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The p -ary linear code C k n , q is defined as the row space of the incidence matrix A of k -spaces and points of PG n , q . It is known that if q is square, a codeword of weight q k q + O q k - 1 exists that cannot be written as a linear combination of at most q rows of A . Over the past few decades, researchers have put a lot of effort towards proving that any codeword of smaller weight does meet this property. We show that if q ⩾ 32 is a composite prime power, every codeword of C k n , q up to weight O q k q is a linear combination of at most q rows of A . We also generalise this result to the codes C j , k n , q , which are defined as the p -ary row span of the incidence matrix of k-spaces and j-spaces, j < k . [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a k-algebra and this new "k-space" becomes a generalization of the corresponding Zariski space. We prove that a k-space is T0, quasi-compact, spectral, and connected. Moreover, we study continuous maps between such k-spaces. We conclude with a question about construction of a sheaf of k-spaces similar to affine schemes. [ABSTRACT FROM AUTHOR]

Liquid sulfur has been studied by density-functional based molecular dynamics simulations at different temperatures ranging from 400 up to 700 K across the well-documented λ transition. Structure models containing either a majority of Sn chains or S8 rings are considered and compared to experimental data from x-ray scattering. The comparison suggests a liquid structure of a majority of twofold sulfur at low temperature, dominated by S8 rings that open progressively upon temperature increase. Typical features associated with such rings are analyzed and indicate that they contribute to a specific third correlating distance in the pair correlation function and to a contribution at low wavevector k in the reciprocal space. The vibrational properties of liquid sulfur are also considered and indicate a contribution at 60 meV that is associated with both chains and rings, albeit the latter lead to a more intense peak at this wavenumber. The underlying network structure also impacts the dynamic properties of the melts which display enhanced dynamic heterogeneities when S8 rings are present. The analysis of the electronic Kohn–Sham energies shows insulating character with a gap of about ≃2.0 eV, albeit the presence of localized mid-gap states is acknowledged that can be associated, in part, with the presence of S6 rings. [ABSTRACT FROM AUTHOR]

Metadynamics (MetaD) is a useful technique to study rare events such as crystallization. It has been only recently applied to study nucleation and crystallization in glass-forming liquids such as silicates, but the optimal set of parameters to drive crystallization and obtain converged free energy surfaces is still unexplored. In this work, we systematically investigated the effects of the simulation conditions to efficiently study the thermodynamics and mechanism of crystallization in highly viscous systems. As a prototype system, we used fused silica, which easily crystallizes to β-cristobalite through MetaD simulations, owing to its simple microstructure. We investigated the influence of the height, width, and bias factor used to define the biasing Gaussian potential, as well as the effects of the temperature and system size on the results. Among these parameters, the bias factor and temperature seem to be most effective in sampling the free energy landscape of melt to crystal transition and reaching convergence more quickly. We also demonstrate that the temperature rescaling from T > Tm is a reliable approach to recover free energy surfaces below Tm, provided that the temperature gap is below 600 K and the configurational space has been properly sampled. Finally, albeit a complete crystallization is hard to achieve with large simulation boxes, these can be reliably and effectively exploited to study the first stages of nucleation. [ABSTRACT FROM AUTHOR]

The rotationally resolved infrared (IR) spectrum of the He– H 3 + complex has been measured in a cryogenic ion trap experiment at a nominal temperature of 4 K. Predissociation of the stored complex has been invoked by excitation of the degenerate ν2 mode of the H 3 + sub-unit using a pulsed optical parametric oscillator system. An assignment of the experimental spectrum became possible through one-to-one correlations with bands of the spectrum theoretically predicted in Paper I [Harding et al., J. Chem. Phys. 156, 144307 (2022)]. 19 bands have been assigned and analyzed, and the energy term diagram of the lower states of this floppy molecular complex has been derived from combination differences (CDs) in the experimental spectrum. Ground state combination differences (GSCDs) reveal a large part of the energy term diagram for the He– H 3 + complex in its vibrational ground state, v = 0. Experimental and theoretical term energies agree within experimental accuracy for the rotational fine structure associated with the total angular momentum quantum number J and the parity e/f as well as for the coarse spacing of the lowest K states of the complex. This favorable comparison shows that the potential energy surface (PES) calculated in Paper I is accurate. The barriers between the three equivalent global minima in this PES are relatively low and the He– H 3 + complex is extremely floppy, with nearly unhindered internal rotation of the H 3 + sub-unit. The resulting Coriolis interactions couple the internal and end-over-end rotation of the complex and contribute significantly to the energy terms. They are observed both in experiment and theory and are, e.g., the origin of different rotational constants for states of e and f parity. Also in this respect, experiment and theory agree very well. Despite the assignment and analysis of many bands of the extremely rich IR spectrum of He– H 3 + , higher levels of excitation, including the complex stretching mode, need further attention. [ABSTRACT FROM AUTHOR]

