For $0<\gamma<180^{\circ}$, a geometric path $P=(p_1,\ldots, p_n)$ is called ANGLE-monotone with width $\gamma$ from $p_1$ to $p_n$ if there exists a closed wedge of ANGLE $\gamma$ such that every directed edge $\overrightarrow{p_ip_{i+1}}$ of~$P$ lies inside the wedge whose apex is $p_i$. A geometric graph $G$ is called ANGLE-monotone with width $\gamma$ if for any two vertices $p$ and $q$ in $G$, there exists an ANGLE-monotone path with width $\gamma$ from $p$ to $q$. In this paper, we show that for any integer $k\geq 1$ and any $i\in\{2, 3, 4, 5\}$, the THETA-graph $\THETA_{4k+i}$ on a set of points in convex position is ANGLE-monotone with width $90^\circ+\frac{i\THETA}{4}$, where $\THETA=\frac{360^{\circ}}{4k+i}$. Moreover, we present two sets of points in the plane, one in convex position and the other in non-convex position, to show that for every $0<\gamma<180^\circ$, the graph $\THETA_4$ is not ANGLE-monotone with width $\gamma$.

The largest class of hyper structures is the Hv-structures, introduced in 1990, which proved to have a lot of applications in mathematics and several applied sciences, as well. Hyper structures are used in the Lie-Santilli theory focusing to the hyper numbers, called e-numbers. We present the appropriate e-hyper structures which are defined using any map, in the sense the derivative map, called ¶-hyper structures.

Introduction: The brain waves pattern in primary insomniacs is different from healthy subjects. Studies have shown that binaural beats can alter the pattern of brain waves in healthy individuals,however, the efficacy of binaural beats in altering the pattern of brain waves in primary insomniacs has not yet been investigated. This study aims to evaluate the efficacy of THETA binaural beat on the absolute power of THETA activity in primary insomniacs. Methods: This study was a randomized clinical trial with experimental and control groups. The primary insomniacs received THETA binaural beats in the experimental group while the control group received white noise. Their brain waves were recorded by electroencephalogram for 25 min,the first 5 min was without stimulus (first block), the next was followed by 15 min of receiving stimulus (binaural beat or white noise), and the last 5 min without stimulus (fifth block). The Matlab software, version R2019a, EEGLAB toolbox, and SPSS software, version 24 were used to analyze the data. Results: The absolute power of THETA activity in the experimental group was significantly higher in the last block compared to the first block in all brain lobes (P<0. 05). The largest changes in THETA activity were in the temporal and parietal lobes, and the last one was in the prefrontal lobe. In the control group, none of the brain lobes showed significant differences in the last block compared to the first block. Conclusion: THETA binaural beat can alter the absolute power of THETA activity in primary insomniacs. The implications of the study are discussed.

Potential fields are due to complex distribution of sources related to susceptibility and mass density variations for magnetic and gravity field respectively. In researching the lateral heterogeneous of different geological bodies, in particular their edge location potential field has some advantages. We mainly refer to linear features such as fault, folding axis, dyke, trend and border of the geological units as well as to circular featurs such as crates and buried voids when mentioning geological bodies.Usually, the edge detection and edge enhancement techniques are used to distinguish between geological bodies with different depths and sizes. Edge detection methods are based on the position of the maximum or zero-crossing points associated with vertical derivative, horizontal derivative and analytic signal filters. These methods which are famed to derivative-based filters use different orders, but some instability may occur in high-order derivatives since any kind of noise or non-harmonic signal will be correspondingly enhanced with desired signals simultaneously. Another difficulty is that in the filtered image, the smaller amplitude features (which may be of considerable importance) may be hard to discern.Other edge detection methods are local phase filters (edge enhancement methods) based on the phase variation of the derivative quantities. The advantage of these filters is their flexibility to produce new filters with most applicability just with partial variation. The edge enhancement methods mainly include Tilt ANGLE (TA), Total Horizontal derivative of the tilt ANGLE (THDR), THETA map, Hyperbolic Tilt ANGLE (HTA) and normalized horizontal derivative.The tilt ANGLE filter did not serve as a very accurate method of locating deep sources. A much improved result came from using the second vertical derivative of the tilt ANGLE in the space domain. Equations (1) to (4) are the local phase filters.T=tan-1[¶f/¶z/Ö[¶f/¶x]2+[¶f/¶y]2]THDR=Ö(¶T/¶x)2+(¶T/¶y)2THETA=cos-1[Ö(¶f/¶x)2+(¶f/¶y)2 / Ö(¶f/¶x)2+(¶f/¶y)2+(¶f/¶z)2]HTA=Â[tanh-1[¶f/¶z/Ö(¶f/¶x)2+(¶f/¶y)2]]Where, f is potential field data, T is tilt ANGLE and x, y, z are three Cartesian components, in the above equation Á denote real part of function.In this paper we applied these filters on synthetic magnetic data from a cylinder model as well as on real magnetic data from the Abadeh quadrANGLE. According to the obtained results the Dehshir-Baft fault in the northeast of the studied and ophiolite outcrops in the southeast are enhanced.

We study the automorphic THETA representation Q2n(r) on the r-fold cover of the symplectic group Sp2n. This representation is ob-tained from the residues of Eisenstein series on this group. Ifr is odd, n£ r<2n, then under a natural hypothesis on the THETA representations, we show that Q2n(r) may be used to construct a globally generic represen-tation s(2r) 2n-r+1 on the 2r -fold cover of Sp 2n-r+1. Moreover, when r=n the Whittaker functions of this representation attached to factorizable data are factorizable, and the unramied local factors may be computed in terms ofn -th order Gauss sums. If n=3 we prove these results, which in that case pertain to the six-fold cover of Sp 4, unconditionally. We expect that in fact the representation constructed here, s(2r) 2n-r+1, is precisely Q(2r) 2n-r+1; that is, we conjecture relations between THETA repre-sentations on different covering groups.

A new class of continuous functions, namely s-R-continuous functions, is introduced. The relations of s-R-continuity with continuity and other variants of continuity are discussed. Basic Properties of s-R-continuous functions are studied. Conditions are established under which every weakly THETA_cl-continuous function is s-R-continuous.