Uniaxial compressive and Brazilian tensile strengths (UCS and BTS) of rocks are considered as the most important properties in the design of most geotechnical projects, such as slope stability and underground excavation, which interact with rocks. Measuring UCS and BTS using standard laboratory tests is time consuming, tedious and expensive. Moreover, it requires a large number of well-prepared rock cores that is not often viable, particularly in soft or highly jointed rock masses. For these reasons, indirect tests such as Schmidt hammer hardness (SH) can be used for prediction of UCS and BTS of rocks. There is a wide variation in the recommended SH test procedures by institutions and researchers. The objective of this study is to evaluate the performance of SH testing procedures for prediction of UCS and BTS of rocks. Accordingly, 22 sandstone samples from Qum Province, Central Iran, were selected and their UCS, BTS and SH values were determined. Using data analysis, the correlation equations have been developed between UCS and BTS with SH. A t– test was performed to check the validity of the correlation equations. The results showed that the SH test procedures that are based on continuous impacts at a point reveal lower values than those based on single impacts. Further, it was found that SH test procedures have different performance for prediction of UCS and BTS of rocks.

Evaluating the cutting rate (CR) of stones is important in the cost estimation and the planning of the stone processing plants. This research used regression models to estimate the stones’ CR based on their physico-mechanical characteristics. Stone processing factories in Mahallat City (Markazi province, Iran) were visited, and the CR of diamond circular saws was recorded on six different travertine stones. Next, the stone block samples were collected from the quarries for laboratory tests. Stones’ porosity (n), uniaxial compressive strength (UCS), and Schmidt hammer hardness (SH) were determined in the laboratory as their physico-mechanical characteristics. Correlation relationships of CR with physico-mechanical characteristics were evaluated using simple and multiple regression analyses, and estimator models were developed. Results showed that multiple regression models are more reliable than simple regression for estimating the stones’ CR. The validity of the developed multiple regression models was verified with the published data of one researcher. The findings indicated that these models are accurate enough for estimating the CR of stones. Consequently, the multiple regression models provide practical advantages for estimating the CR and save time and cost during the planning and design of the stone processing factories.

The Schmidt hammer provides a quick and inexpensive measure of surface hardness that is widely used for estimating the mechanical properties of rock material such as uniaxial compressive strength and Young’ s modulus. On the other hand, Schmidt rebound hardness can be used for a variety of specific applications. In the mining industries, it is used to determine the quality of rock, which is common practice when constructing rock structures such as those found in long wall mining, room and pillar mining, open-pit mining, gate roadways, tunnels, dams, etc. However, a number of issues such as specimen dimensions, water content, hammer type, surface roughness, weathering, testing, data reduction and analysis procedures continue to influence the consistency and reliability of the Schmidt hammer test results. This paper presents: a) a critical review of these basic issues and b) avaluate the effect of temperature, moisture and uniaxial compressive stress on Schmidt hammer hardness. It was found that water content has a significant effect on the Schmidt rebound hardness (SRH) of rocks. So that increase of water content substantially reduced the SRH of samples. Temperature also had a considerable influence on the SRH. However, relationship between SRH decreases with increasing temperature for tested samples were linear. Also tests results showed that uniaxial loading of samples increases the SRH values.

In this paper we introduce and study Besselian g-frames. We show that the kernel of associated synthesis operator for a Besselian g-frame is finite dimensional. We also introduce a-dual of a g-frame and we get some results when we use the Hilbert-Schmidt norm for the members of a g-frame in a finite dimensional Hilbert space.

Brittleness is one of the most important mechanical properties of rock that is widely used in geotechnical designs such as drilling operations and underground structures. So far، standard method for estimating the brittleness of rocks have not been introduced and therefore، indirect methods are used to estimate them. In this research، brittleness of granitic rocks are estimated from Schmidt hardness and petrographic characteristics (minerals mean size and mineralogical composition) using simple and multivariate statistical relationships. To achieve these goals، 10 different granite rocks were collected from quarries Tavandasht and Gosheh around Borujerd (Lorestan province). Based on laboratory tests، brittleness index، Schmidt hammer hardness، minerals mean size and quartz to feldspar ratio were determined. Then، different models were fitted to the data for estimating the brittleness. The results of simple statistical relationships indicate that there are moderate correlations (determination coefficients ranging from 0. 70 to 0. 87) between the brittleness index with Schmidt hammer hardness، minerals mean size and quartz to feldspar ratio. While the multivariate statistical relationships indicate a significant effect of the minerals mean size and quartz to feldspar ratio on the accuracy of correlation between the brittleness index with Schmidt hammer hardness s (with determination coefficients of 0. 91 and 0. 93، respectively).

In this paper, we give some sufficient conditions under which perturbations preserve Hilbert-Schmidt frames. Also show that the canonical dual of a perturbed Hilbert-Schmidt frame is a perturbation of the canonical dual (alternative dual respectively) of the original Hilbert-Schmidt frame and dis-cuss best approximation in the set of all dual Hilbert-Schmidt frames. Next, we apply the woven principle to Hilbert-Schmidt frames and study the stability of weaving Hilbert-Schmidt frames under perturbations. Finally, we present sufficient conditions under which perturbations preserve weaving Hilbert-Schmidt frames and weaving dual Hilbert-Schmidt frames.

In this paper we study a representation of a fuzzy subgroup  m of a group  G, as a product of indecomposable fuzzy subgroups called the components of m. This representation is unique up to the number of components and their isomorphic copies. In the crisp group theory, this is a well-known Theorem attributed to Remak, Krull, and Schmidt. We consider the lattice of fuzzy subgroups and some of their properties to prove this theorem. We illustrate with some examples.

Diatoms are the most diverse group of single-celled algae that contain fucoxanthin, clorophyll a and c. This group is considered as algae, the first group of food chain in aquatic ecosystems, however, a species of them with the scientific name Didymosphenia geminata (Lyngbye) M. Schmidt as an invasive species in North America, Europe, New Zealand and Australia are known to cause many environmental problems. During investigation on diatom flora of the Karaj River, Didymosphenia geminata was found as a dominant species with a relative abundance of 10% in January 2011. The dominance of this species in the Karaj River is a threat for the ecosystem of the river. Therefore, to protect the river ecosystem, monitoring programs should be conducted annually and in case of high prevalence of the species, measures should be taken to clear the species from the river.