Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space [Formula: see text]. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza’s formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

The Volume of the Quiver Vortex Moduli Space

We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with CPN target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

Volume of line bundles via valuation vectors (different from Okounkov bodies)

Up to a factor 1/n!, the volume of a big line bundle agrees with the Euclidean volume of its Okounkov body. The latter is the convex hull of top rank valuation vectors of sections, all with respect to a single flag. In this paper, we give a new volume formula, valid in the ample cone. It is also based on top rank valuation vectors, but mixes data coming from several different flags.

Research on the growth and development of some varieties of Lavandula Angustifolia (Mill.) in the South-East of Romania

The purpose of this paper was to analyze the canopy volume of lavender shrubs as a morphologic trait influenced by variety and also to evaluate the development of the root system on a Chromic Luvisol with an argic B horizon. The crop was established in 2017 at Belciugatele Didactic Station/Moara Domnească Farm (44°30′ North, 26°15′ East) using the following lavender varieties: Sevstopolis, Vera, Hidcote, and Buena Vista. During 2017-2019, plants’ height and diameters were measured at harvest, and these observations were used to determine lavender shrubs’ canopy volume, by applying the derived ellipsoid volume formula. Varieties (A) obtained different values in terms of canopy volumes, and these values also varied under the influence of the different doses and combinations of mineral and organic fertilizers (B). The highest canopy volume was registered by Vera, with values ranging from 0.0557 m3 (1st year, Control) and 0.0944 m3 (3rd year, organic fertilization with manure at 30 t/ha). The evaluation of the root system distribution and development was performed after carrying out a soil profile between two plants, along the row, for each researched variety. Measurements were conducted using a frame of 50/50 cm, and data sampling was performed for every 10 cm layer, by counting and measuring the roots. Based on these observations the root section area (RSA) was determined. The values of this indicator ranged between 283.06 mm2 (Buena Vista) and 378.29 mm2 (Vera).

Rancang Bangun Sistem Monitoring Pengukuran Volume Air Berbasis IoT Menggunakan Arduino Wemos

System The process of counting and storing in a manual water reservoir analysis has a high percentage of error rate compared to an automated system. In a company industry, especially in the WWT (Waste Water Treatment) section, it has several reservoir tanks as stock which are still counted manually. The ultrasonic sensor is placed at the top of the WWT tank in a hanging position. Basically, to measure the volume in a tank only variable height is always changing. So by utilizing the function of the ultrasonic sensor and also the tube volume formula, the stored AIR volume can be monitored in real time based on IoT using the Blynk application. From the sensor, height data is obtained which then the formula is processed by Arduino Wemos and then information is sent to the MySQL database server via the WIFI network.

Enumerating Parking Completions Using Join and Split

Given a strictly increasing sequence $\mathbf{t}$ with entries from $[n]:=\{1,\ldots,n\}$, a parking completion is a sequence $\mathbf{c}$ with $|\mathbf{t}|+|\mathbf{c}|=n$ and $|\{t\in \mathbf{t}\mid t\leqslant i\}|+|\{c\in \mathbf{c}\mid c\leqslant i\}|\geqslant i$ for all $i$ in $[n]$. We can think of $\mathbf{t}$ as a list of spots already taken in a street with $n$ parking spots and $\mathbf{c}$ as a list of parking preferences where the $i$-th car attempts to park in the $c_i$-th spot and if not available then proceeds up the street to find the next available spot, if any. A parking completion corresponds to a set of preferences $\mathbf{c}$ where all cars park. We relate parking completions to enumerating restricted lattice paths and give formulas for both the ordered and unordered variations of the problem by use of a pair of operations termed Join and Split. Our results give a new volume formula for most Pitman-Stanley polytopes, and enumerate the \emph{signature parking functions} of Ceballos and González D'León.

A Comparative study of log volume estimation by using statistical method

Log volume estimation is a measurement of the amount of merchantable volume and precise estimation of log volume plays an important role in sustainable forest management. There are several log volume formula which commonly used in estimating the log volume, however, there are significant differences between each formula. Therefore, this paper evaluates the performance of three different log volume formula which are Smalian’s, Huber’s and Bruce’s formula against several log sectional length. The performance of each log volume formula will be evaluated in terms of the bias, precision and accuracy of the estimation. The result shows that Huber’s formula performs the best for log sectional length of 2 m, 4 m, 6 m and 8 m log sectional length. The log sectional length and prediction accuracy is inversely related whereby the shorter the log sectional length, the better the prediction accuracy is.