Spin wave dynamics form the foundation of spin-torque oscillator (STO) modulation. In addition to the uniform spin wave mode with wave vector k = 0, multiple spin wave modes with different wave vectors and frequencies coexist in the nanoscale STO. To characterize these spin wave modes and the interactions between them, the distribution and evolution of spin wave intensity in k space need to be investigated together with spin wave dispersion of the individual mode, stressing energy degeneracy. To this end, using micromagnetic simulation, we studied the dynamics of the discrete spin wave modes in a nanoscale STO with structure enhancing spin wave propagation. Simulation using the object oriented micromagnetic framework demonstrated that although they were generated with different spin currents, spin wave modes evolved similarly in k space, according to the wave vector. Furthermore, magnetization pinning at the corners of yttrium iron garnet led to two-magnon scattering. This interaction affects spin wave dynamics in two ways: multivalued dispersion occurs with two-magnon scattering and the uniform mode can become weaker than the near-uniform modes (spin wave modes with wave vector close to zero) in the strong excitation region. The latter phenomenon is supported by the results of studies on the spin wave dispersion of the individual mode, which demonstrate the energy degeneracy between the uniform mode and the near-uniform modes. [ABSTRACT FROM AUTHOR]

We establish conditions under which K‐spaces in the limiting real interpolation involving slowly varying functions can be described by means of J‐spaces and we also solve the reverse problem. To this end, we prove several versions of the fundamental lemma of the real interpolation theory. We apply our results to obtain density theorems for the corresponding limiting interpolation spaces. [ABSTRACT FROM AUTHOR]

Herein, the flatband of a W1N2 crystal is theoretically investigated. It is revealed that the flatband can be well-described by a tight-binding model of the N12 skeleton, where the dispersion of the flatband is governed by both the ppσ bonding strength (Vppσ) and the ppπ bonding strength (Vppπ) between the nearest-neighbor N atoms. It is also found that the proper strength of the ppπ bonding between neighboring N atoms plays a prime role in the formation of the flatband. In addition, when the compound is doped with holes, the electrons at the flatband are fully polarized, showing a ferromagnetic character. This behavior has a weak correlation with the on-site Coulomb interaction U. Moreover, several three-dimensional compounds possessing flatbands in the whole k space are predicted. [ABSTRACT FROM AUTHOR]

In this paper, we study the codes Ck(n, q) arising from the incidence of points and k-spaces in PG(n, q) over the field Fp, with q = ph, p prime. We classify all codewords of minimum weight of the dual code Ck(n, q)⊥ in case q ∊ {4,8}. This is equivalent to classifying the smallest sets of even type in PG(n, q) for q ∊{4,8}. We also provide shorter proofs for some already known results, namely, of the best known lower bound on the minimum weight of Ck(n, q)\bot for general values of q, and of the classification of all codewords of Cn-1(n, q) of weight up to 2qn-1. [ABSTRACT FROM AUTHOR]

The problem of finding a nonzero solution of a linear recurrence L y = 0 with polynomial coefficients where y has the form of a definite hypergeometric sum, related to the Inverse Creative Telescoping Problem of Chen and Kauers (2017, Sec. 8) , has now been open for three decades. Here we present an algorithm (implemented in a SageMath package) which, given such a recurrence and a quasi-triangular , shift-compatible factorial basis B = 〈 P k (n) 〉 k = 0 ∞ of the polynomial space K [ n ] over a field K of characteristic zero, computes a recurrence satisfied by the coefficient sequence c = 〈 c k 〉 k = 0 ∞ of the solution y n = ∑ k = 0 ∞ c k P k (n) (where, thanks to the quasi-triangularity of B , the sum on the right terminates for each n ∈ N). More generally, if B is m -sieved for some m ∈ N , our algorithm computes a system of m recurrences satisfied by the m -sections of the coefficient sequence c. If an explicit nonzero solution of this system can be found, we obtain an explicit nonzero solution of L y = 0. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of the Miller basis for S k (N) , the space of cusp forms of weight k and level N. We seek an easy criterion to determine whether or not the space S k (N) admits the Miller basis. In particular, we prove that if the level N is sufficiently composite, then S k (N) does not have the Miller basis. We also prove that there exists a constant w (N) such that S k (N) admits the Miller basis if and only if S k + w (N) (N) admits the Miller basis when k ≥ 4. [ABSTRACT FROM AUTHOR]

