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In this paper, we discuss how pure mathematics and theoretical physics can be applied to the study of language models. Using set theory and analysis, we formulate mathematically rigorous definitions of language models, and introduce the concept of the moduli space of distributions for a language model. We formulate a generalized distributional hypothesis using functional analysis and topology. We define the entropy function associated with a language model and show how it allows us to understand many interesting phenomena in languages. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. Using the entropy function, we formulate a conjecture about AGI. Then, we show how thermodynamics gives us an immediate interpretation to language models. In particular we will define the concepts of partition function, internal energy and free energy for a language model, which offer insights into how language models work. Based on these results, we introduce a general concept of the geometrization of language models and define what is called the Boltzmann manifold. While the current LLMs are the special cases of the Boltzmann manifold., Comment: 18 pages

In this paper, we investigate a novel spatial dataset search paradigm over multiple spatial data sources, which enables users to conduct join and union searches seamlessly. Specifically, we define two search problems called Maximum Intersection Query (MIQ) and Maximum Coverage Query with a Connection constraint (MCQC). To address these problems, we propose a unified Multi-source Spatial Dataset Search (MSDS) framework. In MSDS, we design a multi-layer index to accelerate the MIQ and MCQC. In addition, we prove that the MCQC is NP-hard and design two greedy algorithms to solve the problem. To deal with the constant update of spatial datasets in each data source, we design a dynamic index updating strategy and optimize search algorithms to reduce communication costs and improve search efficiency. We evaluate the efficiency of MSDS on five real-world data sources, and the experimental results show that our framework is able to achieve a significant reduction in running time and communication cost.

In recent work, we conjectured that Calabi-Yau threefolds defined over $\mathbb{Q}$ and admitting a supersymmetric flux compactification are modular, and associated to (the Tate twists of) weight-two cuspidal Hecke eigenforms. In this work, we will address two natural follow-up questions, of both a physical and mathematical nature, that are surprisingly closely related. First, in passing from a complex manifold to a rational variety, as we must do to study modularity, we are implicitly choosing a "rational model" for the threefold; how do different choices of rational model affect our results? Second, the same modular forms are associated to elliptic curves over $\mathbb{Q}$; are these elliptic curves found anywhere in the physical setup? By studying the F-theory uplift of the supersymmetric flux vacua found in the compactification of IIB string theory on (the mirror of) the Calabi-Yau hypersurface $X$ in $\mathbb{P}(1,1,2,2,2)$, we find a one-parameter family of elliptic curves whose associated eigenforms exactly match those associated to $X$. Actually, we find two such families, corresponding to two different choices of rational models for the same family of Calabi-Yaus., Comment: 25 pages, 1 figure, 6 tables. v2: Minor changes, typos corrected

The Fermat type Calabi-Yau $n$-fold, denoted by $\mathscr{F}_n$, is the hypersurface of $\mathbb{P}^{n+1}$ defined by $\sum_{i=0}^{n+1}x_i^{n+2}=0$, which is the smooth fiber over the Fermat point $\psi=0$ of the Fermat pencil $$ \sum_{i=0}^{n+1} x^{n+2}_i -(n+2)\, \psi\, \prod_{i=0}^{n+1} x_i =0. $$ The nowhere vanishing holomorphic $n$-form on $\mathscr{F}_n$ defines an $n+1$ dimensional sub-Hodge structure of $(H^n(\mathscr{F}_n,\mathbb{Q}),F_p)$. In this paper, we will formulate a conjecture which says that this $n+1$ dimensional sub-Hodge structure splits completely into the direct sum of pure Hodge structures with dimensions $\leq 2$, among which is a direct summand $\mathbf{H}^n_{a,1}$ whose Hodge decomposition is $$ \mathbf{H}^n_{a,1}=H^{n,0}(\mathscr{F}_n) \oplus H^{0,n}(\mathscr{F}_n). $$ Using numerical methods, we are able to explicitly construct such a split for the cases where $n=3,4,6$, while we also construct a partial split for the cases where $n=8,10$. For $n=3,4,6,8,10$, we have numerically found that the value of the mirror map $t$ for the Fermat pencil at the Fermat point $\psi=0$ is of the form $$ t|_{\psi=0}=\frac{1}{2}+\xi \,i, $$ where $\xi$ is a real algebraic number that intuitively depends on the integer $n+2$. Furthermore, we have also numerically found that the quotient $c^+(\mathbf{H}^n_{a,1})/c^-(\mathbf{H}^n_{a,1})$ of the Deligne's periods of $\mathbf{H}^n_{a,1}$ is an algebraic number for the cases where $n=3,4,6,8,10$, and in fact we will formulate a stronger conjecture generalizing this observation. We will also show that $\mathbf{H}^4_{a,1}$ satisfies the prediction of Deligne's conjecture., Comment: 48 pages; add a new section about the verification of Deligne's conjecture for the Fermat sextic

