Fourth Root Research Articles

GROUP MEAN CORDIAL LABELING OF SOME QUADRILATERAL SNAKE GRAPHS

Let G be a (p, q) graph and let A be a group. Let f : V (G) −→ A be a map. For each edge uv assign the label [o(f (u))+o(f (v)) / 2]. Here o(f (u)) denotes the order of f (u) as an element of the group A. Let I be the set of all integerslabeled by the edges of G. f is called a group mean cordial labeling if the following conditions hold: (1) For x, y ∈ A, |vf (x) − vf (y)| ≤ 1, where vf (x) is the number of vertices labeled with x. (2) For i, j ∈ I, |ef (i) − ef (j)| ≤ 1, where ef (i) denote the number of edges labeled with i. A graph with a group mean cordial labeling is called a group mean cordial graph. In this paper, we take A as the group of fourth roots of unity and prove that, Quadrilateral Snake, Double Quadrilateral Snake, Alternate Quadrilateral Snake and Alternate Double Quadrilateral Snake are groupmean cordial graphs.

Sums of distinct fifth roots of unity and the regular dodecahedron

Integer linear combinations of cube, fourth, or sixth roots of unity form lattices in the complex plane . In contrast, integer linear combinations of fifth roots of unity do not form a lattice; in fact, they are dense in . Nevertheless, the geometry for fifth roots of unity has considerable structure. Here we consider only sums of distinct fifth roots of unity, and show that 20 of these sums are orthogonal projections of the vertices of a regular dodecahedron. Pentagonal symmetry here is only to be expected, but it is a little surprising to encounter a plane projection of a polyhedron with much richer dodecahedral symmetry.

On Locally Symmetric Polynomial Metrics: Riemannian and Finslerian surfaces

In the paper we investigate locally symmetric polynomial metrics in special cases of Riemannian and Finslerian surfaces. The Riemannian case will be presented by a collection of basic results (regularity of second root metrics) and formulas up to Gauss curvature. In case of Finslerian surfaces we formulate necessary and sufficient conditions for a locally symmetric fourth root metric in 2D to be positive definite. They are given in terms of the coefficients of the polynomial metric to make checking the positive definiteness as simple and direct as possible. Explicit examples are also presented. The situation is more complicated in case of spaces of dimension more than two. Some necessary conditions and an explicit example are given for a positive definite locally symmetric polynomial metric in 3D. Computations are supported by the MAPLE mathematics software (LinearAlgebra).

Baseline cystometric parameters in conscious and anesthetized sheep: experimental data and systematic review.

To characterize cystometry in conscious and anesthetized sheep, including bladder response to sacral root electrical stimulation, thereby providing a baseline set of values. Single-fill cystometries were repeated in adult mule ewes both conscious (n = 5) and under general anesthesia (18) using a commercial system. Parameters including bladder capacity, detrusor (bladder) pressure, urethral opening pressure, bladder compliance, number of nonvoiding detrusor contractions, and bladder pressure change in response to electrical stimulation of the sacral roots under general anesthesia are reported. Pubmed, Embase, and Web of Science databases were searched for studies relating to ovine cystometry, and a systematic review was conducted. In awake sheep, mean ± SD bladder capacity was 79.6 ± 32.2 mL, urethral opening pressure was 26.0 ± 10.7 cm H2O, and compliance was 3.5 ± 1.9 mL/cm H2O. Peak detrusor pressures during micturition reached 57.7 ± 28.3 cm H2O. In anesthetized animals, mean bladder capacity (endpoint, 50 cm H2O) was 333 ± 191 mL, and mean bladder compliance was 7.7 ± 4.9 mL/cm H2O. Values for these parameters from our systematic review are presented for comparison and reference. Electrical stimulation of the second and third sacral roots caused a greater increase in detrusor pressure than stimulation of the first and fourth sacral roots. We present a comprehensive set of data for normal cystometry parameters in sheep, including the first report of detrusor response to sacral root stimulation in anesthetized sheep. This report provides a valuable set of baseline values for a potential translational model of value to neurourologic research and may be a useful reference for clinicians.