We present an analysis of Epoch of Reionization (EoR) data from Phase II of the Murchison Widefield Array using the simpleds delay spectrum pipeline. Prior work analysed the same observations using the fhd /εppsilon imaging pipeline, and so the present analysis represents the first time that both principal types of 21 cm cosmology power spectrum estimation approaches have been applied to the same data set. Our limits on the 21 cm power spectrum amplitude span a range in k space of |$|k| \lt 1 \, h_{100}\, {\rm Mpc}^{-1}$| with a lowest measurement of Δ2(k) ≤ 4.58 × 103 mK2 at |$k = 0.190\, h_{100}\, \rm {Mpc}^{-1}$| and z  = 7.14. In order to achieve these limits, we need to mitigate a previously unidentified common mode systematic in the data set. If not accounted for, this systematic introduces an overall negative bias that can make foreground contaminated measurements appear as stringent, noise-limited constraints on the 21 cm signal amplitude. The identification of this systematic highlights the risk in modelling systematics as positive-definite contributions to the power spectrum and in 'conservatively' interpreting all measurements as upper limits. [ABSTRACT FROM AUTHOR]

The two well-known numerical radius inequalities for the tensor product A ⊗ B acting on H ⊗ K , where A and B are bounded linear operators defined on complex Hilbert spaces H and K , respectively are 1 2 ‖ A ‖ ‖ B ‖ ≤ w (A ⊗ B) ≤ ‖ A ‖ ‖ B ‖ and w (A) w (B) ≤ w (A ⊗ B) ≤ min { w (A) ‖ B ‖ , w (B) ‖ A ‖ }. In this article, we develop new lower and upper bounds for the numerical radius w (A ⊗ B) of the tensor product A ⊗ B and study the equality conditions for those bounds. [ABSTRACT FROM AUTHOR]

Generally, the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models. C*- algebra is being continually used to explain a physical system in quantum field theory and statistical mechanics and has subsequently become an important area of research. The concept of a C*-algebra-valued metric space was introduced in 2014 to generalize the concept of metric space. In fact, It is a generalization by replacing the set of real numbers with a C*-algebra. After that, this line of research continued, where several fixed point results have been obtained in the framework of C*-algebra valued metric, as well as (more general) C*-algebra-valued b-metric spaces and C*-algebra-valued extended b-metric spaces. Very recently, based on the concept and properties of C* -algebras, we have studied the quasi-case of such spaces to give a more general notion of relaxing the triangular inequality in the asymmetric case. In this paper, we first introduce the concept of C*-algebra-valued quasi-controlled K -metric spaces and prove some fixed point theorems that remain valid in this setting. To support our main results, we also furnish some examples which demonstrate the utility of our main result. Finally, as an application, we use our results to prove the existence and uniqueness of the solution to a nonlinear stochastic integral equation. [ABSTRACT FROM AUTHOR]

We prove that the recently introduced spin Benjamin–Ono equation admits a Lax pair and deduce a family of conservation laws that allow proving global wellposedness in all Sobolev spaces H k for every integer k ≥ 2 . We also infer an additional family of matrix-valued conservation laws of which the previous family is just the traces. [ABSTRACT FROM AUTHOR]

We study the Noether-Lefschetz locus of the moduli space M of K 3 [ 2 ] -fourfolds with a polarization of degree 2. Following Hassett's work on cubic fourfolds, Debarre, Iliev, and Manivel have shown that the Noether-Lefschetz locus in M is a countable union of special divisors M d , where the discriminant d is a positive integer congruent to 0 , 2 , or 4 modulo 8. We compute the Kodaira dimensions of these special divisors for all but finitely many discriminants; in particular, we show that for d > 224 and for many other small values of d , the space M d is a variety of general type. [ABSTRACT FROM AUTHOR]

Conventional echo‐planar imaging (EPI) uses a radiofrequency pulse for excitation and a prolonged echo train to sample k space, while off‐resonance and T2* decay effects caused by magnetic susceptibility variation accumulate within each echo, leading to geometric distortion. Multishot EPI methods, which divide k space into segments, can shorten the effective echo spacing and reduce the distortion on EPI images. But multiple shots cost longer scan time and render susceptibility to motion. In this study, we propose a new "multishot" EPI method termed pseudo multishot EPI (pmsEPI), in which phase‐encoding lines are segmented as in multishot EPI but are collected within a single shot. With the magnetization divided into different pathways via interleaved excitation instead of refocusing in a single long echo train, the total phase error accumulation is reduced in each segmented acquisition, thereby improving distortion of the resultant EPI image. The performance of the pmsEPI method is demonstrated by phantom and in vivo brain experiments on a 3‐T scanner. The experimental results show that the distortion displacements of pmsEPI acquisition compared with conventional EPI decrease by 50% with two pseudo shots and 66% with three pseudo shots, validating the ability of the method to obtain images with reduced distortion in a single shot, although magnetization splitting may induce more than 40% SNR loss and minor artifacts. Specifically, the ability of pmsEPI in diffusion‐weighted imaging with different trajectory options is highlighted, and the flexibility is demonstrated in a single‐shot blip up and down acquisition. [ABSTRACT FROM AUTHOR]

