POINTS IN A FOLD

When a page, represented by the interval $[0,1],$ is folded right over left $ n$ times, the right-hand fold contains a sequence of points. We specify these points and the order in which they appear in each fold. We also determine exactly where in the folded structure any point in $[0,1]$ appears and, given any point on the bottom line of the structure, which point lies at each level above it.

A New Way to Compute the Lyapunov Characteristic Exponents for Non-Smooth and Discontinues Dynamical Systems

One of the most important problems of nonlinear dynamics is related to the development of methods concerning the identification of the dynamical modes of the corresponding systems. The classical method is related to the calculation of the Lyapunov characteristic exponents ( LCEs ). Usually, to implement the classical algorithms for the LCEs calculation the smoothness of the right-hand sides of the corresponding equations is required. In this work, we propose a new algorithm for the LCEs computation in systems with strong nonlinearities (these nonlinearities can not be linearized ) including the hysteresis. This algorithm uses the values of the Jacobi matrix in the vicinity of singularities of the right-hand sides of the corresponding equations. The proposed modification of the algorithm is also can be used for systems containing such design hysteresis nonlinearity as the Preisach operator (thus, this modification can be used for all members of the hysteresis family). Moreover, the proposed algorithm can be successfully applied to the well-known chaotic systems with smooth nonlinearities . Examples of dynamical systems with hysteresis nonlinearities , such as the Lorentz system with hysteresis friction and the van der Pol oscillator with hysteresis block, are considered. These examples illustrate the efficiency of the proposed algorithm.

Monitoring of Eccentric Hamstring Strength and Eccentric Derived Strength Ratios in Judokas from a Single Weight Category

Background: This study was designed to perform isokinetic knee testing of male judokas competing in the under 73 kg category. The main aims were: to establish the concentric (CON) and eccentric (ECC) strength profile of hamstrings (H) and CON profile of quadriceps (Q) muscles; to evaluate the differences in CON and ECC peak torques (PT) with various strength ratios and their bilateral asymmetries; the calculation of the dynamic control ratio (DCR) and H ECC to CON ratio (HEC); Methods: 12 judokas competing on a national and international levels with a mean age of 19 ± 4 years, a weight of 75 ± 2 kg and with a height of 176 ± 5 cm were tested. All the subjects were right-hand dominant. Isokinetic testing was performed on iMOMENT, SMM isokinetic machine (SMM, Maribor, Slovenia). The paired t-test was used to determine the difference between paired variables. The level of significance was set at p ≤ 0.05; Results: Statistical differences between left (L) and right (R) Q PT (L 266; R 241 Nm), H ECC PT (L 145; R 169 Nm), HQR (L 0.54; R 0.63), DCR (L 0.55; R 0.70), HEC (L 1.02; R 1.14) and PTQ/BW (L 3.57; R 3.23 Nm/kg) were shown. Bilateral strength asymmetries in CON contraction of 13.52% ± 10.04 % for Q, 10.86% ± 7.67 % for H and 22.04% ± 12.13% for H ECC contraction were shown. Conclusions: This study reports the isokinetic strength values of judokas in the under 73 kg category, emphasising eccentric hamstring strength and eccentric derived strength ratios DCR and HEC. It was shown that asymmetries are better detected using eccentric testing and that the dominant leg in judokas had stronger eccentric hamstring strength resulting in higher DCR and HEC.

Hardy potential versus lower order terms in Dirichlet problems: regularizing effects

<abstract><p>In this paper, dedicated to Ireneo Peral, we study the regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy potentials in the right hand side.</p></abstract>

New Lead Seal of Adrian Komnenos, Protosebastos and Megas Domestikos of the Entire West, Discovered in the Area of Rusokastro in Southeastern Bulgaria

ntroduction. The present article is a primary publication of a newly discovered sphragistic artefact from the area of the medieval fortress of Rusokastro, which was acquired by the Regional Historical Museum in Burgas in 2019. Methods. In their entirety, the Byzantine lead seals are an important primary and reliable source of information on various aspects of Byzantine history. Byzantine seals are especially important from the perspective of prosopography. First of all, they are invaluable evidence of individuals who, in one way or another, participated in the social and political life of the Byzantine Empire. Therefore, the discovery of each new sphragistic monument is of great importance and the available information must be carefully analyzed. Analysis. This interesting artefact is a lead seal of the brother of the Byzantine emperor Alexius I Komnenos (1081–1118), Adrian Komnenos as a protosebastos and megas domestikos of the entire West. The obverse depicts Saint George, nimbate, standing, facing forward, holding a spear in his right hand, and resting his left hand on a shield. Inscription in 7 lines within dotted border on the reverse: + Κ(ύρι)ε β(οή)θ(ει) τῷ σῷ δούλῳ Ἀδριανῷ (πρωτο)σεβαστῷ (καὶ) μ(ε)γ(άλῳ) δομ[ε]στίκῳ πάσης Δύσεως τῷ Κομνην(ῷ). The seal dates from the end of the 11th century. Results. This new Byzantine lead seal, described in this article, increases the number of medieval sphragistic finds in the Rusokastro area, which belong to an undeniably wide chronological range – from the second half of the 8th century to the beginning of the 12th century. Thus, the number of lead seals from the area of Rusokastro grow to six, five of which are Byzantine and one is a Bulgarian imperial seal.

Unified Theory of Gravity and Electromagnetic Field with Problem in Reissner-Nordstrom and Kerr-Newman Solution

Solutions of unified theory equations of gravity and electromagnetism satisfy Einstein-Maxwellequation. Hence, solutions of the unified theory is Reissner-Nordstrom solution in vacuum. We found in revisedEinstein gravity tensor equation, the condition is satisfied by 2-order contravariant metric tensor two timesproduct. In theory, the problem that the special Ricci tensor have two value term is solved by two orientationof rotating (right hand and left hand).

The Mob Rule

In 1966, the Cultural Revolution broke out. Chen Yi, the foreign minister and Zhou Enlai’s right-hand man, sent a message to Jack, through Comrade Xia on a secret errand, that he would grant Jack an exit visa if he applied for one. But soon Mao’s Red Guards ran amok. Chen Yi was pushed aside. Jack, no longer protected by his family’s reputation and his own connections, was assaulted and not allowed to leave the compound of the Foreign Languages Bureau, where he worked on the English edition of the Peking Review. Yuan-tsung, however, was mobile. She went to see the Red Guards rally at Tiananmen; each of the participants held the Little Red Book, the compilation of Mao’s quotes.