In this paper, we will study the connections between the mirror symmetry of K3 surfaces and the geometry of the Legendre family of elliptic curves. We will prove that the mirror map of the Dwork family is equal to the period map of the Legendre family. This result provides an interesting explanation to the modularities of counting functions for K3 surfaces from the mirror symmetry point of view. We will also discuss the relations between the arithmetic geometry of smooth fibers of the Fermat pencil (Dwork family) and that of the smooth fibers of the Legendre family, e.g. Shioda-Inose structures, zeta functions, etc. In particular, we will study the relations between the Fermat quartic, which is modular with a weight-3 modular form $\eta(4z)^6$, and the elliptic curve over $\lambda=2$ of the Legendre family, whose weight-2 newform is labeled as \textbf{32.2.a.a} in LMFDB. We will also compute the Deligne's periods of the Fermat quartic, which are given by special values of the theta function $\theta_3$. Then we will numerically verify that they satisfy the predictions of Deligne's conjecture on the special values of $L$-functions of critical motives., Comment: 30 pages

In this paper, we will study the arithmetic geometry of rank-2 attractors, which are Calabi-Yau threefolds whose Hodge structures admit interesting splits. We will develop methods to analyze the algebraic de Rham cohomologies of rank-2 attractors, and we will illustrate how our methods work by focusing on an example in a recent paper by Candelas, de la Ossa, Elmi and van Straten. We will look at the interesting connections between rank-2 attractors in string theory and Deligne's conjecture on the special values of $L$-functions. We will also formulate several open questions concerning the potential connections between string theory and number theory., Comment: 19 pages. Several important changes. Correct the errors in early versions

Flux compactification of IIB string theory associates special points in Calabi-Yau moduli space to choices of (pairs of) integral three-form fluxes. In this paper, we propose that supersymmetric flux vacua are modular. That is, to a supersymmetric flux vacuum arising in a variety defined over $\mathbb{Q}$, we associate a two-dimensional Galois representation that we conjecture to be modular. We provide numerical evidence for our conjecture by examining flux vacua arising on the octic hypersurface in $\mathbb{P}^{4}(1,1,2,2,2)$., Comment: v2: References added; examples added and removed. 20 pages + appendices, 7 tables

In this paper, we will study the connections between the mirror symmetry of Calabi-Yau threefolds and Deligne's conjecture on the special values of the $L$-functions of critical motives. Using the theory of mirror symmetry, we will develop a method to compute the Deligne's period for a Calabi-Yau threefold in the mirror family of a one-parameter mirror pair. We will give two examples to show how this method works, and we will express the Deligne's period in terms of the classical periods of the threeform. Using this method, we will compute the Deligne's period of a Calabi-Yau threefold studied in a recent paper by Candelas, de la Ossa, Elmi and van Straten. Based on their numerical results, we will explicitly show that this Calabi-Yau threefold satisfies Deligne's conjecture. A second purpose of this paper is to introduce the Deligne's conjecture to the physics community, and provide further evidence that there might exist interesting connections between physics and number theory., Comment: 23 pages

In this paper, we will apply the tools from number theory and modular forms to the study of the Seiberg-Witten theory. We will express the holomorphic functions $a, a_D$, which generate the lattice $Z=n_e a+n_m a_D, (n_e, n_m) \in \mathbb{Z}^2$ of central charges, in terms of the periods of the Legendre family of elliptic curves. Thus we will be able to compute the transformations of the quotient $a_D/a$ under the action of the modular group $\text{PSL}(2,\mathbb{Z})$. We will show the Schwarzian derivative of the quotient $a_D/a$ with respect to the complexified coupling constant is given by the theta functions. We will also compute the scalar curvature of the moduli space of the $N=2$ supersymmetric Yang-Mills theory, which is shown to be asymptotically flat near the perturbative limit., Comment: 17 pages. Correct several typos and errors

The Bombieri-Dwork conjecture predicts that the differential equations satisfied by $G$-functions come from geometry. In this paper, we will look at special $G$-functions whose differential equations have a special singularity with maximally unipotent monodromy. We will formulate a Gamma conjecture about such $G$-functions, which has close connections with the mirror symmetry of Calabi-Yau threefolds and the Gamma conjecture in algebraic geometry. We will provide examples to support this conjecture, which involves numerical computations using Mathematica programs., Comment: 14 pages