We propose a dual-domain deep learning technique for accelerating compressed sensing magnetic resonance image reconstruction. An advanced convolutional neural network with residual connectivity and an attention mechanism was developed for frequency and image domains. First, the sensor domain subnetwork estimates the unmeasured frequencies of k-space to reduce aliasing artifacts. Second, the image domain subnetwork performs a pixel-wise operation to remove blur and noisy artifacts. The skip connections efficiently concatenate the feature maps to alleviate the vanishing gradient problem. An attention gate in each decoder layer enhances network generalizability and speeds up image reconstruction by eliminating irrelevant activations. The proposed technique reconstructs real-valued clinical images from sparsely sampled k-spaces that are identical to the reference images. The performance of this novel approach was compared with state-of-the-art direct mapping, single-domain, and multi-domain methods. With acceleration factors (AFs) of 4 and 5, our method improved the mean peak signal-to-noise ratio (PSNR) to 8.67 and 9.23, respectively, compared with the single-domain Unet model; similarly, our approach increased the average PSNR to 3.72 and 4.61, respectively, compared with the multi-domain W-net. Remarkably, using an AF of 6, it enhanced the PSNR by 9.87 ± 1.55 and 6.60 ± 0.38 compared with Unet and W-net, respectively. [ABSTRACT FROM AUTHOR]

We present a numerical implementation for the Virtual Element Method that incorporates high order spaces. We include all the required computations in order to assemble the stiffness and mass matrices, and right hand side. Convergence of the method is verified for different polygonal partitions. An specific mesh-free application that allows to approximate harmonic functions is discussed, which is based on high-order projections onto polynomial spaces of degree k; this approach only requires to solve a k(k − 1)/2 linear system, reducing significantly the number of operations compared to usual finite or virtual element methods, and can be modified for different virtual spaces and elliptic partial differential equations. [ABSTRACT FROM AUTHOR]

Sphere-bases are constructed for the ℤ 2 vector space formed by the k -dimensional subcomplexes, of n -simplex (or n -cube), for which every (k − 1) -face is contained in a positive even number of k -cells; addition is symmetric difference of the corresponding sets of k -cells. The bases consist of the boundaries of an algorithmically-specified family of k + 1 -simplexes or k + 1 -cubes. Properties of the bases are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we show that the mild solutions to the Navier–Stokes–Vlasov–Fokker–Planck system exist globally in time near a global Maxwellian, provided that we take small-amplitude initial data in the function space L k 1 L T ∞ L v 2 . As a product, we also get the exponential time decay rates for the solutions. Our analysis relies on the refined energy estimates and the low regularity function space L k 1 L T ∞ L v 2 introduced by the work in Duan et al. [Commun. Pure Appl. Math. 74(5), 932–1020 (2021)]. [ABSTRACT FROM AUTHOR]

The application of compressed sensing (CS)-enabled data reconstruction for accelerating magnetic resonance imaging (MRI) remains a challenging problem. This is due to the fact that the information lost in k-space from the acceleration mask makes it difficult to reconstruct an image similar to the quality of a fully sampled image. Multiple deep learning-based structures have been proposed for MRI reconstruction using CS, in both the k-space and image domains, and using unrolled optimization methods. However, the drawback of these structures is that they are not fully utilizing the information from both domains (k-space and image). Herein, we propose a deep learning-based attention hybrid variational network that performs learning in both the k-space and image domains. We evaluate our method on a well-known open-source MRI dataset (652 brain cases and 1172 knee cases) and a clinical MRI dataset of 243 patients diagnosed with strokes from our institution to demonstrate the performance of our network. Our model achieves an overall peak signal-to-noise ratio/structural similarity of 40.92 ± 0.29/0.9577 ± 0.0025 (fourfold) and 37.03 ± 0.25/0.9365 ± 0.0029 (eightfold) for the brain dataset, 31.09 ± 0.25/0.6901 ± 0.0094 (fourfold) and 29.49 ± 0.22/0.6197 ± 0.0106 (eightfold) for the knee dataset, and 36.32 ± 0.16/0.9199 ± 0.0029 (20-fold) and 33.70 ± 0.15/0.8882 ± 0.0035 (30-fold) for the stroke dataset. In addition to quantitative evaluation, we undertook a blinded comparison of image quality across networks performed by a subspecialty trained radiologist. Overall, we demonstrate that our network achieves a superior performance among others under multiple reconstruction tasks. [ABSTRACT FROM AUTHOR]

