Abstract:
We consider a variant of two-point Euclidean shortest path query problem: given a polygonal domain, build a data structure for two-point shortest path query, provided that query points always lie on the boundary of the domain. As a main result, we show that a logarithmic-time query for shortest paths between boundary points can be performed using O~ (n^5) preprocessing time and O(n^5) space where n is the number of corners of the polygonal domain and the O~ notation suppresses the polylogarithmic factor. This is realized by observing a connection between Davenport-Schinzel sequences and our problem in the parameterized space. We also provide a tradeoff between space and query time; a sublinear time query is possible using O(n^{3+epsilon}) space. Our approach also extends to the case where query points should lie on a given set of line segments.

Abstract:
This paper studies the geodesic diameter of polygonal domains having h holes and n corners. For simple polygons (i.e., h = 0), the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time, as known by Hershberger and Suri. For general polygonal domains with h >= 1, however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time O(n^7.73) or O(n^7 (log n + h)). The main difficulty unlike the simple polygon case relies on the following observation revealed in this paper: two interior points can determine the geodesic diameter and in that case there exist at least five distinct shortest paths between the two.

Abstract:
We present an algorithm that computes the geodesic center of a given polygonal domain. The running time of our algorithm is $O(n^{12+\epsilon})$ for any $\epsilon>0$, where $n$ is the number of corners of the input polygonal domain. Prior to our work, only the very special case where a simple polygon is given as input has been intensively studied in the 1980s, and an $O(n \log n)$-time algorithm is known by Pollack et al. Our algorithm is the first one that can handle general polygonal domains having one or more polygonal holes.

Abstract:
In this paper, we show that the $L_1$ geodesic diameter and center of a simple polygon can be computed in linear time. For the purpose, we focus on revealing basic geometric properties of the $L_1$ geodesic balls, that is, the metric balls with respect to the $L_1$ geodesic distance. More specifically, in this paper we show that any family of $L_1$ geodesic balls in any simple polygon has Helly number two, and the $L_1$ geodesic center consists of midpoints of shortest paths between diametral pairs. These properties are crucial for our linear-time algorithms, and do not hold for the Euclidean case.

Abstract:
We establish tight bounds for beacon-based coverage problems, and improve the bounds for beacon-based routing problems in simple rectilinear polygons. Specifically, we show that $\lfloor \frac{n}{6} \rfloor$ beacons are always sufficient and sometimes necessary to cover a simple rectilinear polygon $P$ with $n$ vertices. We also prove tight bounds for the case where $P$ is monotone, and we present an optimal linear-time algorithm that computes the beacon-based kernel of $P$. For the routing problem, we show that $\lfloor \frac{3n-4}{8} \rfloor - 1$ beacons are always sufficient, and $\lceil \frac{n}{4}\rceil-1$ beacons are sometimes necessary to route between all pairs of points in $P$.

Abstract:
The Ulam-Hyers stability problems of the following quadratic equation r 2 f x + y r + r 2 f x - y r = 2 f ( x ) + 2 f ( y ) , where r is a nonzero rational number, shall be treated. The case r = 2 was introduced by J. M. Rassias in 1999. Furthermore, we prove the stability of the quadratic equation by using the fixed point method. 2010 Mathematics Subject Classification: 39B22; 39B52; 39B72.

Abstract:
Use of non-biological agents for mRNA delivery into living systems in order to induce heterologous expression of functional proteins may provide more advantages than the use of DNA and/or biological vectors for delivery. However, the low efficiency of mRNA delivery into live animals, using non-biological systems, has hampered the use of mRNA as a therapeutic molecule. Here, we show that gold nanoparticle-DNA oligonucleotide (AuNP-DNA) conjugates can serve as universal vehicles for more efficient delivery of mRNA into human cells, as well as into xenograft tumors generated in mice. Injections of BAX mRNA loaded on AuNP-DNA conjugates into xenograft tumors resulted in highly efficient mRNA delivery. The delivered mRNA directed the efficient production of biologically functional BAX protein, a pro-apoptotic factor, consequently inhibiting tumor growth. These results demonstrate that mRNA delivery by AuNP-DNA conjugates can serve as a new platform for the development of safe and efficient gene therapy.

