78 research outputs found
Warnaar's bijection and colored partition identities, I
We provide a general and unified combinatorial framework for a number of
colored partition identities, which include the five, recently proved
analytically by B. Berndt, that correspond to the exceptional modular equations
of prime degree due to H. Schroeter, R. Russell and S. Ramanujan. Our approach
generalizes that of S. Kim, who has given a bijective proof for two of these
five identities, namely the ones modulo 7 (also known as the Farkas-Kra
identity) and modulo 3. As a consequence of our method, we determine bijective
proofs also for the two highly nontrivial identities modulo 5 and 11, thus
leaving open combinatorially only the one modulo 23.Comment: Contains the first portion of the first author's MIT senior thesis
(2011). Some minor revisions with respect to the previous version. To appear
in JCT
Ir_urfs_vf: Image Recommendation with User Relevance Feedback Session and Visual Features in Vertical Image Search
In recent years, online shopping has grown exponentially and huge number of images are available online. Hence, it is necessary to recommend various product images to aid the user in effortless and efficient access to the desired products. In this paper, we present image recommendation framework with user relevance feedback session and visual features (IR_URFS_VF) to extract relevant images based on user inputs. User feedback is retrieved from image search history with clicked and un-clicked images. Image features are computed off-line and later used to find relevance between images. The relevance between images is determined by cosine similarity and are ranked based on clicked frequency and similarity score between images. Experiments results show that IR_URFS_VF outperforms CBIR method by providing more relevant ranked images to the user input query
Color Partition Identities Arising from Ramanujan's Theta-Functions
We establish several partition identities with distinct colors that arise from Ramanujan’s theta-function identities and formulas for multipliers in the theory of modular equations. Also, we deduce few partition congruences as a corollary of some partition identities
A note on S-duality for the N=1* Sp(2n) and SO(2n+1) super-Yang-Mills theories
We study the N=1* supersymmetric gauge theories with gauge groups Sp(2n) and
SO(2n+1). These theories are obtained from the corresponding N=4 supersymmetric
Yang-Mills theories via a mass deformation. We show that the number of quantum
vacua in the Sp(2n) theory is equal to the number of quantum vacua in the
SO(2n+1) theory. This constitutes non-trivial support for S-duality between
these theories. The verification of the equality of the number of quantum vacua
involves a rather esoteric identity due to Ramanujan.Comment: 9 pages. v2:clarifying footnote adde
Ising Model Observables and Non-Backtracking Walks
This paper presents an alternative proof of the connection between the
partition function of the Ising model on a finite graph and the set of
non-backtracking walks on . The techniques used also give formulas for
spin-spin correlation functions in terms of non-backtracking walks. The main
tools used are Viennot's theory of heaps of pieces and turning numbers on
surfaces.Comment: 33 pages, 11 figures. Typos and errors corrected, exposition
improved, results unchange
DLCQ Strings, Twist Fields and One-Loop Correlators on a Permutation Orbifold
We investigate some aspects of the relationship between matrix string theory
and light-cone string field theory by analysing the correspondence between the
two-loop thermal partition function of DLCQ strings in flat space and the
integrated two-point correlator of twist fields in a symmetric product orbifold
conformal field theory at one-loop order. This is carried out by deriving
combinatorial expressions for generic twist field correlation functions in
permutation orbifolds using the covering surface method, by deriving the
one-loop modification of the twist field interaction vertex, and by relating
the two-loop finite temperature DLCQ string theory to the theory of Prym
varieties for genus two covers of an elliptic curve. The case of bosonic Z(2)
orbifolds is worked out explicitly and precise agreement between both
amplitudes is found. We use these techniques to derive explicit expressions for
Z(2) orbifold spin twist field correlation functions in the Type II and
heterotic string theories.Comment: 48 pages, 1 figure; v2: typos correcte
Warnaar’s bijection and colored partition identities, II
In our previous paper (J. Comb. Theory Ser. A 120(1):28–38, 2013), we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the Schröter, Russell, and Ramanujan type. The goal of this paper is to use the master bijection of Sandon and Zanello (J. Comb. Theory Ser. A 120(1):28–38, 2013) to show combinatorially several new and highly nontrivial colored partition identities. We conclude by listing a number of further interesting identities of the same type as conjectures.Massachusetts Institute of Technology. Dept. of Mathematics
- …