We formulate a precise conjecture about the size of the L ∞ -mass of the space of Jacobi forms on H n × C g × n of matrix index S of size g. This L ∞ -mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such n, g, S and prove the upper bound in the k aspect when n = 1 , g ≥ 1 . When n = 1 and g = 1 , we make a more refined study of the sizes of the index-(old and) new spaces, the latter via the Waldspurger's formula. Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central L-values L (1 / 2 , f ⊗ χ D) with f varying over newforms of level a prime p and even weight k as k , p → ∞ and D being (explicitly) polynomially bounded by k, p. Here χ D is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight k) is k 5 / 2 by three different methods (with or without the use of central L-values), and show that the size of their pullbacks to the diagonally embedded H × H is k 2 . In an appendix, the same question is answered for the pullbacks of the whole space S k 2 , the size here being k 3 . [ABSTRACT FROM AUTHOR]

Heller (2021) generalized quasi‐ordinal knowledge spaces to polytomous items. Inspired by this paper, we propose CD‐polytomous knowledge space and its polytomous surmise system. A Galois connection is established between the collection K of all polytomous knowledge structures and the collection F1 of particular polytomous attribute functions. The closed elements of the Galois connection are CD‐polytomous knowledge spaces in K and polytomous surmise functions in F1, respectively. With the help of these, this paper provides a characterization of the polytomous knowledge structure corresponding to the polytomous surmise function that is weakly factorial. Based on the finite sets of items and response values, these results generalize the previous approaches for polytomous knowledge spaces. [ABSTRACT FROM AUTHOR]

If T is a bounded linear operator on a Hilbert space H and V is a given linear isometry on a Hilbert space K , we present necessary and sufficient conditions on T in order to ensure the existence of a linear isometry π : H → K such that π T = V * π (i.e., (π , V *) extends T). We parametrize the set of all solutions π of this equation. We show, for example, that for a given unitary operator U on a Hilbert space E and for the multiplication operator by the independent variable M z on the Hardy space H D 2 (D) , there exists an isometric operator π : H → E ⊕ H D 2 (D) such that (π , (U ⊕ M z) *) extends T if and only if T is a contraction, the defect index δ T ≤ dim D and, for some Y : A T → E , (Y , U *) extends the isometric operator A T 1 / 2 h ↦ A T 1 / 2 T h on the space A T = A T H ¯ , where A T is the asymptotic limit associated with T. We also prove that if T is isometric and V is unitary, there exists an isometric operator π : H → K such that (π , V) extends T if and only if (a) the spectral measures of the unitary part of T (in its Wold decomposition) and the restriction of V to one of its reducing subspaces K 0 possess identical multiplicity functions and (b) dim (ker T *) = dim (K 1 ⊖ V K 1) for a certain subspace K 1 of K that contains K 0 and is invariant under V. The precise form of π , in each situation, and characterizations of the minimality conditions are also included. Several examples are given for illustrative purposes. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the convergence theorems in CAT (κ) spaces with k > 0 for total asymptotically nonexpansive mappings which are essentially wider than nonexpansive mappings, asymptotically nonexpansive mapping, and asymptotically nonexpansive mappings in the intermediate sense. Our results generalize, unify, and improve several comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

We study two topologies $ \tau_{KR} $ and $ \tau_K $ on the space of measures on a completely regular space generated by Kantorovich–Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich–Rubinshtein topology $ \tau_{KR} $ coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures, some $ s $-convex measures and stable measures weak convergence implies convergence in the Kantorovich topology. We also obtain an efficiently verified condition for convergence of the barycenters of Radon measures from a sequence or net weakly converging on a locally convex space. As an application it is shown that for weakly convergent logarithmically concave measures and stable measures convergence of their barycenters holds without additional conditions. The same is true for measures given by polynomial densities of a fixed degree with respect to logarithmically concave measures. [ABSTRACT FROM AUTHOR]

