ON THE SMARANDACHE FUNCTION AND SQUARE COMPLEMENTS

The main purpose of this paper is using the elementary method to study the mean value properties of the Smarandache function, and give an interesting asymptotic formula

On a problem of D.H. Lehmer over short intervals

AbstractLet p be an odd prime and a be an integer coprime to p. Denote by N(a,p) the number of pairs of integers b,c with bc≡a (mod p), 1⩽b,c⩽(p−1)2 and with b,c having different parity. The main purpose of this paper is to study the sum ∑a=1p−1(N(a,p)−(p−1)8)2, and obtain a sharp asymptotic formula

On the order of the high-dimensional Cochrane sum and its mean value

AbstractThe main purpose of this paper is to study the high-dimensional Cochrane sum and give a sharp estimate of its order by using properties of hyper-Kloosterman sum and the mean value theorems of Dirichlet L-functions

Sharp uncertainty principles on metric measure spaces

We study the Heisenberg-Pauli-Weyl uncertainty principle and the Caffarelli-Kohn-Nirenberg interpolation inequalities, on metric measure spaces satisfying measure contraction property. Using localization techniques, we show that these inequalities are valid only on volume cones

On the generalized Cochrane sum with Dirichlet characters

In this paper, we defined a new generalized Cochrane sum with Dirichlet characters, and gave the upper bound of the generalized Cochrane sum with Dirichlet characters. Moreover, we studied the asymptotic estimation problem of the mean value of the generalized Cochrane sum with Dirichlet characters and obtained a sharp asymptotic formula for it. By using this asymptotic formula, we also gave the mean value of the generalized Dedekind sum

Transmission of new CRF07_BC Strains with 7 amino acid deletion in Gag p6

A 7 amino acid deletion in Gag p6 (P6delta7) emerged in Chinese prevalent HIV-1 strain CRF07_BC from different epidemic regions. It is important to determine whether this mutation could be transmitted and spread. In this study, HIV-1 Gag sequences from 5 different epidemic regions in China were collected to trace the transmission linkage and to analyze genetic evolution of P6delta7 strains. The sequence analysis demonstrated that P6delta7 is a CRF07_BC specific deletion, different P6delta7 strains could be originated from different parental CRF07_BC recombinants in different epidemic regions, and the transmission of P6delta7 strain has occurred in IDU populations. This is for the first time to identify the transmission linkage for P6delta7 strains and serves as a wake-up call for further monitoring in the future; In addition, P6delta7 deletion may represent an evolutionary feature which might exert influence on the fitness of CRF07_BC strain

Extremely large magnetoresistance in topologically trivial semimetal α\alpha-WP2_2

Extremely large magnetoresistance (XMR) was recently discovered in many non-magnetic materials, while its underlying mechanism remains poorly understood due to the complex electronic structure of these materials. Here, we report an investigation of the $\alpha$-phase WP$_2$, a topologically trivial semimetal with monoclinic crystal structure (C2/m), which contrasts to the recently discovered robust type-II Weyl semimetal phase in $\beta$-WP$_2$. We found that $\alpha$-WP$_2$ exhibits almost all the characteristics of XMR materials: the near-quadratic field dependence of MR, a field-induced up-turn in resistivity following by a plateau at low temperature, which can be understood by the compensation effect, and high mobility of carriers confirmed by our Hall effect measurements. It was also found that the normalized MRs under different magnetic fields has the same temperature dependence in $\alpha$-WP$_2$, the Kohler scaling law can describe the MR data in a wide temperature range, and there is no obvious change in the anisotropic parameter $\gamma$ value with temperature. The resistance polar diagram has a peanut shape when field is rotated in $\textit{ac}$ plane, which can be understood by the anisotropy of Fermi surface. These results indicate that both field-induced-gap and temperature-induced Lifshitz transition are not the origin of up-turn in resistivity in the $\alpha$-WP$_2$ semimetal. Our findings establish $\alpha$-WP$_2$ as a new reference material for exploring the XMR phenomena.Comment: 18 pages, 12 figure

Evidence for chiral superconductivity in Kagome superconductor CsV3Sb5

The interplay among frustrated lattice geometry, nontrivial band topology and correlations yields rich quantum states of matter in Kagome systems. A class of recent Kagome metals, AV3Sb5 (A= K, Rb, Cs), exhibit a cascade of symmetry-breaking transitions, involving 3Q chiral charge ordering, electronic nematicity, roton pair density wave and superconductivity. The interdependence among multiple competing orders suggests unconventional superconductivity, the nature of which is yet to be resolved. Here, we report the electronic evidence for chiral superconducting domains with boundary supercurrent, a smoking-gun of chiral superconductivity, in intrinsic CsV3Sb5 akes. Magnetic field-free superconducting diode effects are observed with its polarity modulated by thermal histories, unveiling a spontaneous time-reversal-symmetry breaking within dynamical order parameter domains in the superconducting phase. Strikingly, the critical current exhibits double-slit superconducting interference patterns, when subjected to external magnetic field. This is attributed to the periodic modulation of supercurrent owing along chiral domain boundaries constrained by fluxoid quantization. Our results provide the direct demonstration of a time-reversal symmetry breaking superconducting order in Kagome systems, opening a potential for exploring exotic physics, e.g. Majorana zero modes, in this intriguing topological Kagome system.Comment: 16 pages,13 figure