Elliptic Gauss Sums and Hecke L-values at s=1

The rationality of the elliptic Gauss sum coefficient is shown. The following is a specific case of our argument. Let f(u)=sl((1-i)\varpi u), where sl() is the Gauss' lemniscatic sine and \varpi=2.62205... is the real period of the elliptic curve y^2=x^3-x, so that f(u) is an elliptic function relative to the period lattice Z[i]. Let \pi be a primary prime of Z[i] such that norm(\pi)\equiv 13\mod 16. Let S be the quarter set mod \pi consisting of quartic residues. Let us define G(\pi):=\sum_{\nu\in S} f(\nu/\pi) and \tilde{\pi}:=\prod_{\nu\in S} f(\nu/\pi). The former G(\pi) is a typical example of elliptic Gauss sum; the latter is regarded as a canonical 4-th root of -\pi: (\tilde{\pi})^4=-\pi. Then we have Theorem: G(\pi)/(\tilde{\pi})^3 is a rational odd integer. G(\pi) appears naturally in the central value of Hecke L associated to the quartic residue character mod \pi, and our proof is based on the functional equation of L and an explicit formula of the root number. In fact, the latter is nothing but the Cassels-Matthews formula on the quartic Gauss sum.Comment: 39 page

Bricks over preprojective algebras and join-irreducible elements in Coxeter groups

A (semi)brick over an algebra $A$ is a module $S$ such that the endomorphism ring $\operatorname{\mathsf{End}}_A(S)$ is a (product of) division algebra. For each Dynkin diagram $\Delta$, there is a bijection from the Coxeter group $W$ of type $\Delta$ to the set of semibricks over the preprojective algebra $\Pi$ of type $\Delta$, which is restricted to a bijection from the set of join-irreducible elements of $W$ to the set of bricks over $\Pi$. This paper is devoted to giving an explicit description of these bijections in the case $\Delta=\mathbb{A}_n$ or $\mathbb{D}_n$. First, for each join-irreducible element $w \in W$, we describe the corresponding brick $S(w)$ in terms of "Young diagram-like" notation. Next, we determine the canonical join representation $w=\bigvee_{i=1}^m w_i$ of an arbitrary element $w \in W$ based on Reading's work, and prove that $\bigoplus_{i=1}^n S(w_i)$ is the semibrick corresponding to $w$.Comment: 37 page

Bipartite Chebyshev polynomials and elliptic integrals expressible by elementary functions

The article is concerned with polynomials $g(x)$ whose graphs are "partially packed" between two horizontal tangent lines. We assume that most of the local maximum points of $g(x)$ are on the first horizontal line, and most of the local minimum points on the second horizontal line, except several "exceptional" maximum or minimum points, that locate above or under two lines, respectively. In addition, the degree of $g(x)$ is exactly the number of all extremum points $+1$. Then we call $g(x)$ a multipartite Chebyshev polynomial associated with the two lines. Under a certain condition, we show that $g(x)$ is expressed as a composition of the Chebyshev polynomial and a polynomial defined by the $x$-component data of the exceptional extremum points of $g(x)$ and the intersection points of $g(x)$ and the two lines. Especially, we study in detail bipartite Chebyshev polynomials, which has only one exceptional point, and treat a connection between such polynomials and elliptic integrals.Comment: 8 page

The Grothendieck groups and stable equivalences of mesh algebras

We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable categories. As an application, we give an complete classification of the mesh algebras up to stable equivalences.Comment: 35 page

Set Cross Entropy: Likelihood-based Permutation Invariant Loss Function for Probability Distributions

We propose a permutation-invariant loss function designed for the neural networks reconstructing a set of elements without considering the order within its vector representation. Unlike popular approaches for encoding and decoding a set, our work does not rely on a carefully engineered network topology nor by any additional sequential algorithm. The proposed method, Set Cross Entropy, has a natural information-theoretic interpretation and is related to the metrics defined for sets. We evaluate the proposed approach in two object reconstruction tasks and a rule learning task.Comment: The source code will be available at https://github.com/guicho271828/perminv . (comment for the revision: the result table was not correctly updated