Motion artifacts deteriorate the quality of magnetic resonance (MR) images. This study proposes a new method to detect phase-encoding (PE) lines corrupted by motion and remove motion artifacts in MR images. 67 cases containing 8710 slices of axial T2-weighted images from the IXI public dataset were split into three datasets, i.e., training (50 cases/6500 slices), validation (5/650), and test (12/1560) sets. First, motion-corrupted k-spaces and images were simulated using a pseudo-random sampling order and random motion tracks. A convolutional neural network (CNN) model was trained to filter the motion-corrupted images. Then, the k-space of the filtered image was compared with the motion-corrupted k-space line-by-line, to detect the PE lines affected by motion. Finally, the unaffected PE lines were used to reconstruct the final image using compressed sensing (CS). For the simulated images with 35%, 40%, 45%, and 50% unaffected PE lines, the mean peak signal-to-noise ratio (PSNRs) of resulting images (mean±standard deviation) were 36.129±3.678, 38.646±3.526, 40.426±3.223, and 41.510±3.167, respectively, and the mean structural similarity (SSIMs) were 0.950±0.046, 0.964±0.035, 0.975±0.025, and 0.979±0.023, respectively. For images with more than 35% PE lines unaffected by motion, images reconstructed with proposed algorithm exhibited better quality than those images reconstructed with CS using 35% under-sampled data (PSNR 37.678±3.261, SSIM 0.964±0.028). It was proved that deep learning and k-space analysis can detect the k-space PE lines affected by motion and CS can be used to reconstruct images from unaffected data, effectively alleviating the motion artifacts. [ABSTRACT FROM AUTHOR]

We study concrete endofunctors of the category of convergence spaces and continuous maps that send initial maps to initial maps or final maps to final maps. The former phenomenon turns out to be fairly common while the latter is rare. In particular, it is shown that the pretopological modification is the coarsest hereditary modifier finer than the topological modifier and this is applied to give a structural interpretation of the role of Fréchet-Urysohn spaces with respect to sequential spaces and of k¹ -spaces with respect to k-spaces. [ABSTRACT FROM AUTHOR]

In this paper, we shall study the polynomial approximation in a more general setting namely to consider the Orlicz-Sobolev spaces W k L M (Ω). We study the local and global interpolation estimate and we will show the finite element error estimate for a nonlinear elliptic problem where we establish a generalization of Cea's Theorem and we prove the modular convergence of the gradient, then we present the existence result and its proof. [ABSTRACT FROM AUTHOR]

Let X be a linear space over K ∈ { R , C } , Y be a real or complex Banach space and f : X n → Y . With some fixed a j i , C i 1 ... i n ∈ K ( j ∈ { 1 , ... , n } , i , i k ∈ { 1 , 2 } , k ∈ { 1 , ... , n } ), we study, using the direct and the fixed point methods, the stability and the general stability of the equation f (a 11 x 11 + a 12 x 12 , ... , a n 1 x n 1 + a n 2 x n 2) = ∑ 1 ≤ i 1 , ... , i n ≤ 2 C i 1 ... i n f (x 1 i 1 , ... , x n i n ) , for all x j i j ∈ X ( j ∈ { 1 , ... , n } , i j ∈ { 1 , 2 } ). Our paper generalizes several known results, among others concerning equations with symmetric coefficients, such as the multi-Cauchy equation or the multi-Jensen equation as well as the multi-Cauchy–Jensen equation. We also prove the hyperstability of the above equation in m-normed spaces with m ≥ 2 . [ABSTRACT FROM AUTHOR]

In this article, we analyse maximal sets of k-spaces, in PG(n, q) and AG(n, q) with n ≥ 2k + t + 3, that pairwise meet in at least a t-space. It is known that for both PG(n, q) and AG(n, q), the largest example is a t-pencil, i.e. the set of all k-spaces containing a fixed t-space. In this paper, we analyse the structure of the second largest maximal example in both PG(n, q) and AG(n, q). [ABSTRACT FROM AUTHOR]

Let p ∈ [ 1 , ∞ ] , q ∈ [ 1 , ∞) , s ∈ Z + : = N ∪ { 0 } , α ∈ R , and β ∈ (0 , 1) . In this article, the authors first find a reasonable version I ~ β of the (generalized) fractional integral I β on the special John–Nirenberg–Campanato space via congruent cubes, J N (p , q , s) α con (R n) , which coincides with the Campanato space C α , q , s (R n) when p = ∞ . To this end, the authors introduce the vanishing moments up to order s of I β . Then the authors prove that I ~ β is bounded from J N (p , q , s) α con (R n) to J N (p , q , s) α + β / n con (R n) if and only if I β has the vanishing moments up to order s. The obtained result is new even when p = ∞ and s ∈ N , namely, the Campanato space. Moreover, the authors show that I β can be extended to a unique continuous linear operator from the Hardy-kind space H K (p , q , s) α + β / n con (R n) , the predual of J N (p ′ , q ′ , s) α + β / n con (R n) with 1 p + 1 p ′ = 1 = 1 q + 1 q ′ , to H K (p , q , s) α con (R n) if and only if I β has the vanishing moments up to order s. The proof of the latter boundedness strongly depends on the dual relation (H K (p , q , s) α con (R n)) ∗ = J N (p ′ , q ′ , s) α con (R n) , the good properties of molecules of H K (p , q , s) α con (R n) , and the crucial criterion for the boundedness of linear operators on H K (p , q , s) α con (R n) . [ABSTRACT FROM AUTHOR]

