Abstract:
We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumar's work on the equivariant cohomology of the affine flag variety. We show that the cap operators for Pieri elements are the same as Pieri operators defined by Berg, Saliola and Serrano. This establishes the affine Pieri rule.

Abstract:
We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutative algebra generated by these operators is isomorphic to the cohomology of the affine flag variety. We show that the cohomology of the affine flag variety is product of the cohomology of an affine Grassmannian and a flag variety, which are generated by MN elements and Dunkl elements respectively. The Schubert classes in cohomology of the affine Grassmannian (resp. the flag variety) can be identified with affine Schur functions (resp. Schubert polynomials) in a quotient of the polynomial ring. Affine Schubert polynomials, polynomial representatives of the Schubert class in the cohomology of the affine flag variety, can be defined in the product of two quotient rings using the Bernstein-Gelfand-Gelfand operators interpreted as divided difference operators acting on the affine Fomin-Kirillov algebra. As for other applications, we obtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. We also define $k$-strong-ribbon tableaux from Murnaghan-Nakayama elements to provide a new formula of $k$-Schur functions. This formula gives the character table of the representation of the symmetric group whose Frobenius characteristic image is the $k$-Schur function.

Abstract:
A centrally symmetric analogue of the cyclic polytope, the bicyclic polytope, was defined in [BN08]. The bicyclic polytope is defined by the convex hull of finitely many points on the symmetric moment curve where the set of points has a symmetry about the origin. In this paper, we study the Barvinok-Novik orbitope, the convex hull of the symmetric moment curve. It was proven in [BN08] that the orbitope is locally $k$-neighborly, that is, the convex hull of any set of $k$ distinct points on an arc of length not exceeding $\phi_k$ in $\mathbb{S}^1$ is a $(k-1)$-dimensional face of the orbitope for some positive constant $\phi_k$. We prove that we can choose $\phi_k $ bigger than $\gamma k^{-3/2} $ for some positive constant $\gamma$.

Abstract:
We present explicit constructions of centrally symmetric 2-neighborly d-dimensional polytopes with about 3^{d/2} = (1.73)^d vertices and of centrally symmetric k-neighborly d-polytopes with about 2^{c_k d} vertices where c_k=3/20 k^2 2^k. Using this result, we construct for a fixed k > 1 and arbitrarily large d and N, a centrally symmetric d-polytope with N vertices that has at least (1-k^2 (gamma_k)^d) binom(N, k) faces of dimension k-1, where gamma_2=1/\sqrt{3} = 0.58 and gamma_k = 2^{-3/{20k^2 2^k}} for k > 2. Another application is a construction of a set of 3^{d/2 -1}-1 points in R^d every two of which are strictly antipodal as well as a construction of an n-point set (for an arbitrarily large n) in R^d with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.

Abstract:
We present explicit constructions of centrally symmetric polytopes with many faces: first, we construct a d-dimensional centrally symmetric polytope P with about (1.316)^d vertices such that every pair of non-antipodal vertices of P spans an edge of P, second, for an integer k>1, we construct a d-dimensional centrally symmetric polytope P of an arbitrarily high dimension d and with an arbitrarily large number N of vertices such that for some 0 < delta_k < 1 at least (1-delta_k^d) {N choose k} k-subsets of the set of vertices span faces of P, and third, for an integer k>1 and a>0, we construct a centrally symmetric polytope Q with an arbitrary large number N of vertices and of dimension d=k^{1+o(1)} such that least (1 - k^{-a}){N choose k} k-subsets of the set of vertices span faces of Q.

Abstract:
We consider the convex hull B_k of the symmetric moment curve U(t)=(cos t, sin t, cos 3t, sin 3t, ..., cos (2k-1)t, sin (2k-1)t) in R^{2k}, where t ranges over the unit circle S= R/2pi Z. The curve U(t) is locally neighborly: as long as t_1, ..., t_k lie in an open arc of S of a certain length phi_k>0, the convex hull of the points U(t_1), ..., U(t_k) is a face of B_k. We characterize the maximum possible length phi_k, proving, in particular, that phi_k > pi/2 for all k and that the limit of phi_k is pi/2 as k grows. This allows us to construct centrally symmetric polytopes with a record number of faces.

