Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity

Touchard-Riordan-like formulas are some expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove these kind of formulas, related with integer partitions. This gives a new perspective on the original result of Touchard and Riordan. But the main goal is to give a combinatorial proof of a Touchard-Riordan--like formula for q-secant numbers discovered by the first author. An interesting limit case of these objects can be directly interpreted in terms of partitions, so that we obtain a connection between the formula for q-secant numbers, and a particular case of Jacobi's triple product identity. Building on this particular case, we obtain a "finite version" of the triple product identity. It is in the form of a finite sum which is given a combinatorial meaning, so that the triple product identity can be obtained by taking the limit. Here the proof is non-combinatorial and relies on a functional equation satisfied by a T-fraction. Then from this result on the triple product identity, we derive a whole new family of Touchard-Riordan--like formulas whose combinatorics is not yet understood. Eventually, we prove a Touchard-Riordan--like formula for a q-analog of Genocchi numbers, which is related with Jacobi's identity for (q;q)^3 rather than the triple product identity.Comment: 29 page

On the Boundary Dynamics of Chern-Simons Gravity

We study Chern-Simons theory with a complex G_C or a real G x G gauge group on a manifold with boundary - this includes Lorentzian and Euclidean (anti-) de Sitter (E/A)dS gravity for G=SU(2) or G=SL(2,R). We show that there is a canonical choice of boundary conditions that leads to an unambiguous, fully covariant and gauge invariant, off-shell derivation of the boundary action - a G_C/G or G WZW model, coupled in a gauge invariant way to the boundary value of the gauge field. In particular, for (E/A)dS gravity, the boundary action is a WZW model with target space (E/A)dS_3, reminiscent of a worldsheet for worldsheet mechanism. We discuss in some detail the properties of the boundary theories that arise and we confront our results with various related constructions in the literature.Comment: 22 pages, LaTeX2e, v2: JHEP3.cls, references and a footnote adde

The timelike half-supersymmetric backgrounds of N=2, D=4 supergravity with Fayet-Iliopoulos gauging

Subject to some relatively mild assumptions, we derive the complete form of all timelike half-supersymmetric solutions to N=2, D=4 gauged supergravity coupled to an arbitrary number of abelian vector multiplets. This is done using spinorial geometry techniques. Explicit examples are given for a simple prepotential. Among the solutions, there are near-horizon geometries of extremal rotating BPS black holes still to be discovered, with a nontrivial dependence of the scalar fields on one of the horizon coordinates.Comment: 38 pages, uses JHEP3.cl

Three Recitations on Holonomic Systems and Hypergeometric Series

A tutorial on what later became to be known as WZ theory, as well as a motivated account of the seminal Gosper algorithm.Comment: Plain Te

S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality

A recent claim that the S-duality between 4d SUSY gauge theories, which is AGT related to the modular transformations of 2d conformal blocks, is no more than an ordinary Fourier transform at the perturbative level, is further traced down to the commutation relation [P,Q]=-i\hbar between the check-operator monodromies of the exponential resolvent operator in the underlying Dotsenko-Fateev matrix models and beta-ensembles. To this end, we treat the conformal blocks as eigenfunctions of the monodromy check operators, what is especially simple in the case of one-point toric block. The kernel of the modular transformation is then defined as the intertwiner of the two monodromies, and can be obtained straightforwardly, even when the eigenfunction interpretation of the blocks themselves is technically tedious. In this way, we provide an elementary derivation of the old expression for the modular kernel for the one-point toric conformal block.Comment: 15 page

Glimpses of the Third Millennium BC in the Carpathian Basin

The relative and absolute chronology of the cultural groups of the 3rd millennium BC is a particularly exciting field of prehistoric research because this period spans the assumed boundary of two major periods — the final phase of the Copper Age and the initial phase of the Early Bronze Age. The transition from one major archaeological period to the next no doubt involved major structural changes. The transition period roughly spanning the middle third of the 3rd millennium BC, i.e. the period between the close of the Late Copper Age and the Early Bronze Age 2a in Hungary, corresponds to what is known as the Late Eneolithic in Central Europe and is generally divided into three main phases: 1. the late Baden, 2. the post-Baden/Vučedol, and 3. the post-Vučedol period. The transition in the Carpathian Basin had a mosaic patterning: the rhythm of change varied from region to region and the transformation in each major region followed a different cultural trajectory (Fig. 1, Table 1). This period is here examined in the light of two recently discovered finds from Hungary (Fig. 2, Fig. 4). The finds described here offered a glimpse into the transitional period between the Copper Age and the Bronze Age, with a focus on southern Transdanubia. The more or less identical artefacts and decorative styles appearing roughly synchronously in several regions reflect the period’s cultural contacts spanning extensive territories. The mapping of these communications networks and the clarification of the period’s finer chronological details based on the growing corpus of data will no doubt remain one of the priorities of future research. | A Kr. e. 3. évezred relatív és abszolút kronológiai viszonyainak kutatása különösen izgalmas területe az őskor vizsgálatának, hiszen két nagy korszak feltételezett határán, a rézkor kései és a bronzkor kezdeti fázisainak megismerését foglalja magában. A nagy régészeti korszak definíciók azt sejtetik, hogy fontos strukturális változások állnak mögöttük. A Kr. e. 3. évezred középső harmadára tehető átmeneti időszakot, vagyis a magyarországi késő rézkor vége és kora bronzkor 2a közötti korszakot, közép-európai szemszögből tekintve a késő eneolitikum időszakát három főbb szakaszra tagoltan vizsgálhatjuk: 1. késő Baden, 2. post-Baden/ Vučedol korszak, 3. post-Vučedol korszak. A Kárpát-medence térségében egy mozaikos jellegű, fokozatos, területenként változóan eltérő ütemű és tendenciájú átalakulásnak lehetünk tanúi (Fig. 1, Table 1). Ezúttal két magyarországi lelet alapján vizsgáljuk a korszakot (Fig. 2, Fig. 4). A bemutatott leletek segítségével a rézkor és bronzkor átmeneti időszakának néhány kérdését tekintjük át, különösen a Dél-Dunántúl területére fókuszálva. A kor nagy távolságokat átfogó kapcsolatrendszerének maradandó lenyomatát őrzik azok a tárgyak/stíluslemek, melyek több régióban közel egyidőben és hasonló jellegben találhatók meg. Ezek alapján fontos feladat lesz ennek/ezeknek a kommunikációs hálózat(ok)nak a további feltérképezése és a folyamatosan gyarapodó adatok alapján kronológiai összefüggéseik tisztázása

Learnable PINs: Cross-Modal Embeddings for Person Identity

We propose and investigate an identity sensitive joint embedding of face and voice. Such an embedding enables cross-modal retrieval from voice to face and from face to voice. We make the following four contributions: first, we show that the embedding can be learnt from videos of talking faces, without requiring any identity labels, using a form of cross-modal self-supervision; second, we develop a curriculum learning schedule for hard negative mining targeted to this task, that is essential for learning to proceed successfully; third, we demonstrate and evaluate cross-modal retrieval for identities unseen and unheard during training over a number of scenarios and establish a benchmark for this novel task; finally, we show an application of using the joint embedding for automatically retrieving and labelling characters in TV dramas.Comment: To appear in ECCV 201