We provide methods to construct explicit examples of K 3 surfaces. This leads to unirational constructions of Noether–Lefschetz divisors inside the moduli space of K 3 surfaces of genus g. We implement Mukai's unirationality construction of the moduli spaces of K 3 surfaces of genus g ∈ { 6 , ... , 10 , 12 } , and we also present a new constructive proof of the unirationality of the moduli space of K 3 surfaces of genus 11. Furthermore, we show the existence of three unirational hypersurfaces in any moduli space of K 3 surfaces of genus g. [ABSTRACT FROM AUTHOR]

With the development of the Internet of Things (IoT) and its applications, a large amount of data is generated regularly. If this information is used by malicious attackers, it will be a great disaster for the relevant users. In this regard, this article focuses on the user's identity privacy issues involved in the IoT. By protecting the user's identity privacy, the attacker cannot associate the obtained data with the user's real identity, and so achieve the purpose of protecting the user. This article uses the features of blockchain that cannot be tampered with nor forged to strengthen the reliability of the system. The proposed scheme saves the transaction information of user information through the Hyperledger and uses the ring signature method to obscure the real identity. A key generator is used to generate system public parameters and ring membership information required for signature. Users can use this information to hide their identity in a ring group of n users so that other users can only guess the true identity of the user with a probability of 1/n. Additionally, the method of aggregated signature is used to shorten the time and space required for k signature verification to 1/k, which greatly improves the efficiency. Finally, this article also uses an accountability mechanism to punish some attackers who attempt to waste system resources by revealing the real identity of the attacker and refusing to serve him. In this paper, GO language is used to write chain code to realize the proposed algorithm, and a prototype system is built through HyperLeger Fabric blockchain network, and the prototype system is verified by experiment. The correctness and efficiency of the above scheme are also proved through theoretical analysis and experiments. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the mixed central bounded mean oscillation space CMO ˙ q → (R n) , which is a new central version of the bounded mean oscillation spaces BMO. The pre-dual of CMO ˙ q → (R n) is proved to be the mixed Herz–Hardy space HA q → (R n). Moreover, we also give a characterization of CMO ˙ q → (R n) via the boundedness of the commutators of n-dimensional Hardy operator and its dual operator on mixed Herz spaces K ˙ q → α , p (R n). [ABSTRACT FROM AUTHOR]

In this paper, we give the nth derivertive criterion for functions belonging to function spaces H K 2 . Furthermore, some sufficient conditions for coefficients of the linear differential equation f (k) + A k − 1 (z) f (k − 1) + ⋯ + A 1 (z) f ′ + A 0 (z) f = A k (z) are found such that all solutions belong to the H K 2 spaces, where A j (z) are analytic functions in D , j = 0 , ... , k. [ABSTRACT FROM AUTHOR]

Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with k = 2 , 3 distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds. [ABSTRACT FROM AUTHOR]

Fully-dense ZrB 2 and HfB 2 composites were elaborated by Spark Plasma Sintering with 20 vol% TaSi 2 to replace SiC in the aim of improving their oxidation resistance. This study follows the one carried out on (Zr/Hf)B 2 –SiC systems in the same oxidation conditions. The oxidation behavior of ZrB 2 –20%TaSi 2 and HfB 2 -20%TaSi 2 in air plasma conditions, at 1000 Pa total pressure and from 1950 K up to 2600 K, was studied using the MESOX facility implemented at the focus of the 6 kW Odeillo solar furnace. The mass variation of the samples during an oxidation duration of 300 s on a temperature plateau was followed. A four step oxidation mechanism is proposed. A good oxidation behavior, with a slight mass gain, has been identified up to 2170 K due to the formation of a refractory mixed oxide (Hf/Zr) 6 Ta 2 O 17 at the surface of the samples. An improvement of the oxidation resistance by 200 K is obtained compared to the composites with SiC as additional compound. Moreover, as emissivity is an important parameter for space applications and that oxidation plays a huge role on the evolution of the emissivity, the normal spectral emissivity was measured on several pre-oxidized samples using atomic oxygen (representative surfaces) and the normal total emissivity was calculated. Finally, for both the compositions, the emissivity is comprised between 0.80 and 0.90 in the temperature range 1300–2200 K. [ABSTRACT FROM AUTHOR]

