DISTORTION THEOREMS FOR CLASSES OF g-PARAMETRIC STARLIKE MAPPINGS OF REAL ORDER IN C^(n)

In this paper,we define the class S_(g)^(BX)of g-parametric starlike mappings of real order γ on the unit ball BX in a complex Banach space X,where g is analytic and satisfies certain conditions.By establishing the distortion theorem of the Fr´echet-derivative type of S_(g)^(BX)with a weak restrictive condition,we further obtain the distortion results of the Jacobi-determinant type and the Fr´echet-derivative type for the corresponding classes(compared with S_(g)^(BX))defined on the unit polydisc(resp.unit ball with the arbitrary norm)in the space of n-dimensional complex variables,n≥2.Our results extend the classic distortion theorem of holomorphic functions from the case in one-dimensional complex space to the case in the higher dimensional complex space.The main theorems also generalize and improve some recent works.

Dilation,discrimination and Uhlmann's theorem of link products of quantum channels

We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways,discuss the discrimination of quantum channels,and show that the distinguishability can be improved by self-linking each quantum channel n times as n grows.We also find that the maximum value of Uhlmann's theorem can be achieved for diagonal channels.

New Asymptotic Results on Fermat-Wiles Theorem

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

Local tetragonal distortion of Pt alloy catalysts for enhanced oxygen reduction reaction efficiency

Platinum-based alloy nanoparticles are the most attractive catalysts for the oxygen reduction reaction at present,but an in-depth understanding of the relationship between their short-range structural information and catalytic performance is still lacking.Herein,we present a synthetic strategy that uses transition-metal oxide-assisted thermal diffusion.PtCo/C catalysts with localized tetragonal distortion were obtained by controlling the thermal diffusion process of transition-metal elements.This localized structural distortion induced a significant strain effect on the nanoparticle surface,which further shortened the length of the Pt-Pt bond,improved the electronic state of the Pt surface,and enhanced the performance of the catalyst.PtCo/C catalysts with special short-range structures achieved excellent mass activity(2.27 Amg_(Pt)^(-1))and specific activity(3.34 A cm^(-2)).In addition,the localized tetragonal distortion-induced surface compression of the Pt skin improved the stability of the catalyst.The mass activity decreased by only 13%after 30,000 cycles.Enhanced catalyst activity and excellent durability have also been demonstrated in the proton exchange membrane fuel cell configuration.This study provides valuable insights into the development of advanced Pt-based nanocatalysts and paves the way for reducing noble-metal loading and increasing the catalytic activity and catalyst stability.

Linear magnetoresistance and structural distortion in layered SrCu_(4-x)P_(2) single crystals

We report a systematic study on layered metal SrCu_(4-x)P_(2) single crystals via transport, magnetization, thermodynamic measurements and structural characterization. We find that the crystals show large linear magnetoresistance without any sign of saturation with a magnetic field up to 30T. We also observe a phase transition with significant anomalies in resistivity and heat capacity at T_(p)~140 K. Thermal expansion measurement reveals a subtle lattice parameter variation near Tp, i.e.,?L_(c)/L_(c)~0.062%. The structural characterization confines that there is no structure transition below and above T_(p). All these results suggest that the nonmagnetic transition of SrCu_(4-x)P_(2) could be associated with structural distortion.

Engineering Thermoelectric Performance of α-GeTe by Ferroelectric Distortion

The rhombohedralα-GeTe can be approximated as a slightly distorted rock-salt structure along its[111]direction and possesses superb thermoelectric performance.However,the role of such a ferroelectric-like structural distortion on its transport properties remains unclear.Herein,we performed a systematic study on the crystal structure and electronic band structure evolutions of Ge_(1-x)Sn_(x)Te alloys where the degree of ferroelectric distortion is continuously tuned.It is revealed that the band gap is maximized while multiple valence bands are converged at x=0.6,where the ferroelectric distortion is the least but still works.Once undistorted,the band gap is considerably reduced,and the valence bands are largely separated again.Moreover,near the ferro-to-paraelectric phase transition Curie temperature,the lattice thermal conductivity reaches its minima because of significant lattice softening enabled by ferroelectric instability.We predict a peak ZT value of 2.6 at 673 K inα-GeTe by use of proper dopants which are powerful in suppressing the excess hole concentrations but meanwhile exert little influence on the ferroelectric distortion.