Electrochemical oxidation of ATMP (Amino Trimethylene Phosphonic Acid) in low-concentrated acidic electrolyte on graphite and glassy carbon electrodes using cyclic voltammetry

This study investigates the irreversible electrochemical oxidation of Amino Trimethylene Phosphonic Acid (ATMP) in ClO4−–electrolyte on original unmodified graphite electrodes (GRE) and glassy carbon electrodes (GCE) using cyclic voltammetry (CV) as the main analytical technique. The oxidation reaction starts at ≈ +0.7 V on GRE and ≈ +0.92 V on GCE vs. Ag/AgCl (3M KCl) respectively. Four distinct electrode/electrolyte systems were selected for a comprehensive analysis of charge transfer and mass transport processes at non-Faradaic and Faradaic current regions at various scan rates υ (SR). These systems comprised of GRE/ClO4−–electrolyte, GRE/ATMP–ClO4−–electrolyte, GCE/ClO4−–electrolyte, and GCE/ATMP–ClO4−–electrolyte. Charging currents in the so-called double-layer region of the CV are found to scale linearly with SR on GCE (Adsorption mechanism); instead, the charging currents obey a nearly square root law on GRE (Diffusion mechanism). The electrochemical oxidation kinetics of ATMP in ClO4−–electrolyte using GRE and GCE obey nearly the fourth root law of the maximum oxidation current density (at the upper reverse potential) vs. SR. Finally, we identified the major transformation products generated from the ATMP electrochemical oxidation: ortho-phosphate (o− PO4), iminodimethylphosphonic acid (IDMP), and aminomethylphosphonic acid (AMPA).

On parity decision trees for Fourier-sparse Boolean functions

We study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Our contributions are as follows: Let \(f : \mathbb {F}_2^n \rightarrow \lbrace -1, 1\rbrace\) be a Boolean function with Fourier support 𝒮 and Fourier sparsity k .

Assessing spontaneous sensory neuron activity using in vivo calcium imaging.

Heightened spontaneous activity in sensory neurons is often reported in individuals living with chronic pain. It is possible to study this activity in rodents using electrophysiology, but these experiments require great skill and can be prone to bias. Here, we have examined whether in vivo calcium imaging with GCaMP6s can be used as an alternative approach. We show that spontaneously active calcium transients can be visualised in the fourth lumbar dorsal root ganglion (L4 DRG) through in vivo imaging in a mouse model of inflammatory pain. Application of lidocaine to the nerve, between the inflamed site and the DRG, silenced spontaneous firing and revealed the true baseline level of calcium for spontaneously active neurons. We used these data to train a machine learning algorithm to predict when a neuron is spontaneously active. We show that our algorithm is accurate in 2 different models of pain: intraplantar complete Freund adjuvant and antigen-induced arthritis, with accuracies of 90.0% ±1.2 and 85.9% ±2.1, respectively, assessed against visual inspection by an experienced observer. The algorithm can also detect neuronal activity in imaging experiments generated in a different laboratory using a different microscope configuration (accuracy = 94.0% ±2.2). We conclude that in vivo calcium imaging can be used to assess spontaneous activity in sensory neurons and provide a Google Colaboratory Notebook to allow anyone easy access to our novel analysis tool, for the assessment of spontaneous neuronal activity in their own imaging setups.

Group Difference Cordial Labeling in Graphs

. Let group .For
 denotes 
 . 
 which is the group of fourth roots of unity, that is cyclic with generators i and -i. We proved that path graph, bistar graph, and Crown graph are We Ladder graph, and star graph is a difference cordial graph.

Ensuring the Quality of Control of the Crankshaft of the ZMZ Engine at Repair Enterprises

An assessment of the quality of control of the parameters of the crankshaft of the ZMZ engine at the repair company was carried out. The metrological support system of the enterprise must guarantee the accuracy and quality of the repair of the crankshaft of the ZMZ engine, excluding the possibility of marriage during subsequent assembly. When the engines are repaired, the crankshafts are defective, after which they are replaced with new ones or processed for repair size. After processing the crankshaft for the repair size, it is necessary to carry out control according to six parameters. It was revealed that in order to measure the diameters of the main and connecting rod necks of the crankshaft, as well as their ovality, it is necessary to replace the micrometer 75-0.01 used at the enterprise, the maximum measurement error of which is twice as large as the permissible one, with a lever bracket 75- 0.002, in order to reduce the measurement error and improve the quality of control. To control the greatest beating of the second, third and fourth root necks relative to the first and fifth root necks, the hour-type indicator used at enterprises fully satisfies the specified conditions, the maximum error does not exceed the permissible one. To control the axial clearance of the already assembled crankshaft along the thrust bearing, 10-0.01 is also used, which provides the specified measurement accuracy with a three-fold margin. When measuring the length of the third root neck, the use of a caliper 125-0.05 leads to an excess of the permissible measurement error by 3 times, but there are no other more accurate universal measuring instruments for monitoring this parameter, because control is carried out by special sponges for measuring holes that are available only for calipers. It is recommended to design a gauge for the purpose of measuring the length of the third root neck.

