On one extremal property of Korovkin's means

We point out that$$\inf\limits_{L \in L_n} \sup\limits_{\substack{f \in C_{2\pi}\\f \ne const}} \frac{\max \| f(x) - L(f, x) \|}{\omega^*_2(f, \pi/n + 1)} = \frac{1}{2}$$where $C_{2\pi}$ is the space of periodic continuous functions on real domain, $L_n$ is the set of linear operators that map $C_{2\pi}$ to the set of trigonometric polynomials of order no greater than $n$ ($n = 0,1,\ldots$), $\omega_2(f, t) = \sup\limits_{x, |h| \leqslant t} |f(x-h) - 2f(x) + f(x+h)|$, $\omega^*_2(f, t)$ is the concave hull of the function $\omega_2(f, t)$. In this equality, the infimum is attained for Korovkin's means.

Traffic Sign Detection and Classification based on Combination of MSER Features and Multi-language OCR

Road signs are so important because they help preserve safe driving conditions; they also influence the safety of drivers and pedestrians. Without these signs, no one would know the driving speed limit, on which direction to drive down a road, any upcoming hazard, or whether they are approaching a merge. It would be chaotic to drive in such situations. Moreover, these signs help new drivers to find their way in the absence of navigators. Therefore, traffic sign recognition takes a critical place in computer vision applications to develop an effective algorithm. In order to tackle this challenge, we proposed the use of Multi-language Traffic Sign Detection and Classification. One of our contributions in this work is that, instead of using the standard grayscale image, we used the RGB colored image. This image is converted into the 2D highest-level grayscale image using the largest values of each pixel in the RGB channels. The novel generated image has the strongest features of the RGB image that make the features distinct and more informative in the classification step. Consider that, in general, the traffic sign has two colors only, the foreground (text location) and background (non-text location). The Maximally Stable Extremal Regions (MSER) used to extract features from the 2D image where the locations of interest are well-identified exclusively by an extremal property of the intensity function in the location and on its outer boundary. The geometrical properties and thinning operations were used to remove the non-text locations. A multi-language OCR was used to understand multi-language. This proposed method has been tested using 240 images which were collected from the Internet and two datasets. The experimental results demonstrated the performance of the proposed method where the traffic sign detected in 92% of the tested images with a very high percentage of localization.

Spectrally Extremal Vertices, Strong Cospectrality, and State Transfer

In order to obtain perfect state transfer between two sites in a network of interacting qubits, their corresponding vertices in the underlying graph must satisfy a property called strong cospectrality. Here we determine the structure of graphs containing pairs of vertices which are strongly cospectral and satisfy a certain extremal property related to the spectrum of the graph. If the graph satisfies this property globally and is regular, we also show that the existence of a partition of the vertex set into pairs of vertices at maximum distance admitting perfect state transfer forces the graph to be distance-regular. Finally, we present some new examples of perfect state transfer in simple graphs constructed with our technology. In particular, for odd distances, we improve the known trade-off between the distance perfect state transfer occurs in simple graphs and the size of the graph.