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Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which not only gives, as a special case, a famous modular relation between the Rogers-Ramanujan functions $G(q)$ and $H(q)$ but also a relation between two fifth order mock theta functions and $G(q)$ and $H(q)$. We generalize Ramanujan's relation with the help of a parameter $s$ to get an infinite family of such identities. Our result shows that a theta function can always be ``dissected'' as a finite sum of products of generalized Rogers-Ramanujan functions. Several well-known results are shown to be consequences of our theorem, for example, a generalization of the Jacobi triple product identity and Andrews' relation between two of his generalized third order mock theta functions. We give enough evidence, through asymptotic analysis as well as by other means, to show that the identities we get from our main result for $s>2$ transcend the modular world and hence look difficult to be written in the form of a modular relation. Using asymptotic analysis, we also offer a clinching evidence that explains how Ramanujan may have arrived at his generalized modular relation., Comment: 23 pages, submitted for publication. Comments are welcome

The Automated Audio Captioning (AAC) task aims to describe an audio signal using natural language. To evaluate machine-generated captions, the metrics should take into account audio events, acoustic scenes, paralinguistics, signal characteristics, and other audio information. Traditional AAC evaluation relies on natural language generation metrics like ROUGE and BLEU, image captioning metrics such as SPICE and CIDEr, or Sentence-BERT embedding similarity. However, these metrics only compare generated captions to human references, overlooking the audio signal itself. In this work, we propose MACE (Multimodal Audio-Caption Evaluation), a novel metric that integrates both audio and reference captions for comprehensive audio caption evaluation. MACE incorporates audio information from audio as well as predicted and reference captions and weights it with a fluency penalty. Our experiments demonstrate MACE's superior performance in predicting human quality judgments compared to traditional metrics. Specifically, MACE achieves a 3.28% and 4.36% relative accuracy improvement over the FENSE metric on the AudioCaps-Eval and Clotho-Eval datasets respectively. Moreover, it significantly outperforms all the previous metrics on the audio captioning evaluation task. The metric is opensourced at https://github.com/satvik-dixit/mace

Spin gapless semiconductors based Heusler alloys are the special class of materials due to their unique band structure, high spin polarization and high Curie temperature. These materials exhibit a distinct electronic structure: a nonzero band gap in one spin channel while the other spin channel remains gapless, making them highly suitable for tunable spintronics. In this study, a comprehensive analysis of structural, magnetic, thermoelectric, and transport properties of the quaternary Heusler alloy CoFeMnSn is conducted. X-ray diffraction and Neutron diffraction analyses confirm a well ordered structure with partial antisite disorder between Co, Fe and Mn, Sn atoms. Magnetic studies show that the material exhibits room-temperature ferromagnetism, with a Curie temperature of around 660 K. Notably, we observe an anomalous Hall effect linked to intrinsic mechanisms driven by Berry curvature, underscoring the intricate relationship between structural disorder and electronic behavior. Transport measurements also highlight the impact of antisite disorder on the systems, with resistivity decreasing as temperature increases. These insights position CoFeMnSn as a promising material for future spintronic devices and advanced technological applications., Comment: 9 pages, 9 Figures

Desert locusts are notorious agriculture pests prompting billions in losses and global food scarcity concerns. With billions of these locusts invading agrarian lands, this is no longer a thing of the past. This study taps into the existing doppler weather radar (DWR) infrastructure which was originally deployed for meteorological applications. This study demonstrates a systematic approach to distinctly identify and track concentrations of desert locust swarms in near real time using single polarization radars. Findings reveal the potential to establish early warning systems with lead times of around 7 hours and spatial coverage of approximately 100 kilometers. Embracing these technological advancements are crucial to safeguard agricultural landscapes and upload global food security., Comment: 18 pages, 5 figures

Magnesium sulfate (${ \rm MgSO_4 }$) is used as an additive to reduce capacity fading in rechargeable zinc-ion batteries. This study investigates the ion pairing and solvation structure of ${ \rm Zn^{2+}-SO_4^{2-} }$ and ${ \rm Mg^{2+}-SO_4^{2-} }$ in mixtures of ${ \rm [ZnSO_4]{2M} + [MgSO_4]{0M} }$, ${ \rm [ZnSO_4]{1M} + [MgSO_4]{1M} }$, and ${ \rm [ZnSO_4]{0M} + [MgSO_4]{2M} }$. Using molecular dynamics simulations, we evaluated dipole dynamics, spatial distributions, and coordination of ions through radial distribution functions, running coordination numbers, and potentials of mean force. The results reveal that increasing ${ \rm MgSO_4 }$ concentration disrupts ${ \rm Zn^{2+} }$ self-association, while ${ \rm Mg^{2+} }$ shows enhanced association. The analysis indicates different solvation structures, with ${ \rm Zn^{2+} }$ maintaining a more ordered coordination with ${ \rm SO_4^{2-} }$ than ${ \rm Mg^{2+} }$. Preferential binding analysis highlights stronger ${ \rm Zn^{2+} }$ affinity for ${ \rm SO_4^{2-} }$, shedding light on ion interaction dynamics in mixed electrolytes.