Bulging Distortion of Austenitic Stainless Steel Sheet on the Partially Penetrated Side of Non-Penetration Lap Laser Welding Joint

Non-penetration laser welding of lap joints in austenitic stainless steel sheets is commonly preferred in fields where the surface quality is of utmost importance.However,the application of non-penetration welded austenitic stainless steel parts is limited owing to the micro bulging distortion that occurs on the back surface of the partial penetration side.In this paper,non-penetration lap laser welding experiments,were conducted on galvanized and SUS304 austenitic stainless steel plates using a fiber laser,to investigate the mechanism of bulging distortion.A comparative experiment of DC01 galvanized steel-Q235 carbon steel lap laser welding was carried out,and the deflection and distortion profile of partially penetrated side of the sheets were measured using a noncontact laser interferometer.In addition,the cold-rolled SUS304 was subjected to heat holding at different temperatures and water quenching after bending to characterize its microstructure under tensile and compressive stress.The results show that,during the heating stage of the thermal cycle of laser lap welding,the partial penetration side of the SUS304 steel sheet generates compressive stress,which extrudes the material in the heat-affected zone to the outside of the back of the SUS304 steel sheet,thereby forming a bulge.The findings of these experiments can be of great value for controlling the distortion of the partial penetrated side of austenitic stainless steel sheet during laser non-penetration lap welding.

Diophantine equations and Fermat's last theorem for multivariate(skew-)polynomials

Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.

AAK Theorem and a Design of Multidimensional BIBO Stable Filters

Rational approximation theory occupies a significant place in signal processing and systems theory. This research paper proposes an optimal design of BIBO stable multidimensional Infinite Impulse Response filters with a realizable (rational) transfer function thanks to the Adamjan, Arov and Krein (AAK) theorem. It is well known that the one dimensional AAK results give the best approximation of a polynomial as a rational function in the Hankel semi norm. We suppose that the Hankel matrix associated to the transfer function has a finite rank.

Some convergence theorems of fuzzy concave integral on fuzzyσ-algebra

In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.

Cross-section distortion and springback characteristics of double-cavity aluminum profile in force controlled stretch-bending

3D elastic-plastic FE model for simulating the force controlled stretch-bending process of double-cavity aluminum profile was established using hybrid explicit−implicit solvent method.Considering the computational accuracy and efficiency,the optimal choices of numerical parameters and algorithms in FE modelling were determined.The formation mechanisms of cross-section distortion and springback were revealed.The effects of pre-stretching,post-stretching,friction,and the addition of internal fillers on forming quality were investigated.The results show that the stress state of profile in stretch-bending is uniaxial with only a circumferential stress.The stress distribution along the length direction of profile is non-uniform and the maximum tensile stress is located at a certain distance away from the center of profile.As aluminum profile is gradually attached to bending die,the distribution characteristic of cross-section distortion along the length direction of profile changes from V-shape to W-shape.After unloading the forming tools,cross-section distortion decreases obviously due to the stress relaxation,with a maximum distortion difference of 13%before and after unloading.As pre-stretching and post-stretching forces increase,cross-section distortion increases gradually,while springback first decreases and then remains unchanged.With increasing friction between bending die and profile,cross-section distortion slightly decreases,while springback increases.Cross-section distortion decreases by 83%with adding PVC fillers into the cavities of profile,while springback increases by 192.2%.

Small Modular Solutions to Fermat’s Last Theorem

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.

Computational Quantification of Map Projection Distortion by Fractal Dimension of Coastlines

Maps, essential tools for portraying the Earth’s surface, inherently introduce distortions to geographical features. While various quantification methods exist for assessing these distortions, they often fall short when evaluating actual geographic features. In our study, we took a novel approach by analyzing map projection distortion from a geometric perspective. We computed the fractal dimensions of different stretches of coastline before and after projection using the divide-and-conquer algorithm and image processing. Our findings revealed that map projections, even when preserving basic shapes, inevitably stretch and compress coastlines in diverse directions. This analysis method provides a more realistic and practical way to measure map-induced distortions, with significant implications for cartography, geographic information systems (GIS), and geomorphology. By bridging the gap between theoretical analysis and real-world features, this method greatly enhances accuracy and practicality when evaluating map projections.