The wall-chamber structures of the real Grothendieck groups

For a finite-dimensional algebra $A$ over a field $K$ with $n$ simple modules, the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}:=K_0(\operatorname{\mathsf{proj}} A) \otimes_\mathbb{Z} \mathbb{R} \cong \mathbb{R}^n$ gives stability conditions of King. We study the associated wall-chamber structure of $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ by using the Koenig--Yang correspondences in silting theory. First, we introduce an equivalence relation on $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ called TF equivalence by using numerical torsion pairs of Baumann--Kamnitzer--Tingley. Second, we show that the open cone in $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ spanned by the g-vectors of each 2-term silting object gives a TF equivalence class, and this gives a one-to-one correspondence between the basic 2-term silting objects and the TF equivalence classes of full dimension. Finally, we determine the wall-chamber structure of $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ in the case that $A$ is a path algebra of an acyclic quiver.Comment: 31 page

The Role of Head-Up Display in Computer- Assisted Instruction

We investigated the role of HUDs in CAI. HUDs have been used in various situations in daily lives by recent downsizing and cost down of the display devices. CAI is one of the promising applications for HUDs. We have developed an HUD-based CAI system for effectively presenting instructions of the equipment in the transportable earth station. This chapter described HUDs in CAI from a viewpoint of human-computer interaction based on the development experience.Comment: www.sciyo.co

Characterization of Product Measures by Integrability Condition

It is natural to ask whether "positivity" of white noise operators can be discussed in some sense and characterized. To answer this question, we consider the Gel'fand triple over the Complex Gaussian space (\ce'_c,\m_c), i.e. \ce'_c=\ce'+i\ce' equipped with the product measure \m_c=\m'\times\m' where \m' is the Gaussian measure on \ce' with variance 1/2 (Section \ref{sec:2-2}). Following AKK's Legendre transform technique, we have \cw_{u_1,u_2}\subset L^2(\ce'_c,\m_c)\subset [\cw]^{*}_{u_1,u_2} for functions $u_1,u_2\in C_{+,1/2}$ satisfying (U0)(U2)(U3). Several examples for $u_1, u_2$ are given in Section \ref{sec:2-3}. We remark that Ouerdiane \cite{oue} studied a special case $u_1(r^2)=u_2(r^2)=\exp(k^{-1}r^k)$, where $1\leq k\leq 2$. In Section \ref{sec:3}, the characterization theorem for measures can be extended to the case of positive product Radon measures on \ce'\times \ce'. In addition, the notion of pseudo-positive operators is naturally introduced via kernel theorem and characterized by an integrability condition. Lemma \ref{lem:3-2} plays crucial roles in Section \ref{sec:3}.Comment: To appear in Quantum Information III, T. Hida and K. Saito (eds), (World Scientific) 2001, (Volterra Preprint No. 436, 2000

Photo-Realistic Blocksworld Dataset

In this report, we introduce an artificial dataset generator for Photo-realistic Blocksworld domain. Blocksworld is one of the oldest high-level task planning domain that is well defined but contains sufficient complexity, e.g., the conflicting subgoals and the decomposability into subproblems. We aim to make this dataset a benchmark for Neural-Symbolic integrated systems and accelerate the research in this area. The key advantage of such systems is the ability to obtain a symbolic model from the real-world input and perform a fast, systematic, complete algorithm for symbolic reasoning, without any supervision and the reward signal from the environment.Comment: The dataset generator is available at https://github.com/ibm/photorealistic-blocksworl

Integral Transform and Segal-Bargmann Representation Associated to q-Charlier Polynomials

Let $\mu_p^{(q)}$ be the q-deformed Poisson measure in the sense of Saitoh Yoshida and $\nu_p$ be the measure given by Equation \eqref{eq:nu-q}. In this short paper, we introduce the q-deformed analogue of the Segal-Bargmann transform associated with $\mu_p^{(q)}$. We prove that our Segal-Bargmann transform is a unitary map of $L^2(\mu_p^{(q)})$ onto the q-deformed Hardy space ${\cal H}^2(\nu_q)$. Moreover, we give the Segal-Bargmann representation of the multiplication operator by $x$ in $L^2(\mu_p^{(q)})$, which is a linear combination of the q-creation, q-annihilation, q-number, and scalar operators.Comment: Accepted for the publication in "Quantum Information IV", T. Hida and K. Saito (eds.), World Scientific. Minor misprints have been fixed. Reference information has been update