Many students struggle with math word problems (MWPs), often finding it difficult to identify key information and select the appropriate mathematical operations. Schema-based instruction (SBI) is an evidence-based strategy that helps students categorize problems based on their structure, improving problem-solving accuracy. Building on this, we propose a Schema-Based Instruction Retrieval-Augmented Generation (SBI-RAG) framework that incorporates a large language model (LLM). Our approach emphasizes step-by-step reasoning by leveraging schemas to guide solution generation. We evaluate its performance on the GSM8K dataset, comparing it with GPT-4 and GPT-3.5 Turbo, and introduce a "reasoning score" metric to assess solution quality. Our findings suggest that SBI-RAG enhances reasoning clarity and facilitates a more structured problem-solving process potentially providing educational benefits for students., Comment: Accepted to the 4th MATH-AI Workshop at NeurIPS'24

We investigate two asymmetric loss functions, namely LINEX loss and power divergence loss for optimal spatial prediction with area-level data. With our motivation arising from the real estate industry, namely in real estate valuation, we use the Zillow Home Value Index (ZHVI) for county-level values to show the change in prediction when the loss is different (asymmetric) from a traditional squared error loss (symmetric) function. Additionally, we discuss the importance of choosing the asymmetry parameter, and propose a solution to this choice for a general asymmetric loss function. Since the focus is on area-level data predictions, we propose the methodology in the context of conditionally autoregressive (CAR) models. We conclude that choice of the loss functions for spatial area-level predictions can play a crucial role, and is heavily driven by the choice of parameters in the respective loss.

The Axion Dark Matter eXperiment is sensitive to narrow axion flows, given axions compose a fraction of the dark matter with a non-negligible local density. Detecting these low-velocity dispersion flows requires a high spectral resolution and careful attention to the expected signal modulation due to Earth's motion. We report an exclusion on the local axion dark matter density in narrow flows of $\rho_a \gtrsim 0.03\,\mathrm{GeV/cm^3}$ and $\rho_a \gtrsim 0.004\,\mathrm{GeV/cm^3}$ for Dine-Fischler-Srednicki-Zhitnitski and Kim-Shifman-Vainshtein-Zakharov axion-photon couplings, respectively, over the mass range $3.3-4.2\,\mu\text{eV}$. Measurements were made at selected resolving powers to allow for a range of possible velocity dispersions., Comment: 7 pages, 3 figures

In this paper, we establish the existence of positive ground state solutions for a class of mixed Schr\"{o}dinger systems with concave-convex nonlinearities in $\mathbb{R}^2$, subject to $L^2$-norm constraints; that is, \[ \left\{ \begin{aligned} -\partial_{xx} u + (-\Delta)_y^s u + \lambda_1 u &= \mu_1 u^{p-1} + \beta r_1 u^{r_1-1} v^{r_2}, && -\partial_{xx} v + (-\Delta)_y^s v + \lambda_2 v &= \mu_2 v^{q-1} + \beta r_2 u^{r_1} v^{r_2-1}, && \end{aligned} \right. \] subject to the $L^2$-norm constraints: \[ \int_{\mathbb{R}^2} u^2 \,\mathrm{d}x\mathrm{d}y = a \quad \text{and} \quad \int_{\mathbb{R}^2} v^2 \,\mathrm{d}x\mathrm{d}y = b, \] where $(x,y)\in \mathbb{R}^2$, $u, v \geq 0$, $s \in \left(1/2, 1 \right)$, $\mu_1, \mu_2, \beta > 0$, $r_1, r_2 > 1$, the prescribed masses $a, b > 0$, and the parameters $\lambda_1, \lambda_2$ appear as Lagrange multipliers. Moreover, the exponents $p, q, r_1 + r_2$ satisfy: \[ \frac{2(1+3s)}{1+s} < p, q, r_1 + r_2 < 2_s, \] where $2_s = \frac{2(1+s)}{1-s}$. To obtain our main existence results, we employ variational techniques such as the Mountain Pass Theorem, the Pohozaev manifold, Steiner rearrangement, and others, consolidating the works of Louis Jeanjean et al. \cite{jeanjean2024normalized}.

Speaker verification systems have seen significant advancements with the introduction of Multi-scale Feature Aggregation (MFA) architectures, such as MFA-Conformer and ECAPA-TDNN. These models leverage information from various network depths by concatenating intermediate feature maps before the pooling and projection layers, demonstrating that even shallower feature maps encode valuable speaker-specific information. Building upon this foundation, we propose a Multi-scale Feature Contrastive (MFCon) loss that directly enhances the quality of these intermediate representations. Our MFCon loss applies contrastive learning to all feature maps within the network, encouraging the model to learn more discriminative representations at the intermediate stage itself. By enforcing better feature map learning, we show that the resulting speaker embeddings exhibit increased discriminative power. Our method achieves a 9.05% improvement in equal error rate (EER) compared to the standard MFA-Conformer on the VoxCeleb-1O test set.

We derive Vorono\"{\dotlessi} summation formulas for the Liouville function $\lambda(n)$, the M\"{o}bius function $\mu(n)$, and for $d^{2}(n)$, where $d(n)$ is the divisor function. The formula for $\lambda(n)$ requires explicit evaluation of certain infinite series for which the use of the Vinogradov-Korobov zero-free region of the Riemann zeta function is indispensable. Several results of independent interest are obtained as special cases of these formulas. For example, a special case of the one for $\mu(n)$ is a famous result of Ramanujan, Hardy, and Littlewood. Cohen type and Ramanujan-Guinand type identities are established for $\lambda(n)$ and $\sigma_a(n)\sigma_b(n)$, where $\sigma_s(n)$ is the generalized divisor function. As expected, infinite series over the non-trivial zeros of $\zeta(s)$ now form an essential part of all of these formulas. A series involving $\sigma_a(n)\sigma_b(n)$ and product of modified Bessel functions occurring in one of our identities has appeared in a recent work of Dorigoni and Treilis in string theory. Lastly, we obtain results on oscillations of Riesz sums associated to $\lambda(n), \mu(n)$ and of the error term of Riesz sum of $d^2(n)$ under the assumption of the Riemann Hypothesis, simplicity of the zeros of $\zeta(s)$, the Linear Independence conjecture, and a weaker form of the Gonek-Hejhal conjecture., Comment: 45 pages, submitted for publication