Avaliação de métodos de transformação para comparações de grupos sobre pressupostos violados: taxa de erro tipo I e poder do teste

ABSTRACT In this study, some transformation methods that are applied when the assumptions of analysis of variance are not met are evaluated in terms of type I error rate and the test power, under circumstances with different distributions, number of groups, number of observations, variance ratios, and different standard deviation differences. The data set used in the study consisted of random numbers generated from N (0,1), and χ2(3) distributions using the random function of the Numpy library in the Python programming language. The logarithmic, square root and root transformations were evaluated on ANOVA based on simulation combinations. It was observed that the transformation techniques of taking the square root after adding 0.5 and 0.375 to the data were relatively more reliable compared to other transformations in terms of type I error rate. However, in every case, type I error rate determined at the beginning of the experiment increased both before and after the transformation was applied. In particular, interestingly, the third and fourth degree root transformations gave better results of test power in the right skewed distribution. In addition, we compared the transformation techniques in question to determine the normality of the data and the homogeneity of variances by a real data.

Effects of aconitine on the respiratory activity of brainstem-spinal cord preparations isolated from newborn rats.

Aconitine is a sodium channel opener, but its effects on the respiratory center are not well understood. We investigated the dose-dependent effects of aconitine on central respiratory activity in brainstem-spinal cord preparations isolated from newborn rats. Bath application of 0.5-5μM aconitine caused an increase in respiratory rhythm and decrease in the inspiratory burst amplitude of the fourth cervical ventral root (C4). Separate application of aconitine revealed that medullary neurons were responsible for the respiratory rhythm increase, and neurons in both the medulla and spinal cord were involved in the decrease of C4 amplitude by aconitine. A local anesthetic, lidocaine (100μM), or a voltage-dependent sodium channel blocker, tetrodotoxin (0.1μM), partially antagonized the C4 amplitude decrease by aconitine. Tetrodotoxin treatment tentatively decreased the respiratory rhythm, but lidocaine tended to further increase the rhythm. Treatment with 100μM riluzole or 100μM flufenamic acid, which are known to inhibit respiratory pacemaker activity, did not reduce the respiratory rhythm enhanced by aconitine + lidocaine. The application of 1μM aconitine depolarized the preinspiratory, expiratory, and inspiratory motor neurons. The facilitated burst rhythm of inspiratory neurons after aconitine disappeared in a low Ca2+/high Mg2+ synaptic blockade solution. We showed the dose-dependent effects of aconitine on respiratory activity. The antagonists reversed the depressive effects of aconitine in different manners, possibly due to their actions on different sites of sodium channels. The burst-generating pacemaker properties of neurons may not be involved in the generation of the facilitated rhythm after aconitine treatment.

Cannabinoid receptors involved in descending inhibition on spinal seizure-like activity in the phrenic output.

Seizure-like burst activities are induced by blockade of GABAA and/or glycine receptors in various spinal ventral roots of brainstem-spinal cord preparation from neonatal rodents. We found that this is not applicable to the phrenic nerve and that a new inhibitory descending pathway may suppress seizure-like activity in the phrenic nerve. Experiments were performed in brainstem-spinal cord preparation from newborn rats (age: 0-1 day). Left phrenic nerve and right C4 activities were recorded simultaneously. When GABAA and glycine receptors were blocked by 10 μM bicuculline and 10 μM strychnine (Bic+Str), seizure-like burst activities appeared in the fourth cervical ventral root (C4) but not the phrenic nerve. After making a transverse section at C1, the inspiratory burst activity disappeared from both C4 and the phrenic nerve, whereas seizure-like activity appeared in both nerves. We hypothesized that inhibitory descending pathways other than those via GABAA and/or glycine receptors (from the medulla to the spinal cord) work to avoid disturbance of regular respiratory-related diaphragm contraction by seizure-like activity. We found that cannabinoid receptor antagonist, AM251 was effective for the induction of seizure-like activity by Bic+Str in the phrenic nerve in brainstem-spinal cord preparation. Cannabinoid receptors may be involved in this descending inhibitory system.