In this paper, we will apply the ideas from the mirror symmetry of Calabi-Yau threefolds to study the modular forms and one-parameter family of K3 surfaces found by Beukers and Peters, which provide enlightenment to the two mysterious sequences constructed by Ap\'ery in his proof of the irrationality of $\zeta(3)$. We will construct a fourth order differential operator and a prepotential from the canonical solutions of this differential operator. The third derivative of this prepotential with respect to the mirror map defines a Yukawa coupling that is a weight-4 modular form. The instanton expansion of this Yukawa coupling yields integral instanton numbers, which are also periodic with period 6., Comment: 16 pages. Major corrections. Many new results

In this paper we will study the double zeta values $\zeta(k,m)$ using Picard-Fuchs equation. We will give a very efficient method to evaluate $\zeta(k,1)$ (resp. $\zeta(k,2)$) in terms of the products of zeta values $\zeta(2),\zeta(3),\cdots$ when $k$ is even (resp. odd), which admits immediate generalization to arbitrary double zeta values. Moreover, this method provides new insights into the nature of double zeta values, which further can be generalized to arbitrary multiple zeta values., Comment: 22 pages

In the mirror symmetry of Calabi-Yau threefolds, the instanton expansion of the prepotential has a constant term that is a rational multiple of $\zeta(3)/(2 \pi i)^3$, the motivic origin of which has been carefully studied in the author's paper with M. Kim. The Gamma conjecture claims that higher zeta values in fact appear in the theory of Calabi-Yau $n$-folds for $n \geq 4$. Moreover, a natural question is whether they also have a motivic origin. In this paper, we will study the limit mixed Hodge structure (MHS) of the Fermat pencil of Calabi-Yau $n$-folds at the large complex structure limit, which is actually a mixed Hodge-Tate structure. We will compute the period matrix of this limit MHS for the cases where $n=4,5,6,7,8,9,10,11,12$ by numerical method, and our computations have shown the occurrence of higher zeta values, which provide evidence to the Gamma conjecture. Furthermore, we will also provide a motivic explanation to our numerical results., Comment: 22 pages

In this paper, we present an application of mirror symmetry to arithmetic geometry. The main result is the computation of the period of a mixed Hodge structure, which lends evidence to its expected motivic origin. More precisely, given a mirror pair $(M,W)$ of Calabi-Yau threefolds, the prepotential of the complexified Kahler moduli space of $M$ admits an expansion with a constant term that is frequently of the form $$-3\, \chi (M) \,\zeta(3)/(2 \pi i)^3+r,$$ where $r \in \mathbb{Q}$ and $\chi(M)$ is the Euler characteristic of $M$. We focus on the mirror pairs for which the deformation space of the mirror threefold $W$ forms part of a one-parameter algebraic family $W_{\varphi}$ defined over $\mathbb{Q}$ and the large complex structure limit is a rational point. Assuming a version of the mirror conjecture, we compute the limit mixed Hodge structure on $H^3(W_{\varphi})$ at the large complex structure limit. It turns out to have a direct summand expressible as an extension of $\mathbb{Q}(-3)$ by $\mathbb{Q}(0)$ whose isomorphism class can be computed in terms of the prepotential of $M$, and hence, involves $\zeta(3)$. By way of Ayoub's works on the motivic nearby cycle functor, this reveals in precise form a connection between mirror symmetry and a variant of the Hodge conjecture for mixed Tate motives., Comment: 57 pages

Small objects detection is a challenging task in computer vision due to its limited resolution and information. In order to solve this problem, the majority of existing methods sacrifice speed for improvement in accuracy. In this paper, we aim to detect small objects at a fast speed, using the best object detector Single Shot Multibox Detector (SSD) with respect to accuracy-vs-speed trade-off as base architecture. We propose a multi-level feature fusion method for introducing contextual information in SSD, in order to improve the accuracy for small objects. In detailed fusion operation, we design two feature fusion modules, concatenation module and element-sum module, different in the way of adding contextual information. Experimental results show that these two fusion modules obtain higher mAP on PASCALVOC2007 than baseline SSD by 1.6 and 1.7 points respectively, especially with 2-3 points improvement on some smallobjects categories. The testing speed of them is 43 and 40 FPS respectively, superior to the state of the art Deconvolutional single shot detector (DSSD) by 29.4 and 26.4 FPS. Code is available at https://github.com/wnzhyee/Feature-Fused-SSD. Keywords: small object detection, feature fusion, real-time, single shot multi-box detector, Comment: Artificial Intelligence;8 pages,8 figures