Abstract:
The notion of Image partition regularity near zero was first introduced by De and Hindman. It was shown there that like image partition regularity over $\mathbb{N}$ the main source of infinite image partition regular matrices near zero are Milliken- Taylor matrices. But Milliken- Taylor matrices are far apart to have images in central sets. In this regard the notion of centrally image partition regularity was introduced. In the present paper we propose the notion centrally partition regular matrices near zero for dense sub semigroup of $(\ber^+,+)$ which are different from centrally partition regular matrices unlike finite cases.

Abstract:
Fully idempotent near-rings are defined and characterized which yields information on the lattice of ideals of fully idempotent rings and near-rings. The space of prime ideals is topologized and a sheaf representation is given for a class of fully idempotent near-rings which includes strongly regular near-rings.

Abstract:
We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where "one piece is big" is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer's proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.

Abstract:
Denote by $\mathcal T_n$ and $\mathcal S_n$ the full transformation semigroup and the symmetric group on the set $\{1,\ldots,n\}$, and $\mathcal E_n=\{1\}\cup(\mathcal T_n\setminus \mathcal S_n)$. Let $\mathcal T(X,\mathcal P)$ denote the set of all transformations of the finite set $X$ preserving a uniform partition $\mathcal P$ of $X$ into $m$ subsets of size $n$, where $m,n\geq2$. We enumerate the idempotents of $\mathcal T(X,\mathcal P)$, and describe the subsemigroup $S=\langle E\rangle$ generated by the idempotents $E=E(\mathcal T(X,\mathcal P))$. We show that $S=S_1\cup S_2$, where $S_1$ is a direct product of $m$ copies of $\mathcal E_n$, and $S_2$ is a wreath product of $\mathcal T_n$ with $\mathcal T_m\setminus \mathcal S_m$. We calculate the rank and idempotent rank of $S$, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of $\mathcal E_n$.

Abstract:
We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated by the idempotents of the semigroup $T(X,P)$, which consists of all transformations of the finite set $X$ preserving a non-uniform partition $P$. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.

Abstract:
We prove that a certain matrix, which is not image partition regular over R near zero, is image partition regular over N. This answers a question of De and Hindman.

Abstract:
Near-duplicate image detection is a necessary operation to refine image search results for efficient user exploration. The existences of large amounts of near duplicates require fast and accurate automatic near-duplicate detection methods. We have designed a coarse-to-fine near duplicate detection framework to speed-up the process and a multi-modal integra-tion scheme for accurate detection. The duplicate pairs are detected with both global feature (partition based color his-togram) and local feature (CPAM and SIFT Bag-of-Word model). The experiment results on large scale data set proved the effectiveness of the proposed design.

Abstract:
A practical image stitching method, named band-type optimum partition method or BOP method in short, is introduced in this study. By the use of multiple cut lines and band-type borders, this method achieves 2 major advantages. First, it reduces the discontinuities appearing on the boundaries of the overlap region. Second, it eliminates the undesired ghost image effect, which is caused by misalignment of the input images. Besides, the BOP method is implemented by evaluating the sum of similarity and smoothness costs on a group of pixels instead of a single pixel. Finally, experimental results are presented to demonstrate the success of the BOP method.

Abstract:
We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if $\pi:A\to M_n(\mathbb C)$ is a finite dimensional representation of a Hopf $C^*$-algebra, we prove that the idempotent state associated to its Hopf image $A'$ must be the convolution Ces\`aro limit of the linear functional $\phi=tr\circ\pi$. We discuss then some consequences of this result, notably to inner linearity questions.

Abstract:
A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed.