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In this paper, for each of six families of three-valued \begin{document}$ m $\end{document} -sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring \begin{document}$ R = \mathbb{F}_2+u\mathbb{F}_2 $\end{document} , where \begin{document}$ u^2 = 0. $\end{document} The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are \begin{document}$ [2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)] $\end{document} for some explicit \begin{document}$ r. $\end{document}

Let f be a non-degenerate Laurent polynomial over a finite field and L ⁎ ( f , T ) the associated L-function of the toric exponential sums of f. The Newton polygon can be used to study the p-adic valuations of roots or poles of L ⁎ ( f , T ) , which has a lower bound called the Hodge polygon that depends only on the Newton polytope of f. These two polygons coincide when the coefficients of f are not zeros of the certain Hasse polynomial. Using the Dwork trace formula, we find a formula for the Hasse polynomial of the slope one side for a class of Laurent polynomials, which generalizes the results of Zhang, Feng and Chen.

This paper considers a privacy-concerned distributed optimization problem over multi-agent networks, in which malicious agents exist and try to infer the privacy information of the normal ones. We propose a novel dual averaging algorithm which involves the use of a correlated perturbation mechanism to preserve the privacy of the normal agents. It is shown that our algorithm achieves deterministic convergence under arbitrary initial conditions and the privacy preservation is guaranteed. Moreover, a probability density function of the perturbation is given to maximize the degree of privacy measured by the trace of the Fisher information matrix. Finally, a numerical example is provided to illustrate the effectiveness of our algorithm.

Since the development of higher local class field theory, several explicit reciprocity laws have been constructed. In particular, there are formulas describing the higher-dimensional Hilbert symbol given, among others, by M. Kurihara, A. Zinoviev and S. Vostokov. K. Kato also has explicit formulas for the higher-dimensional Kummer pairing associated to certain (one-dimensional) $p$-divisible groups. In this paper we construct an explicit reciprocity law describing the Kummer pairing associated to any (one-dimensional) formal group. The formulas are a generalization to higher-dimensional local fields of Kolyvagin's reciprocity laws. The formulas obtained describe the values of the pairing in terms of multidimensional $p$-adic differentiation, the logarithm of the formal group, the generalized trace and the norm on Milnor K-groups. In the second part of this paper, we will apply the results obtained here to give explicit formulas for the generalized Hilbert symbol and the Kummer pairing associated to a Lubin-Tate formal group. The results obtained in the second paper constitute a generalization to higher local fields, of the formulas of Artin-Hasse, K. Iwasawa and A. Wiles., The stronger reciprocity laws in this new version cover arbitrary higher local fields, as opposed to only standard higher local fields

The relative geometry of the anchors and the emitter plays important role in the estimation accuracy of the emitter location. In this paper, it is assumed that there are a number of anchors with known location in a 2D wireless network. Each anchor either measures the distance between itself and the emitter or it measures the look angle to the emitter. Knowing the location of the anchors, the emitter location can be estimated using these noisy measurements. It is assumed that one wants to add a number of new anchors to this network to improve the accuracy of estimating the emitter location. In this paper, we have used the frame theory to obtain an algorithm which proposes the optimum location of the new anchors in the network. The optimization criterion for anchors placement is maximizing the Fisher information matrix (FIM) determinant, which is equivalent to minimizing the trace of the Cramer Rao Lower Bound (CRLB) in our assumed problem. Numerical simulations confirm the ability of the proposed algorithm to find the optimal location for new anchors.

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

The core challenge of seismic data interpolation is how to capture latent spatial-temporal relationships between unknown and known traces in 3-D space. The prevailing tensor-based interpolation schemes seek a globally low-rank approximation to mine the high-dimensional relationships hidden in 3-D seismic data. However, when the low-rank assumption is violated for data involving complex geological structures, the existing interpolation schemes fail to precisely capture the trace relationships, which may influence the interpolation results. As an alternative, this article presents a basic deep tensor autoencoder (DTAE) and two variants to implicitly learn a data-driven, nonlinear, and high-dimensional mapping to explore the complicated relationship among traces without the need for any underlying assumption. Then, tensor backpropagation (TBP), which can be essentially viewed as a tensor version of traditional backpropagation (BP), is introduced to solve for the new model parameters. For ease of implementation, a mathematical relationship between tensor and matrix autoencoders is constructed by taking advantage of the properties of a tensor-tensor product. Based on the derived relationship, the DTAE weight parameters are inferred by applying a matrix autoencoder to each frontal slice in the discrete cosine transform (DCT) domain, and this process is further summarized into a general theoretical and practical framework. Finally, the performance benefits of the proposed DTAE-based method are demonstrated in experiments with both synthetic and real field seismic data.