Kurtosis-based vs volatility-based asset allocation strategies: Do they share the same properties? A first empirical investigation

Using a sample of international equity markets over the period 2001–2020, this paper aims to empirically investigate the implications in terms of asset allocation and the key properties of kurtosis-based strategies compared to the more traditional volatility-based strategies for financial portfolios construction. Furthermore, the contribution demonstrates that the portfolio recommended by the novel Kurtosis-based Risk Parity strategy introduced by Braga et al. (2023) admits to being interpreted as an intermediate portfolio between the Minimum-Kurtosis portfolio and the Equally Weighted portfolio in terms of the fourth root of the portfolio fourth moment.

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ via a logarithmic Kazhdan–Lusztig correspondence. We first investigate the representation theory of $$\overline{\mathcal {U}}_{\textsf{i}}^H\!(\mathfrak {sl}_3)$$ , the unrolled restricted quantum group of $$\mathfrak {sl}_3$$ at fourth root of unity. In particular, we analyse its finite-dimensional weight category, determining Loewy diagrams for all projective indecomposables and decomposing all tensor products of irreducibles. Our motivation is that this category is conjecturally braided tensor equivalent to a category of $$W^0_{\!A_2}(2)$$ -modules. Here, $$W^0_{\!A_2}(2)$$ is an orbifold of the octuplet vertex algebra $$W_{\!A_2}(2)$$ of Semikhatov, the latter being the natural $$\mathfrak {sl}_3$$ -analogue of the well known triplet algebra. Moreover, $$W^0_{\!A_2}(2)$$ is the parafermionic coset of the affine vertex algebra $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ . We formulate an explicit conjecture relating the representation theory of $$W^0_{\!A_2}(2)$$ and $$\overline{\mathcal {U}}_{\textsf{i}}^H\!(\mathfrak {sl}_3)$$ and work out the resulting structures of the corresponding $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ -modules. In particular, we obtain conjectural Loewy diagrams for the latter’s projective indecomposables and decompositions for the fusion products of its irreducibles. These products coincide with those recently computed via Verlinde’s formula. Finally, we give analogous results for $$W_{\!A_2}(2)$$ .

On Einstein-reversible mth root Finsler metrics

The theory of mth root Finsler metrics has been applied to Biology, Ecology, Gravitation, Seismic ray theory, etc. It is regarded as a direct generalization of Riemannian metric in a sense, namely, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature faithfully reveals the local geometric properties of a Riemann–Finsler metric. The reversibility of Riemannian and Ricci curvatures of Finsler metrics is an essential concept in Finsler geometry. Here, we study the Riemannian curvature of the class of third and fourth root [Formula: see text]-metrics. Then, we find the necessary and sufficient condition under which a cubic and fourth root [Formula: see text]-metric be Einstein-reversible.

The 3-D problem of temperature and thermal flux distribution around defects with temperature-dependent material properties

The analytical solution of 3-D heat conduction problem, including the temperature and thermal flux fields, is one of the important problems that have not been completely solved in solid mechanics. Considering the temperature dependence of material parameters makes the problem more difficult. In this paper, we first reduce the 3-D temperature-dependent heat conduction problem to the solution of 3-D Laplace equation by introducing the intermediate function. Then, the generalized ternary function is proposed, and the general solution of 3-D Laplace equation is given. Finally, the analytical solutions of three specific problems are obtained and the corresponding temperature-thermal flux fields are discussed. The results show that the thermal flux field of 3-D temperature dependent problem is the same as the classical constant thermal conductivity approach result, while the temperature field is different from the classical result. Thermal flux at a planar defect boundary has r?1/2 singularity, and its intensity is proportional to the fourth root of defect width. On the other hand, when blocked by a planar defect, the thermal flux distribution will re-adjusted so that it overflows at the same rate from all parts of the planar defect boundary.