Abstract:
The endoplasmic reticulum (ER) is an organelle that functions to synthesize, fold, and transport proteins. ER stress is a key link between type 2 diabetes (T2D), obesity, and insulin resistance. In this study, we investigated the effect of WHW on the ER stress response and the insulin signaling pathway in 3T3-L1 preadipocytes. 3T3-L1 preadipocytes were differentiated into adipocytes, and ER stress was then induced by treatment with tunicamycin. ER stress-induced adipocytes were treated with different concentrations of WHW for 24？h. The expression of ER stress-related molecules such as X-box-binding protein-1 (XBP-1), glucose-regulated protein 78 (GRP78), C/EBP-homologous protein 10 (CHOP10), and eukaryotic initiation factor 2α (eIF2α) and signaling molecules such as phosphatidylinositol 3-kinase (PI3K), insulin receptor substrates-1 (IRS-1), and c-Jun N-terminal protein kinase (JNK) were investigated. WHW significantly inhibited the expression of XBP-1, GRP78, CHOP10, and eIF2α in ER stress-induced 3T3-L1 adipocytes. WHW also increased the PI3K expression and the IRS-1 phosphorylation but decreased the phosphorylation of JNK in ER stress-induced 3T3-L1 adipocytes. Our results indicate that WHW inhibits ER stress in adipocytes by suppressing the expression of ER stress-mediated molecules and the insulin signaling pathway, suggesting that WHW may be an attractive therapeutic agent for managing T2D. 1. Introduction Obesity is the leading risk factor for the development of many life-threatening diseases, particularly insulin resistance and type 2 diabetes (T2D) [1]. Although the mechanisms by which obesity contributes to insulin resistance and T2D remain the subject of intensive investigation, recent studies suggest that endoplasmic reticulum (ER) stress plays a major role in mediating obesity-induced insulin resistance and T2D [2]. The ER is a critical intracellular organelle that coordinates the synthesis, folding, and transport of proteins [3, 4]. A variety of biochemical or pathophysiological stimuli can interrupt the protein folding process in the ER by disrupting protein glycosylation, disulfide bond formation, or the ER calcium pool. These disruptions can cause the accumulation of unfolded or misfolded proteins in the ER lumen, a condition termed as ER stress [5]. The presence of unfolded proteins in the ER is sensed, and activation of the unfolded protein response (UPR) is regulated by ER chaperones and folding enzymes, such as glucose-regulated protein 78 (GRP78), X-box-binding protein-1 (XBP-1), C/EBP-homologous protein 10 (CHOP10), and eukaryotic

Abstract:
The aperture angle alpha(x, Q) of a point x not in Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the plane with respect to an inscribed convex polygon Q of C is the minimum aperture angle of any x in C Q with respect to Q. We show that for any compact convex set C in the plane and any k > 2, there is an inscribed convex k-gon Q of C with aperture angle approximation error (1 - 2/(k+1)) pi. This bound is optimal, and settles a conjecture by Fekete from the early 1990s. The same proof technique can be used to prove a conjecture by Brass: If a polygon P admits no approximation by a sub-k-gon (the convex hull of k vertices of P) with Hausdorff distance sigma, but all subpolygons of P (the convex hull of some vertices of P) admit such an approximation, then P is a (k+1)-gon. This implies the following result: For any k > 2 and any convex polygon P of perimeter at most 1 there is a sub-k-gon Q of P such that the Hausdorff-distance of P and Q is at most 1/(k+1) * sin(pi/(k+1)).

Abstract:
For a polygonal domain with $h$ holes and a total of $n$ vertices, we present algorithms that compute the $L_1$ geodesic diameter in $O(n^2+h^4)$ time and the $L_1$ geodesic center in $O((n^4+n^2 h^4)\alpha(n))$ time, where $\alpha(\cdot)$ denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in $O(n^{7.73})$ or $O(n^7(h+\log n))$ time, and compute the geodesic center in $O(n^{12+\epsilon})$ time. Therefore, our algorithms are much faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on $L_1$ shortest paths in polygonal domains.