Arrays deployed in antenna-array-based direct position determination (DPD) systems are conventionally placed parallel to each other with invariant orientations, which may be inappropriate. This paper proposes a novel array orientation adjustment method to achieve optimal DPD performance, in which we seek optimal array orientations that maximize the trace of the Fisher information matrix (FIM) and then minimize localization uncertainty. The optimization problem is unconstrained, multivariable, and can be converted into multiple tractable single-variable optimization problems. An exact solution can be obtained by a straightforward but computationally tedious one-dimensional exhaustive search. To reduce the computational load, we then derive an asymptotic solution in cases where the arrays consist of massive antennas. Numerical examples demonstrate the effectiveness and feasibility of the proposed method in resolution and precision enhancement, especially in a low signal-to-noise ratio (SNR).

Band selection is an effective means to alleviate the curse of dimensionality in hyperspectral data. Many methods select a compact and low redundant band subset, which is inadequate as it may degrade the classification performance. Instead, more emphasis shall be put on selecting representative bands. In this article, we propose a robust unsupervised band selection method to address this issue. Our method reveals bandwise representativeness based on the comprehensive interband neighborhood structure. It incorporates an interband neighborhood graph into a sparse self-contained regression model in order to provide a reasonable measure for bandwise representativeness. The derived coefficient matrix not only uncovers bandwise importance values but also is coherent to the generalized interband local neighborhood structure. For constructing the interband neighboring structural graph, an integrated multigraph model is employed to achieve better generalization performance. It combines the benefit of multiple graphs but is insusceptible to the defects of a single one. To enhance the reliability of this model, a joint trace minimum and nonnegative constraint is imposed on the coefficient matrix. Accordingly, a multigraph integrated embedding and robust self-contained regression model (MGRSR) is formulated. In addition, an iterative update algorithm is developed to solve the problem. Comparative experiments on three hyperspectral data sets illustrate that MGRSR is robust to various data and has superior performance compared with several state-of-the-art methods.

As a very attractive clean energy, hydrogen has a high energy density and great potential to achieve zero pollution emission. Therefore, the preparation of hydrogen evolution electrocatalysts with excellent performance is an urgent task to ameliorate the global energy shortage and environmental pollution. Here, a trace amount of NiP2 coupled with CoMoP nanosheets (NCMP) was synthesized by the one-step hydrothermal method and low-temperature phosphidation. Studies have found that although the dosage of NiP2 is very low, its appearance has been efficient to improve the hydrogen evolution reaction (HER) performance of CoMoP, which may be induced by the synergistic effect of the two different components NiP2 and CoMoP. To find the superior catalyst, the effect of Ni content on the catalyst performance is also studied, and it is found that when the dosage of Ni is 0.02 mM, NCMP-2 (2 means 0.02 mM) displays the most outstanding overpotential (10 mA cm–2) of 46 mV.

The periodic tables of transition metal thiophosphates MPS3, transition metal dichalcogenides MX2 and other materials, the origin of chemical elements and toxic trace elements in dried mushrooms are provided. The effective nucleus-electron attraction is proportional to the effective nuclear charge (Zeff) and inversely proportional to the effective principal quantum number (n*). The periodic arch is one of many modern visual displays that have been developed to augment the traditional periodic table of the chemical elements. The table is related to the multiparameter optimisation of N atom, nuclear magnetic resonance and everyday life. Educational activities were developed and evaluated. Se proporcionan las tablas periódicas de los tiofosfatos de metales de transición MPS3, los dicalcogenuros de metales de transición MX2 y otros materiales, el origen de los elementos químicos y elementos traza tóxicos en setas secas. La atracción efectiva núcleo-electrón es proporcional a la carga nuclear efectiva (Zeff) e inversamente proporcional al número cuántico principal efectivo (n*). El arco periódico es una de las muchas presentaciones visuales modernas que se han desarrollado para aumentar la tabla periódica tradicional de los elementos químicos. La tabla se relaciona con la optimización multiparamétrica del átomo de N, la resonancia magnética nuclear y la vida cotidiana. Se desarrollan actividades educativas con evaluación. Ciencias Experimentales

L'article pertany al grup de recerca Combinatorics, Coding and Security Group (CCSG) A subset of a vector space Fn q is additive if it is a linear space over the field Fp, where q = pe, p prime, and e > 1. Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these bounds, additive GH codes are constructed. Moreover, for the case e = 2, it is shown that the given bounds are tight and it is possible to construct an additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.

