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Ewing's sarcomas/primitive neuroectodermal tumors (PNET) are a group of tumors characterized by small round cells that rarely affect the bone and soft tissues of the nasal cavities. Herein, we describe the case of a 38-year-old man who presented with a one-year history of epistaxis, headache, hyposmia, anosmia, night sweats, weight loss, hearing loss, dysphagia, and nasal blockage. A computed tomography (CT) scan and magnetic resonance imaging (MRI) studies revealed a lesion in the posterior nasal cavities with extension into the sphenoid and ethmoid sinuses. Histological examination of a biopsied specimen obtained via flexible fiberoptic endoscopy showed small dark round cells with scanty cytoplasm arranged in solid nests and pseudorosette formations. The diagnosis of Ewing's sarcoma was confirmed by the positive reactive pattern of the neoplastic cells for CD99 by immunohistochemistry studies and also identification of a chromosomal translocation of FLI1 in PCR studies. Cervical lymph nodes and bone marrow involvement were also observed. The patient was treated by adjuvant chemotherapy including vincristine, endoxan, and adriamycin for 6 cycles followed by radiotherapy (eleven courses). Unfortunately, before completion of treatment, the patient died due to disease progression and brain metastasis.

Background: Glioblastoma multiforme with a primitive neuronal component is a rare entity, with few cases reported in the literature. Case Description: A patient who had a supratentorial glioblastoma multiforme with a primitive neuronal component developed spinal metastasis during the disease course. With his history of leukemia during childhood, he was likely exposed to therapeutic ionizing brain radiation, which could have increased the risk of developing brain cancer in adulthood. Conclusions: The range of incidence rates of dissemination in the literature is 2%–4%, typically in cases of cerebellar glioblastoma multiforme, but as high as 25% in autopsy series. Our case highlights several other topics in the literature, such as immunohistochemical patterns that differ between the primary tumor and spinal metastases and dissemination locations, typically leptomeningeal or ventricular invasion. Key words: Component, Glioblastoma, Metastases, Neuronal, Primitive, Spinal

A nonnegative square matrix A is primitive if some power A k > 0 (that is, A k is entrywise positive). The least such k is called the exponent of A . In [2], Akelbek and Kirkland defined the scrambling index of a primitive matrix A , which is the smallest positive integer k such that any two rows of A k have at least one positive element in a coincident position. In this paper, we give a relation between the scrambling index and the exponent for symmetric primitive matrices, and determine the scrambling index set for the class of symmetric primitive matrices. We also characterize completely the symmetric primitive matrices in this class such that the scrambling index is equal to the maximum value.

In [B. Cheng, B. Liu, The base sets of primitive zero-symmetric sign pattern matrices, Linear Algebra Appl. 428 (2008) 715–731], Cheng and Liu studied the bases of primitive zero-symmetric sign pattern matrices. The sharp upper bound of the bases was obtained. In this paper, we characterize the sign pattern matrices with the sharp bound.

In 1994, Z. Li, F. Hall and C. Eschenbach extended the concept of the index of convergence from nonnegative matrices to powerful sign pattern matrices. Recently, Jiayu Shao and Lihua You studied the bases of non-powerful irreducible sign pattern matrices. In this paper, the local bases, which are generalizations of the base, of primitive non-powerful signed digraphs are introduced, and sharp bounds for local bases of primitive non-powerful signed digraphs are obtained. Furthermore, extremal digraphs are described.

A two-colored digraph is a digraph whose arcs are colored red or blue. A two-colored digraph is primitive provided that there exist nonnegative integers h and k with h + k > 0 such that for each pair ( i , j ) of vertices there is an ( h , k ) -walk from i to j in D. The exponent of D is the minimum value of h + k taken over all such h and k. In this paper, we consider a class of special primitive two-colored digraphs whose uncolored digraphs have n + s vertices and consist of one n-cycle and one ( n - t ) -cycle for t ⩾ 1 . We give bounds on the exponents and characterize the extreme two-colored graphs, which generalizes the results in [Y. Gao, Y. Shao, Exponents of two-colored digraphs with two cycles, Linear Algebra Appl. 407 (2005) 263–276; Y. Gao, Y. Shao, Exponents of a class two-colored digraphs, Linear and Multilinear Algebra 53(3) (2005) 175–188].