Abstract:
Selective loss of neurons, abnormal protein deposition and neuroinflammation are the common pathological features of neurodegenerative diseases, and these features are closely related to one another. In Parkinson's disease, abnormal aggregation and deposition of α-synuclein is known as a critical event in pathogenesis of the disease, as well as in other related neurodegenerative disorders, such as dementia with Lewy bodies and multiple system atrophy. Increasing evidence suggests that α-synuclein aggregates can activate glial cells to induce neuroinflammation. However, how an inflammatory microenvironment is established and maintained by this protein remains unknown. Findings from our recent study suggest that neuronal α-synuclein can be directly transferred to astrocytes through sequential exocytosis and endocytosis and induce inflammatory responses from astrocytes. Here we discuss potential roles of astrocytes in a cascade of events leading to α-synuclein-induced neuroinflammation.

Abstract:
Selective loss of neurons, abnormal protein deposition and neuroinflammation are the common pathological features of neurodegenerative diseases, and these features are closely related to one another. In Parkinson's disease, abnormal aggregation and deposition of α-synuclein is known as a critical event in pathogenesis of the disease, as well as in other related neurodegenerative disorders, such as dementia with Lewy bodies and multiple system atrophy. Increasing evidence suggests that α-synuclein aggregates can activate glial cells to induce neuroinflammation. However, how an inflammatory microenvironment is established and maintained by this protein remains unknown. Findings from our recent study suggest that neuronal α-synuclein can be directly transferred to astrocytes through sequential exocytosis and endocytosis and induce inflammatory responses from astrocytes. Here we discuss potential roles of astrocytes in a cascade of events leading to α-synuclein-induced neuroinflammation.

Abstract:
We report here a case of an inflammatory myofibroblastic tumor in the retroperitoneum, which mimicked a germ cell tumor of the undescended testis. A 75-year-old healthy man presented with a palpable abdominal mass. On the computed tomography image, there was large, well-defined soft tissue mass in the left side of the retroperitoneum, and there was no visible left testis or seminal vesicle. After contrast enhancement, the mass appeared to be relatively homogeneous, considering its large size. With ultrasonography, it appeared as a well-defined, hypoechoic mass with intratumoral vascularity. This solid mass was surgically diagnosed as an inflammatory myofibroblastic tumor.

Abstract:
Background In sparse-view CT imaging, strong streak artifacts may appear around bony structures and they often compromise the image readability. Compressed sensing (CS) or total variation (TV) minimization-based image reconstruction method has reduced the streak artifacts to a great extent, but, sparse-view CT imaging still suffers from residual streak artifacts. We introduce a new bone-induced streak artifact reduction method in the CS-based image reconstruction. Methods We firstly identify the high-intensity bony regions from the image reconstructed by the filtered backprojection (FBP) method, and we calculate the sinogram stemming from the bony regions only. Then, we subtract the calculated sinogram, which stands for the bony regions, from the measured sinogram before performing the CS-based image reconstruction. The image reconstructed from the subtracted sinogram will stand for the soft tissues with little streak artifacts on it. To restore the original image intensity in the bony regions, we add the bony region image, which has been identified from the FBP image, to the soft tissue image to form a combined image. Then, we perform the CS-based image reconstruction again on the measured sinogram using the combined image as the initial condition of the iteration. For experimental validation of the proposed method, we take images of a contrast phantom and a rat using a micro-CT and we evaluate the reconstructed images based on two figures of merit, relative mean square error and total variation caused by the streak artifacts. Results The images reconstructed by the proposed method have been found to have smaller streak artifacts than the ones reconstructed by the original CS-based method when visually inspected. The quantitative image evaluation studies have also shown that the proposed method outperforms the conventional CS-based method. Conclusions The proposed method can effectively suppress streak artifacts stemming from bony structures in sparse-view CT imaging.