This paper proposes an improved stochastic estimation method based on contour integral to estimate the number of critical eigenvalues. The stochastic estimation method transforms the eigenvalue numbers' calculation into summing the traces of a series of inverse matrices. And then, a stochastic strategy is used to estimate the trace of the inverse matrix. However, the method results in an unacceptable error when the state matrices of many practical power systems are ill-conditioned. To solve this problem, the improved algorithm is proposed by formulating a satisfactory matrix for trace calculation artificially instead of randomly with the help of the well-known greedy algorithm based on graph coloring. It is proved to be much more reliable when dealing with ill matrix. The proposed method based on contour integral is very suitable for large-scale power systems due to the element distribution of descriptor matrices makes it possible to construct only one fixed matrix at all integral points. Moreover, the calculation efficiency is further improved by calculating implicitly without destroying the sparsity of descriptor systems. Numerical experiments indicate that the proposed method can significantly improve accuracy without considerably increasing the calculation time. And the approximate eigenvalue number can fully meet the needs of subsequent algorithms based on contour integral to calculate the specific eigenvalues.

The composite quantile regression (CQR) estimator is a robust and efficient alternative to the ordinary least squares estimator and single quantile regression estimator in linear models. For massive data, two different optimal subsampling probabilities through minimizing the trace of the asymptotic variance–covariance matrix for a linearly transformed parameter estimator and the asymptotic mean squared error of the resultant estimator are proposed to downsize the data volume and reduce the computational burden. Furthermore, to improve the efficiency of the ordinary CQR, the optimal subsampling for the weighted CQR estimator is also studied. We also propose iterative subsampling procedures based on two optimal subsampling probabilities to estimate the variance–covariance matrix. The finite-sample performance of the proposed estimators is studied through simulations and an application to household electric power consumption data is also presented.

We consider a boundary value problem with boundary conditions involving the tangential part of the vector of normal derivatives of the solution. We establish the existence of a weak solution. We show that the trace of the normal derivative vector on the boundary can be understood in a certain sense (the trace lemma) and, as a consequence, prove the existence of a generalized solution and establish a connection between the generalized and weak solutions to the problem.

A conical poly(ethylene terephthalate) nanopore surface modified with a p-tert-butylcalix[4]arene crown is adopted to theoretically model the detection of trace concentrations of cesium ions in an ...

Given $$m, n, q\in \mathbb {N}$$ such that q is a prime power and $$m\ge 3$$ , $$a\in \mathbb {F}_q$$ , we establish a sufficient condition for the existence of a primitive pair $$(\alpha , f(\alpha ))$$ in $$\mathbb {F}_{q^m}$$ such that $$\alpha $$ is normal over $$\mathbb {F}_q$$ and $$\text {Tr}_{\mathbb {F}_{q^m}/\mathbb {F}_q}(\alpha ^{-1})=a$$ , where $$f(x)\in \mathbb {F}_{q^m}(x)$$ is a rational function of degree sum n. Further, when $$n=2$$ and $$q=5^k$$ for some $$k\in \mathbb {N}$$ , such a pair definitely exists for all (q, m) apart from at most 20 choices.

In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr's inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n,\ n\ge 2$.

Trace elements are essential nutrients for plant growth. An accurate prediction of soil available trace elements is necessary for scientifically based fertilization and soil environmental protection. In total, 670 surface soil samples were collected across the study area to analyze the relationship between soil properties and available trace elements (Cu, Zn, Fe, Mn, B, and Mo). Moreover, the Bayesian maximum entropy (BME) technique was employed to predict the spatial distribution of the trace elements by combining soil property information as auxiliary data. In addition, the prediction accuracy of the BME technique was compared with that of the conventional cokriging (CK) method. The results showed that soil macronutrients, especially organic matter, available P, available K and slow-release K, were significantly correlated with the content of the available trace elements. Soil texture also had great influences on trace elements and was generally represented as clay > heavy loam > medium loam > light loam. The BME technique combined with auxiliary soil property information (both categorical and numerical) performed better than the traditional CK technique, which was supported by the smaller MAE and RMSE and higher R2 from the ten-fold cross validation. By mapping the spatial prediction error difference between the BME and CK methods across the study area, in comparison to the CK method, the BME technique provided consistently more accurate spatial predictions of trace element concentrations. In conclusion, with the advantages of combining categorical and numerical auxiliary data, the BME technique is a suitable spatial prediction method, and the results obtained in local regions could provide an important reference for the scientific application of microfertilizers.

