

The hypergeometric distribution is a popular distribution, whose properties have been extensively investigated. Generating functions of this distribution, such as the probability-generating function, the moment-generating function, and the characteristic function, are known to involve the Gauss hypergeometric function. We elucidate that the existing formula of generating functions is incomplete and provide the corrected formula. In view of the utility and applicability of the hypergeometric distribution, the exact and correct foundation is crucially important.

We give a construction for spherical 3-designs. This construction is a generalization of Bondarenko's work.

We study classical solutions in the Weyl-transverse (WTDiff) gravity coupled to an electro-magnetic field in four space-time dimensions. The WTDiff gravity is invariant under both the local Weyl (conformal) transformation and the volume preserving diffeormorphisms (transverse diffeomorphisms) and is known to be equivalent to general relativity at least at the classical level (perhaps even in the quantum regime). In particular, we find that only in four space-time dimensions, the charged Reissner-Nordstrom black hole metric is a classical solution when it is expressed in the Cartesian coordinate system.

The ground state of the multiple-spin exchange model with up to the six-spin exchange interactions on a triangular lattice in the magnetic field is investigated within the mean-field approximation. By comparing the phase diagrams of systems with and without the five- and six-spin interactions, we find that the five- and six-spin interactions expand the ferromagnetic phase. In contrast, the antiferromagnetic-like phases are contracted. A phase with a 1/2 plateau is also contracted by the five- and six-spin interactions. In addition to the results that have not yet been reported for systems with up to the four-spin interactions, we obtain new phases for systems with up to the six-spin interactions. In particular, a 1/6 plateau appears in the low magnetic field near the ferromagnetic phase in the magnetization process. This phase has a twelve-sublattice structure in which a unit cell consists of seven spins parallel to the magnetic field and five spins antiparallel to it.

Using scanning tunneling spectroscopy in ultra-high vacuum at low temperature (T = 0.3 K) and high magnetic fields (B < 12 T), we directly probe electronic wave functions across an integer quantum Hall transition. In accordance with theoretical predictions, we observe the evolution from localized drift states in the insulating phases to branched extended drift states at the quantum critical point. The observed microscopic behavior close to the extended state indicates points of localized quantum tunneling, which are considered to be decisive for a quantitative description of the transition.

In a previous paper, I found that the Weyl group $W(F_4)$ and Barns-Wall Lattice $BW_{16}$ can be constructed using the rank $2$ tensor of the quaternion. In the present paper, I describe how I were able to construct an algebra, which is the subalgebra of the direct product of Hurwitz Quaternionic integers $\mathscr{H}^4$, isomorphic to the automorphism $\text{Aut}(BW_{16})$ order $2^{21} \cdot 3^5 \cdot 5^2 \cdot 7$ of Barns Wall Lattice $BW_{16}$ by functionally extending the rank of the tensor product of quaternions to $4$.

The charge density wave (CDW) state is a widespread phenomenon in low-dimensional metals/semimetals. The spectral weight of the associated folded bands (shadow bands) can be an intriguing trigger leading to additional Fermi surface instability and unexplored phase transitions. The rare earth tri-telluride CeTe3 exhibits a single CDW stabilized below ~400 K and antiferromagnetism below ~3 K. The distinct periodicities between the Te-square net, the CeTe block layer, and the CDW give rise to rich shadow band formations. In this work, we reveal the predominant scattering between the original and shadow bands at 4 K, with the scattering within the original bands being relatively suppressed at Fermi energy. This unconventional quasi-particle scattering collectively underscores the vital role of the shadow bands' spectral weight and the hidden matrix element effect, which are crucial for controlling electronic properties in this system. Furthermore, our finding points to the existence of rich and unexplored Fermi surface instabilities, which potentially play a role in controlling the nature of long-range antiferromagnetism at lower temperatures in the presence of finite charge-spin interaction.

We consider a coupling of conformal gravity to the classically scale-invariant B-L extended standard model which has been recently proposed as a phenomenologically viable model realizing the Coleman-Weinberg mechanism of breakdown of the electro-weak symmetry. As in a globally scale-invariant dilaton gravity, it is also shown in a locally scale-invariant conformal gravity that without recourse to the Coleman-Weinberg mechanism, the B-L gauge symmetry is broken in the process of spontaneous symmetry breakdown of the local scale invariance (Weyl invariance) at the tree level and as a result the B-L gauge field becomes massive via the Higgs mechanism. As a bonus of conformal gravity, the massless dilaton field does not appear and the parameters in front of the non-minimal coupling of gravity are completely fixed in the present model. This observation clearly shows that the conformal gravity has a practical application even if the scalar field does not possess any dynamical degree of freedom owing to the local scale symmetry.

Let $\Lambda$ be any integral lattice in Euclidean space. It has been shown that for every integer $n>0$, there is a hypersphere that passes through exactly $n$ points of $\Lambda$. Using this result, we introduce new lattice invariants and give some computational results related to two-dimensional Euclidean lattices of class number one.