Whole Perfect Vectors and Fermat’s Last Theorem

A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.

Vanishing Theorems for p-Harmonic Forms on Submanifolds in Spheres

In this paper,we present some vanishing theorems for p-harmonic forms on-super stable complete submanifold M immersed in sphere Sn+m.When 2≤1≤n-2,M has a flat normal bundle.Assuming that M is a minimal submanifold andδ>1(n-1)p2/4n[p-1+(p-1)2kp],we prove a vanishing theorem for p-harmonicℓ-forms.

A Study on High School Mathematics Teaching Design Based on Teaching for Robust Understanding: Taking the Cosine Theorem as an Example

Teachers’teaching behavior plays a crucial role in students’development,and there are problems in the current teaching behavior of mathematics teachers such as ignoring students’cognitive needs,lack of equal opportunities for students’classroom performance as well as lack of formative evaluation of students.In order to solve the phenomenon,this paper analyzes and explains how to promote teaching based on the Teaching for Robust Understanding(TRU)evaluation framework with the goal of focusing on the development of all students,taking the teaching design of The Cosine Theorem as an example,and provides ideas and methods for first-line high school mathematics teachers.

The Njiki’s Fundamental Theorem-Definition on Fractions in the Mathematical Set ℚ and by Extension in ℝ and ℂ, for the Purpose of Leading to the Construction of Some Algebraic Structures as Its Theoretical Applications and for the Practical Ones

The purpose of the research in the NJIKI’s fundamental THEOREM-DEFINITION on fractions in the mathematical set ℚand by extension in ℝand ℂand in order to construct some algebraic structures is about the proved EXISTENCE and the DEFINITION by NJIKI of two INNOVATIVE, IMPORTANT and TEACHABLE operations of addition or additive operations, in ℚ, marked ⊕and +α,β, and taken as VECTORIAL, TRIANGULAR, of THREE or PROPORTIONAL operations and in order to make THEM not be different from the RATIONAL ONE, +, but to bring much more and new information on fractions, and, by extension in ℝand ℂ. And the very NJIKI’s fundamental THEOREM-DEFINITION having many APPLICATIONS in the everyday life of the HUMAN BEINGS and without talking about computer sciences, henceforth being supplied with very interesting new ALGORITHMS. And as for the work done in the research, it will be waiting for its extension to be done after publication and along with the research results concerned.

DISTORTION THEOREMS FOR SUBCLASSES OF STARLIKE MAPPINGS ALONG A UNIT DIRECTION IN C^n

In this paper, we obtain a distortion theorem of Jacobian matrix for biholomorphic subclasses of starlike mappings along a unit direction on the unit polydisc. These results extend the classical distortion theorem of starlike mappings to higher dimensions. We then give an upper bound estimate for biholomorphic subclasses of starlike mappings along a unit direction in a complex Banach space.

GROWTH AND DISTORTION THEOREMS FOR ALMOST STARLIKE MAPPINGS OF COMPLEX ORDER λ

In this article, the sharp growth theorem for almost starlike mappings of complex order λ is given firstly. Secondly, distortion theorem along a unit direction is also established as the application of the growth theorem. In particular, using our results can reduce to some well-known results.

SHARP DISTORTION THEOREMS FOR A CLASS OF BIHOLOMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES

In this paper,we first establish the sharp growth theorem and the distortion theorem of the Frechet derivative for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in C^(n) with some restricted conditions.We next give the distortion theorem of the Jacobi determinant for biholomorphic mappings defined on the unit ball of C^(n) with an arbitrary norm and the unit polydisk in C^(n) under certain restricted assumptions.Finally we obtain the sharp Goluzin type distortion theorem for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in C^(n) with some additional conditions.The results derived all reduce to the corresponding classical results in one complex variable,and include some known results from the prior literature.