Let $F$ be an infinite field and let $f$ be a nonzero multilinear polynomial with coefficients in $F$. We prove that for every positive integer $d$ there exists a positive integer $s$ such that $f(M_{s}(F))$, the image of $f$ in $M_{s}(F)$, contains all trace zero $d \times d$ matrices. In particular, the image of $f$ in the algebra of all finitary matrices contains all trace zero finitary matrices., 11 pages, 0 figures, submited to Linear Algebra and Its Applications

We define a family of pseudodifferential operators on the Hilbert space $$L^2({\mathbb{Q}_p})$$ of complex valued square-integrable functions on the $$p$$ -adic number field $${\mathbb{Q}_p}$$ . These generalise the Vladimirov derivative, using the Dirichlet and other suitable multiplicative characters.The Riemann zeta-function and the related Dirichlet $$L$$ -functions can be expressed as a trace of these operators on a subspace of $$L^2({\mathbb{Q}_p})$$ . We also extend this to the $$L$$ -functions associated with modular (cusp) forms. Wavelets on $${\mathbb{Q}_p}$$ are common sets of eigenfunctions of all these operators.

In pH 3.1 HCOOH-HCOONa buffer solution, liquid crystal 5CB was used as the probe to generate a strong resonance Rayleigh scattering (RRS) peak at 450 nm. Gold nanoparticles (AuNPs) could catalyse the reaction of phosphomolybdic acid (PMo)-formic acid (FA) to form phosphomolybdenum blue (PMoB), which caused the RRS strength of PMo at 450 nm to decrease linearly. Hg2+ could have an electrostatic substitution reaction with AuNPs, thereby inhibiting the catalytic effect of AuNPs, and the RRS peak was enhanced. In the range of 2.5 × 10−9 mol/L–4.5 × 10−8 mol/L Hg2+, as the concentration of Hg2+ increased, the catalytic effect of AuNPs gradually weakened, and the peak value of RRS at 450 nm (ΔI) was linearly increased. The regression equation is ΔI450nm = 27.988C−5.6159, and the detection limit is 1.0 nmol/L. The method has been used to detect Hg2+ in wastewater with satisfactory results.

The paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in ℝ3. The method employs parametric P k -P k−1 finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin–Helmholtz instability problem on the unit sphere.

In the present paper, we have generalized the behaviors of the transit-decelerating to accelerating FRW cosmological model in f(R, T) gravity theory, where R and T are Ricci scalar and trace of the energy–momentum tensor, respectively. The solution of the corresponding field equations is obtained by assuming a linear function of the Hubble parameter H (i.e., q = c1 + c2H), which gives a time-dependent deceleration parameter [Formula: see text], where c3 and c2 are arbitrary integrating constants (Tiwari et al. Eur. Phys. J. Plus, 131, 447 (2016); Ibid. 132, 126 (2017)). There are two scenarios in which we explain the particular form of scale factor thus obtained: (i) by using the recent constraints from observational Hubble data (OHD) and joint light curves (JLA) data, which show a cosmic deceleration to acceleration and (ii) by using new constraints from supernovae type Ia union data, which show accelerating expansion of the universe (q < 0) throughout its evolution. We have observed that EoS parameter, energy density parameters, and important cosmological planes yield results compatible with the modern observational data. For the derived models, we have calculated various physical parameters as luminosity distance, distance modulus, and apparent magnitude versus redshift for both current supporting observations.

The aim of this note is to give a formula expressing the trace form associated with the 27 lines of a cubic surface.

We study the sequence of momenta of the Coxeter transformation Φ G → associated to a bipartite quiver G → (that is, the sequence of traces of the powers of Φ G → ). Among other results, we derive A'Campo like formulas for such momenta in terms of the spectrum of the corresponding underlying graph G of G → (using Chebyshev polynomials), and analyze the maximal number of spikes in a tree graph (that is, eigenvalues with norm greater than 1), to construct general bounds for the Coxeter momenta. We apply the main results of the paper to produce formulas and bounds for the trace of the exponential of Coxeter matrices of tree graphs (called Coxeter-Estrada index in the paper).

