ON THE 82-TH SMARANDACHE’S PROBLEM

The main purpose of this paper is using the elementary method to study the asymptotic properties of the integer part of the k-th root positive integer, and give two interesting asymptotic formulae

Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators

The surface of a three-dimensional topological electron system often hosts symmetry-protected gapless surface states. With the effect of electron interactions, these surface states can be gapped out without symmetry breaking by a surface topological order, in which the anyon excitations carry anomalous symmetry fractionalization that cannot be realized in a genuine two-dimensional system. We show that for a mirror-symmetry-protected topological crystalline insulator with mirror Chern number $n=4$, its surface can be gapped out by an anomalous $\mathbb Z_2$ topological order, where all anyons carry mirror-symmetry fractionalization $M^2=-1$. The identification of such anomalous crystalline symmetry fractionalization implies that in a two-dimensional $\mathbb Z_2$ spin liquid the vison excitation cannot carry $M^2=-1$ if the spinon carries $M^2=-1$ or a half-integer spin.Comment: 6+8 pages, 2 figures. v2: added a new section in the supplemental material, the journal reference and some other change

The Ringel--Hall Lie algebra of a spherical object

For an integer $w$, let \cs_w be the algebraic triangulated category generated by a $w$-spherical object. We determine the Picard group of \cs_w and show that each orbit category of \cs_w is triangulated and is triangle equivalent to a certain orbit category of the bounded derived category of a standard tube. When $n=2$, the orbit category \cs_w/\Sigma^2 is 2-periodic triangulated, and we characterize the associated Ringel--Hall Lie algebra in the sense of Peng and Xiao.Comment: 26page

Detecting crystal symmetry fractionalization from the ground state: Application to Z2\mathbb Z_2 spin liquids on the kagome lattice

In quantum spin liquid states, the fractionalized spinon excitations can carry fractional crystal symmetry quantum numbers, and this symmetry fractionalization distinguishes different topologically ordered spin liquid states. In this work we propose a simple way to detect signatures of such crystal symmetry fractionalizations from the crystal symmetry representations of the ground state wave function. We demonstrate our method on projected $\mathbb Z_2$ spin liquid wave functions on the kagome lattice, and show that it can be used to classify generic wave functions. Particularly our method can be used to distinguish several proposed candidates of $\mathbb Z_2$ spin liquid states on the kagome lattice.Comment: main text: 6 pages, 1 figure. supplemental material: 8 pages, 2 figures. Added a few references and the journal referenc

Fire resistance of steel beam to square CFST column composite joints using RC slabs: Experiments and numerical studies

In this paper, experimental investigation and numerical simulation of steel beam to square concrete-filled steel tube (CFST) column composite joints that use reinforced concrete (RC) slabs subjected to localized and global fire conditions are presented. Eight joints were tested under the ISO 834 fire standard, and the effect of different parameters including the load ratio of beams, the beam-to-column ratio of linear stiffness, and different fire scenarios was studied during testing. The failure patterns and the thermal responses of the structural members including the temperature distribution, axial displacement of columns, vertical deflection of the beam ends, and fire resistance of the joints were recorded and discussed. The results show that tube buckling of the square CFST columns, flange buckling of the steel beams, and separation between the top flange of the steel beams and the RC slabs were the primary failure patterns of this type of joint. Moreover, the temperatures of structural members within the connection zone were lower than those in the other regions. Compared with other factors, the load ratio of the beams demonstrated a significant influence on the displacement of the structural members and the fire resistance of the joints. A three-dimensional finite element analysis (FEA) model was built to simulate the fire performance of this type of composite joint. The simulation results were compared to the test results in terms of failure patterns, temperature distributions, displacements, and fire resistances, and good agreement in general was achieved. Finally, the FEA model was adopted to examine the effect of parameters on the fire resistance of the composite joints with axial and flexural constraints applied at the ends of the beam