We analyze the observational constraints on brane-world cosmology whereby the universe is described as a three-brane embedded in a five-dimensional anti-de Sitter space. In this brane-universe cosmology, the Friedmann equation is modified by the appearance of extra terms which derive from existence of the extra dimensions. In the present work we concentrate on the ``dark radiation'' term which diminishes with cosmic scale factor as $a^{-4}$. We show that, although the observational constraints from primordial abundances allow only a small contribution when this term is positive, a much wider range of negative values is allowed. Furthermore, such a negative contribution can reconcile the tension between the observed primordial $\he4$ and D abundances. We also discuss the possible constraints on this term from the power spectrum of CMB anisotropies in the limit of negligible cosmological perturbation on the brane world. We show that BBN limits the possible contribution from dark radiation just before the nucleosynthesis epoch to lie between -65% and $+5$ of the background photon energy density. Combining this with the CMB constraint reduces this range to between -24% and $+3.5$ at the $2\sigma$ confidence level.

CeTe3 is a unique platform to investigate the itinerant magnetism in a van der Waals (vdW) coupled metal. Despite chemical pressure being a promising route to boost quantum fluctuation in this system, a systematic study on the chemical pressure effect on Ce3+(4f1) states is absent. Here, we report on the successful growth of a series of Se doped single crystals of CeTe3. We found a fluctuation driven exotic magnetic rotation from the usual easy-axis ordering to an unusual hard-axis ordering. Unlike in localized magnetic systems, near-critical magnetism can increase itinerancy hand-in-hand with enhancing fluctuation of magnetism. Thus, seemingly unstable hard-axis ordering emerges through kinetic energy gain, with the self-consistent observation of enhanced magnetic fluctuation (disorder). As far as we recognize, this order-by-disorder process in fermionic system is observed for the first time within vdW materials. Our finding opens a unique experimental platform for direct visualization of the rich quasiparticle Fermi surface deformation associated with the Fermionic order-by-disorder process. Also, the search for emergent exotic phases by further tuning of quantum fluctuation is suggested as a promising future challenge.

The Horton-Strahler analysis is a graph-theoretic method to measure the bifurcation complexity of branching patterns, by defining a number called the order to each branch. The main result of this paper is a large deviation theorem for the number of branches of each order in a random binary tree. The rate function associated with a large deviation cannot be derived in a closed form; instead, asymptotic forms of the rate function are given.

The observation of cyclotron resonance in ultra-clean crystals of URu2Si2 [S. Tonegawa et al., PRL 109, 036401 (2012)] provides another route besides quantum oscillations to the determination of the bulk electronic structure in the hidden order phase. We report detailed analyses of the resonance lines, which fully resolve the cyclotron mass structure of the main Fermi surface sheets. A particular focus is given to the anomalous splitting of the sharpest resonance line near the [110] direction under in-plane magnetic-field rotation, which implies peculiar electronic structure in the hidden order phase. The results under the field rotation from [110] toward [001] direction reveal that the splitting is a robust feature against field tilting from the basal plane. This is in sharp contrast to the reported frequency branch alpha in the quantum oscillation experiments showing a three-fold splitting that disappears by a small field tilt, which can be explained by the magnetic breakdown between the large hole sphere and small electron pockets. Our analysis of the cyclotron resonance profiles reveals that the heavier branch of the split line has a larger scattering rate, providing evidence for the existence of hot-spot regions along the [110] direction. These results are consistent with the broken fourfold rotational symmetry in the hidden-order phase, which can modify the interband scattering in an asymmetric manner. We also extend our measurements down to 0.7 K, which results in the observation of cyclotron resonance in the superconducting state, where novel effects of vortex dynamics may enter. We find that the cyclotron mass undergoes no change in the superconducting state. In contrast, the quasiparticle scattering rate shows a rapid decrease below the vortex-lattice melting transition temperature, which supports the formation of quasiparticle Bloch state in the vortex lattice phase.

We study the low-temperature electrical and thermal conductivity of CoSi and Co$_{1-x}$M$_x$Si alloys (M = Fe, Ni; $x \leq$ 0.06). Measurements show that the low-temperature electrical conductivity of Co$_{1-x}$Fe$_{x}$Si alloys decreases at $x >$ 0.01 by an order of magnitude compared with that of pure CoSi. It was expected that both the lattice and electronic contributions to thermal conductivity would decrease in the alloys. However, our experimental results revealed that at temperatures below 20K the thermal conductivity of Fe- and Ni-containing alloys is several times larger than that of pure CoSi. We discuss possible mechanisms of the thermal conductivity enhancement. The most probable one is related to the dominant scattering of phonons by charge carriers. We propose a simple theoretical model that takes into account the complex semimetallic electronic structure of CoSi with nonequivalent valleys, and show that it explains well the increase of the lattice thermal conductivity with increasing disorder and the linear temperature dependence of the thermal conductivity in the Co$_{1-x}$Fe$_x$Si alloys below 20K.