Best estimate plus uncertainty for the safety assessment of nuclear power plant transient requires, among others, estimating the probability density function (PDF) of physical model parameters in t...

Incorporating the arc-length constraint, the dynamic relaxation strategy has been widely used to trace full equilibrium path in the post-buckling analysis of structures. This combined numerical scheme has been shown to be successful for solving snap-through problems, but its applicability to snap-back problems has been rarely investigated and remains unclear. This paper proposes a direct and more general finite-difference equation to investigate the numerical stability of this combined numerical scheme, which is dominated by the spectral radius of amplification matrix. And a key discovery of this paper is that a first minor of the tangent stiffness matrix is always negative once snap back occurs. Due to this negative minor stiffness, the spectral radius is invariably greater than one, resulting in unconditional instability, which demonstrates the invalidity of dynamic relaxation arc-length method for snap-back problems. These important conclusions are corroborated by the numerical results of three representative examples in one-, two- and three-dimensional spaces.

In this paper we investigate the nuclear trace of vector‐valued Fourier multipliers on the torus and its applications to the index theory of periodic pseudo‐differential operators. First we characterise the nuclearity of pseudo‐differential operators acting on Bochner integrable functions. In this regards, we consider the periodic and the discrete cases. We go on to address the problem of finding sharp sufficient conditions for the nuclearity of vector‐valued Fourier multipliers on the torus. We end our investigation with two index formulae. First, we express the index of a vector‐valued Fourier multiplier in terms of its operator‐valued symbol and then we use this formula for expressing the index of certain elliptic operators belonging to periodic Hormander classes.

This paper develops nonasymptotic growth and concentration bounds for a product of independent random matrices. These results sharpen and generalize recent work of Henriksen-Ward, and they are similar in spirit to the results of Ahlswede-Winter and of Tropp for a sum of independent random matrices. The argument relies on the uniform smoothness properties of the Schatten trace classes., Comment: 21 pages

We fully describe the doubly stochastic orbit of a self-adjoint element in the noncommutative $$L_1$$ -space affiliated with a semifinite von Neumann algebra, which answers a problem posed by Alberti and Uhlmann (Stochasticity and partial order: doubly stochastic maps and unitary mixing. VEB Deutscher Verlag der Wissenschaften, Berlin, 1982) in the 1980s, extending several results in the literature. It follows further from our methods that, for any $$\sigma $$ -finite von Neumann algebra $${{\mathcal {M}}}$$ equipped with a semifinite infinite faithful normal trace $$\tau $$ , there exists a self-adjoint operator $$y\in L_1({{\mathcal {M}}},\tau )$$ such that the doubly stochastic orbit of y does not coincide with the orbit of y in the sense of Hardy–Littlewood–Polya, which confirms a conjecture by Hiai (J Math Anal Appl 127:18–48, 1987). However, we show that Hiai’s conjecture fails for non- $$\sigma $$ -finite von Neumann algebras. The main result of the present paper also answers the (noncommutative) infinite counterparts of problems due to Luxemburg (Proc Symp Anal Queen’s univ 83–144, 1967) and Ryff (Pac J Math 13:1379–1386, 1963) in the 1960s.

A cyclic code has been one of the most active research topics in coding theory because they have many applications in data storage systems and communication systems. They have efficient encoding and decoding algorithms. This paper explains the construction of a family of cyclic codes from sequences generated by a trace of a monomial over finite fields of odd characteristics. The parameter and some examples of the codes are presented in this paper.

We establish similarity between some tripotents and idempotents on a Hilbert space $\mathcal{H}$ and obtain new results on differences and commutators of idempotents P and Q . In the unital case, the difference $P-Q$ is associated with the difference $A_{P, Q}$ of another pair of idempotents. Let $\varphi$ be a trace on a unital $C^*$ -algebra $\mathcal{A}$ , $\mathfrak{M}_{\varphi}$ be the ideal of definition of the trace $\varphi$ . If $P-Q \in \mathfrak{M}_\varphi$ , then $A_{P, Q} \in \mathfrak {M}_\varphi$ and $\varphi (A_{P, Q}) = \varphi (P-Q) \in \mathbb{R}$ . In some cases, this allowed us to establish the equality $\varphi (P-Q) = 0$ . We obtain new identities for pairs of idempotents and for pairs of isoclinic projections. It is proved that each operator $A \in \mathcal{B} (\mathcal{H})$ , $\dim \mathcal{H} = \infty$ , can be presented as a sum of no more than 50 commutators of idempotents from $\mathcal{B} (\mathcal{H})$ . It is shown that the commutator of an idempotent and an arbitrary element from an algebra $\mathcal{A}$ cannot be a nonzero idempotent. If $\mathcal{H}$ is separable and $\dim \mathcal{H} = \infty$ , then each skew-Hermitian operator $T \in \mathcal {B} (\mathcal{H})$ can be represented as a sum $T = \sum_{k = 1}^4 [A_k, B_k]$ , where $A_k, B_k \in \mathcal{B} (\mathcal {H})$ are skew-Hermitian.