In [J.Y. Shao, L.H. You, Bound on the base of irreducible generalized sign pattern matrices, Discrete Math., in press], Shao and You extended the concept of the base from powerful sign pattern matrices to non-powerful (and generalized) sign pattern matrices. In this paper, we study the bases of primitive zero-symmetric sign pattern (and generalized sign pattern) matrices. Sharp upper bounds of the bases are obtained. We also show that there exist no “gaps” in the base sets of the classes of such matrices.

Two kinds of optimal convergence orders in L 1 -norm to a self-similar solution are proved or conjectured for various evolutionary problems so far. The first convergence order is of the magnitude of the similarity solution itself and the second one is of order 1 / t . Employing a potential comparison technique to scalar conservation laws we may easily see that these asymptotic convergence orders are related to space and time translation of potentials. We present the technique clearly in the simple setting of scalar conservation laws in one space dimension.

Let b = b ( A ) be the Boolean rank of an n × n primitive Boolean matrix A and exp( A ) be the exponent of A . Then exp( A ) ⩽ ( b − 1) 2 + 2, and the matrices for which equality occurs have been determined in [D.A. Gregory, S.J. Kirkland, N.J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra Appl. 217 (1995) 101–116]. In this paper, we show that for each 3 ⩽ b ⩽ n − 1, there are n × n primitive Boolean matrices A with b ( A ) = b such that exp( A ) = ( b − 1) 2 + 1, and we explicitly describe all such matrices.

In [J.Y. Shao, L.H. You, Bound on the base of irreducible generalized sign pattern matrices, Discrete Math., in press], Shao and You extended the concept of the base from powerful sign pattern matrices to non-powerful (and generalized) sign pattern matrices. In this paper, we study the base for primitive non-powerful nearly reducible sign pattern (and generalized sign pattern) matrices. We obtain sharp upper bounds, together with complete characterization of the equality cases of the base for primitive nearly reducible sign pattern (and generalized sign pattern) matrices. We also show that there exist “gaps” in the base set of the classes of such matrices.

The Fibonacci language F u , v is the set of all Fibonacci words, where the first word and the second word in the Fibonacci sequence are u and v, respectively. We show that the language F u , v is context-free free. We also show that F u , v is not dense if the word uv contains at least two distinct letters. Let w i denote the ith Fibonacci word. When considering the Fibonacci language F a , b for two distinct letters a and b, we show that for k ⩾2 and 1⩽ i k , the word w i w i is not a prefix of the Fibonacci word w k . We also show that for k ⩾2, the language F k ={w nk | n⩾1} is a code.

In this paper we analyze properties of transition matrices T∈ R n,n of regular Markov chains whose exponent (index of primitivity) is bounded below by ⌊((n−1) 2 +1)/2⌋+2 . Our investigation leads to a proof that for such a T , the corresponding condition number K (T):= max 1⩽i,j⩽n |Q # i,j | , where Q=I−T . We go on to show that if E∈ R n,n is a perturbation of T such that T+E is also a transition matrix for a regular Markov chain and π and π are the stationary distribution vectors of T and T+E , respectively, then ∥π− π ∥ ∞ ∞ and ∥π− π ∥ 1 1 . Many of the techniques used to establish these results are combinatorial in nature.

A word v is said to be a proper d-factor of a word u if v ≠ u and u = vx = yv for some words x,y. The family of words which have i distinct proper d-factors is denoted by D(i). According to the number of distinct proper d-factors of words, the free semigroup X+ generated by X can be expressed as the disjoint union of D(i)'s. Words in D(1) are called d-minimal words. d-Minimal words are often called non-overlapping words, dipolar words or unborded words. In this paper, we study the relationship between Di(1) and D(i) concerning the basic properties of decompositions and catenations of words. We give characterizations of words in D2(1)∩D(1) and D(2). We also show that sets Di(1)⧹D(j) and D(j)⧹Di(1) are disjunctive. It is known that every disjunctive language is dense but not regular. We obtain the results that X+D(1) and X+D(2) are regular but X+D(i) is disjunctive for every i ⩾ 4. Served as an example of disjunctive d-minimal context-free languages, a disjunctive d-minimal context-free language is constructed. Moreover, we show that the well-known Dyck language is a free semigroup generated by a d-minimal bifix code. The languages of which the catenations consist of d-minimal words are studied in this paper too. That is, some properties of d-minimality-annihilators of languages are investigated.