Justification of the choice of means of controlling the parameters of the crankshaft after processing for the repair size

An assessment of the quality of the control of the parameters of the crankshaft after processing for the repair size was carried out. The company's metrological support system must guarantee the accuracy and quality of the crankshaft repair, eliminating the possibility of defects during subsequent assembly. When repairing engines, defects in the crankshafts are detected, after which they are replaced with new ones or processed according to the repair size. After processing the crankshaft for the repair size, it is necessary to carry out control according to six parameters. It was revealed that to measure the diameters of the main and connecting rod necks of the crankshaft of the Zavolzhsky Engine Plant, as well as their ovality, it is necessary to replace the micrometer 50-75 (0.01) used at the enterprise, the maximum measurement error of which is twice as large as permissible, with a lever bracket 50-75 (0.002) to reduce the measurement error and improve the quality of control. To control the greatest beating of the second, third and fourth root necks relative to the first and fifth root necks, the hour-type indicator used at enterprises fully satisfies the specified conditions, the maximum error does not exceed the permissible one. To control the axial clearance of the already assembled crankshaft along the thrust bearing, dial indicator 0-10 (0.01) is also used, which provides the specified measurement accuracy with a three-fold margin. When measuring the length of the third root neck, the use of a caliper calipers 0-125 (0.05) leads to an excess of the permissible measurement error by 3 times, but there are no other more accurate universal measuring instruments to control this parameter, since the control is carried out by special sponges for measuring holes, which are available only for calipers. It is recommended to design a gauge to measure the length of the third root neck.

Voltage–Current Characteristic of Free Burning Arcs in SF₆ Alternative Gas Mixtures

Voltage–current characteristics of free burning arcs in SF6 and air have been known for decades. As the demand for an SF6-free solution is increasing, there is an accompanying need to determine arc parameters in the alternative gases. An unblown arc experiment has been established to determine the voltage–current characteristics of SF6 alternative gases, which have not yet been thoroughly studied. In this experiment, free burning arc measurements were performed in a number of gases under consideration of SF6 alternatives, including CO2 and mixtures of CO2/O2 with and without C4F7N or C5F10O additives at the concentrations of up to 10 %. Measurements were also performed in air and SF6 for comparison. Arc voltage was measured in each gas at pressures ranging from 1- to 5-bar absolute, and electrode separations ranging from 20 to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$95 \mathrm {~mm}$ </tex-math></inline-formula> . Voltage–current characteristic measurements for air and SF6 show good agreement with previously published results. A linear relationship of the arc voltage to the arc length is shown as well as the fourth root dependence of the arc voltage on the gas pressure. It was shown that neither the O2 nor the fluorinated additives to CO2 have any significant influence on the voltage–current characteristic. The minimum arc voltage in all measured gases was slightly higher than in SF6, but the arc in SF6 was the least stable and had the highest elongations resulting in high-voltage peaks. The arc voltage in air had a similar minimum value to the CO2-based gases, but the arc was much more stable, resulting in lower effective voltage, especially at low currents.

On -gain graphs with few positive eigenvalues

Let T4={1,−1,i,−i} be the group of fourth roots of unit. A T4-gain graph is a graph where each orientation of an edge is given a complex unit in T4, which is the inverse of the complex unit assigned to the opposite orientation. In this paper, we characterize the structure of the T4-gain graphs with exactly one positive eigenvalue and determine the T4-gain graphs with cut vertices having exactly two positive eigenvalues. Our results extend some parallel ones about mixed graphs and signed graphs.

Effect of initial pressure on the propagation characteristics of supersonic turbulent jets in fuel jet ignition

The effects of initial pressure (Pini) on the propagation characteristics of supersonic turbulent jets (STJ) were studied experimentally in a constant volume chamber. Based on the top vortex motion, three types of flame propagation modes are identified: Mode 1, the vortex in the middle of the jet propagates; Mode 2, the vortex in the middle of the jet moves to the top and the tail; Mode 3, jet tail flame extinguished. Each mode follows a 3-stage propagation method: (a) a jet flame with a with a sharp drop in the flame tip velocity (VFTV); (b) a self-accelerating mushroom-shaped flame followed by turbulent flame with low frequency oscillation; (c) a turbulent flame with a drop in VFTV. The momentum dissipation causes the velocity to drop in stage I and stage III, and stage II is controlled by the eddy motion of the mushroom-shaped flame tip and the pulsation of the flow field. The experimental results show that higher Pini can stimulate a stronger interaction mechanism between flow field oscillation in the main combustion chamber (MCC) and chemical heat release oscillation in the pre-chamber (PCC). Based on an evaluation of the model correlations, we found that the jet penetration length showed a linear relationship with the product of the relative pressure fourth root and the time square root.