In this paper, we consider $C^*$ -algebras generated by Markov operators induced by a probabilistic dynamical system. It is shown that in some cases these algebras are isomorphic to Toeplitz, Cuntz, and Cuntz–Toeplitz algebras. By the probabilistic dynamic system, the Markov operator is constructed, and it is shown that it can be extended to a quantum channel on a convex set of positive operators with unit trace.

Let A {\mathcal {A}} be a von Neumann algebra and P A {\mathcal {P}}_{\mathcal {A}} the manifold of projections in A {\mathcal {A}} . There is a natural linear connection in P A {\mathcal {P}}_{\mathcal {A}} , which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of C n \mathbb {C}^n . In this paper we show that two projections p , q p,q can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of A {\mathcal {A}} ), if and only if p ∧ q ⊥ ∼ p ⊥ ∧ q , \begin{equation*} p\wedge q^\perp \sim p^\perp \wedge q, \end{equation*} where ∼ \sim stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal geodesic is unique if and only if p ∧ q ⊥ = p ⊥ ∧ q = 0 p\wedge q^\perp = p^\perp \wedge q=0 . If A {\mathcal {A}} is a finite factor, any pair of projections in the same connected component of P A {\mathcal {P}}_{\mathcal {A}} (i.e., with the same trace) can be joined by a minimal geodesic. We explore certain relations with Jones’ index theory for subfactors. For instance, it is shown that if N ⊂ M {\mathcal {N}}\subset {\mathcal {M}} are II 1 _1 factors with finite index [ M : N ] = t − 1 [{\mathcal {M}}:{\mathcal {N}}]={\mathbf {t}}^{-1} , then the geodesic distance d ( e N , e M ) d(e_{\mathcal {N}},e_{\mathcal {M}}) between the induced projections e N e_{\mathcal {N}} and e M e_{\mathcal {M}} is d ( e N , e M ) = arccos ⁡ ( t 1 / 2 ) d(e_{\mathcal {N}},e_{\mathcal {M}})=\arccos ({\mathbf {t}}^{1/2}) .

Rickart $$C^*$$ -algebras are unital and satisfy polar decomposition. We proved that if a unital $$C^*$$ -algebra $${\mathcal {A}}$$ satisfies polar decomposition and admits “good” faithful tracial states then $${\mathcal {A}}$$ is a Rickart $$C^*$$ -algebra. Via polar decomposition we characterized tracial states among all states on a Rickart $$C^*$$ -algebra. We presented the triangle inequality for Hermitian elements and traces on Rickart $$C^*$$ -algebra. For a block projection operator and a trace on a Rickart $$C^*$$ -algebra we proved a new inequality. As a corollary, we obtain a sharp estimate for a trace of the commutator of any Hermitian element and a projection. Also we give a characterization of traces in a wide class of weights on a von Neumann algebra.

Using the thermal-entangled state representations, we obtain the analytical evolution of quantum states simultaneously undergoing two kinds of quantum noises that can be represented as the infinitive Kraus operator-sum representation. We also prove the normalization and trace preservation of Kraus operators. Besides, we obtain the evolutions of Wigner distribution and photon number distribution of any quantum state when simultaneously passing through two kinds of quantum noises.

In this paper, we construct several infinite families of codes over the chain ring $$R={\mathbb {F}}_2[u]/\langle u^k\rangle $$ , i.e., $$R={\mathbb {F}}_2+u{\mathbb {F}}_2+\cdots +u^{k-1}{\mathbb {F}}_2, u^{k}=0$$ by employing simplicial complexes. When the simplicial complexes are all generated by a single maximal element, we compute the homogeneous weight distributions of these classes of codes. By the Gray map, we obtain that some classes of codes are minimal and some classes of these codes are distance optimal. The codewords of these codes are shown to be minimal for inclusion of supports, a fact favorable to an application to secret sharing schemes.

Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups [3,2]. In this article, we prove similar result on isolating the cuspidal spectrum in [3] for the function field case., The paper has been accepted for publication in SCIENCE CHINA Mathematics. A gap in the previous version is fixed

R.G. and Perez proved that under certain conditions the test ideal of a module closure agrees with the trace ideal of the module closure. We use this fact to compute the test ideals of various rings with respect to the closures coming from their indecomposable maximal Cohen–Macaulay modules. We also give an easier way to compute the test ideal of a hypersurface ring in three variables coming from a module with a particular type of matrix factorization.

In this paper, two classical inverse spectral problems are investigated, namely, the inverse second-order Sturm-Liouville problem and the inverse fourth-order Sturm-Liouville problem. Based on Lidskii’s theorem, we derive trace formulas showing relations between the unknown coefficients and eigenvalues explicitly for both problems. According to those trace formulas, two efficient algorithms are proposed to recover the symmetric potential from one spectrum for second-order Sturm-Liouville problem and two coefficients simultaneously from three spectra for fourth-order Sturm-Liouville problem, respectively. Numerical results are presented to illustrate the effectiveness of the proposed reconstruction algorithms.

Fabricating fine equiaxed grains without undesirable secondary phases is highly challenging for additively manufactured Ti−6Al−4V alloy. The reference amount of Ni addition, which can achieve grain refinement without secondary phase formation, is 0.9 wt. % based on Thermo-Calc calculation. The Ti−6Al−4V−0.9Ni alloy produced by laser-based directed energy deposition demonstrate refined microstructure and an ultrahigh yield strength (1309 MPa). A modified quantitative model is proposed to analyse the strengthening mechanism, and the results demonstrate that the yield strength increment is mainly ascribed to the refined α phase. This work can contribute to the development of customised titanium alloy using additive manufacturing.

For an elliptic curve $$E/{\mathbb {Q}}$$ , let $$a_p$$ denote the trace of its Frobenius endomorphism over $${\mathbb {F}}_p$$ , where p is a prime of good reduction for E. Hasse’s theorem asserts that $$|a_p| \le 2\sqrt{p}$$ . In this paper we establish average asymptotics for primes p for which $$a_p \in \left( 2\sqrt{p} - f(p), 2\sqrt{p}\right) $$ or $$a_p \in \left( c\sqrt{p} - f(p), c\sqrt{p}\right) $$ , where $$f(x) = o(\sqrt{x})$$ is a function satisfying mild growth conditions and $$0< c < 2$$ is a constant.

This paper studies and supplements the proofs of the properties of the Kronecker Product of two matrices of different orders. We observe the relation between the singular value decomposition of the matrices and their Kronecker product and the relationship between the determinant, the trace, the rank and the polynomial matrix of the Kronecker products. We also establish the best least square solutions of the Kronecker product system of equations by using modified QR-algorithm.

Let $${\mathscr {M}}$$ be a semi-finite von Neumann algebra with a faithful normal semi-finite trace $$\tau $$ and $$L^{p}({\mathscr {M}})$$ the non-commutative $$L^p$$ space associated with $$({\mathscr {M}},\tau )$$ . We extend star partial order $$\overset{*}{\le }$$ and diamond order $$\le ^{\diamond }$$ to $$L^{p}({\mathscr {M}})$$ and present some properties about several bounds of these two partial orders. We establish a result about the existence of the star infimum and supremum. We also prove that a subset with an upper bound must have a minimal upper bound in $$L^p({\mathscr {M}})$$ under the diamond order in the case of finite von Neumann algebra $${\mathscr {M}}$$ . However, we give an example and show that this result may fail if $${\mathscr {M}}$$ is not finite. Moreover, we characterize the forms of all norm closed hereditary subspaces in $$L^{p}({\mathscr {M}})$$ under these two partial orders.

Given a finite sequence of vectors in a finite dimensional Hilbert space, we present a method to obtain a tight frame by appending some vectors with prescribed norms. The approach, based on the alternating projection technique, is an iterative process. Von Neumann trace inequality and the polar decomposition of the matrix are employed to derive the method. The approach can be easily implemented and is numerically stable. The local convergence of the approach is achieved. Numerical examples are given to show the performance of the method.