We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of Berkovich non-archimedean analytic spaces. The motivation for our construction is Tony Yue Yu's non-archimedean enumerative geometry in Gromov—Witten theory. The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson—Thomas invariants. In this paper we give the moduli construction over a non-archimedean field K . We use the machinery of formal schemes, that is, we define and construct the formal moduli stack of (semi)-stable coherent sheaves over a discrete valuation ring R, and taking generic fiber we get the non-archimedean analytic moduli space of semistable coherent sheaves over the fractional non-archimedean field K . We generalize Joyce's d-critical scheme structure in [37] or Kiem—Li's virtual critical manifolds in [38] to the world of formal schemes, and Berkovich non-archimedean analytic spaces. As an application, we provide a proof for the motivic localization formula for a d-critical non-archimedean K -analytic space using global motive of vanishing cycles and motivic integration on oriented formal d-critical schemes. This generalizes Maulik's motivic localization formula for the motivic Donaldson—Thomas invariants. [ABSTRACT FROM AUTHOR]

We consider the initial value problem for the Landau–Lifshitz equation with a helicity term (chiral interaction term), which arises from the Dzyaloshinskii–Moriya interaction energy. We prove that it is well-posed locally in-time in the space k ̄ + H s for s ≥ 3 with s ∈ Z and k ̄ = (0 , 0 , 1) t . We also show that if we further assume that the solution is homotopic to constant maps, then local well-posedness holds in the space k ̄ + H s for s > 2 with s ∈ R. Our proof is based on two different approaches: One is the geometric energy method by McGahagan and the other is via the modified Schrödinger map equation. In the present analysis, we exploit a special structure of the helicity term, which enables us to overcome the difficulty in the quadratic derivative nonlinearity of the helicity term. [ABSTRACT FROM AUTHOR]

Certain bounded linear operators T on a complex Hilbert space H which have 2-isometric liftings S on another space K ⊃ H are being investigated. We refer also to a more special type of liftings, as well as to those which additionally meet the condition S ∗SH ⊂ H. Furthermore we describe other types of dilations for T, which are close to 2-isometries. Among these we refer to expansive (concave) operators and also to Brownian unitary dilations. Different matrix representations for such operators are obtained, where matrix entries involve contractive operators. [ABSTRACT FROM AUTHOR]

We obtain the necessary and sufficient conditions for the bilinear Dunkl–Hausdorff operators on the product Lebesgue spaces and (central) Morrey spaces. We also obtain the sharp constant for the bilinear Dunkl–Hausdorff operators on the weighted Lebesgue spaces L k p (1 ≤ p < ∞) when the matrix A in the definition of operators is a scalar matrix. [ABSTRACT FROM AUTHOR]

Given an affine algebra R = P / I , where P = K [ x 1 , ... , x n ] is a polynomial ring over a field K and I is an ideal in  P , we study re-embeddings of the affine scheme Spec (R) , i.e. presentations R ≅ P ′ / I ′ such that P ′ is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials f i in the ideal I which are coherently separating in the sense that they are of the form f i = z i − g i with an indeterminate z i which divides neither a term in the support of g i nor in the support of f j for j ≠ i. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of I. The dimension of the cotangent space of  R at a K -linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of R and found an optimal re-embedding. [ABSTRACT FROM AUTHOR]

In this paper, using some properties about Toeplitz kernels, we present some results about finite-rank properties of the commutator [ A f , A g ] . Firstly, we show that [ A B n , A v ∗ ] must have a finite rank on the model space K u 2 , where B n is a finite Blaschke product and v is an inner function. Next, we present that when ker T u ¯ B n is an invariant subspace of T ϕ ∗ , then [ A B n , A ϕ ∗ ] has a finite rank on K u 2 for ϕ ∈ H ∞ . Finally, we prove that [ A B n , A ϕ ∗ ] must have a finite rank on K u 2 when u = B n u 1 for an inner function u 1 . [ABSTRACT FROM AUTHOR]

A t-intersecting constant dimension subspace code C is a set of k-dimensional subspaces in a projective space PG (n , q) , where distinct subspaces intersect in exactly a t-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same t-space. The sunflower bound states that such a code is a sunflower if | C | > q k + 1 - q t + 1 q - 1 2 + q k + 1 - q t + 1 q - 1 + 1 . In this article we will look at the case t = 0 and we will improve this bound for q ≥ 9 : a set S of k-spaces in PG (n , q) , q ≥ 9 , pairwise intersecting in a point is a sunflower if | S | > 2 q 6 + 4 q 3 - 5 q q k + 1 - 1 q - 1 2 . [ABSTRACT FROM AUTHOR]

Let G be a finite group with given subgroups H and K. Let π be an irreducible complex representation of G such that its space of H -invariant vectors as well as the space of K -invariant vectors are both one dimensional. Let v H (resp. v K) denote an H -invariant (resp. K -invariant) vector of unit norm in a given G -invariant inner product 〈 , 〉 π on π. Our interest is in computing the square of the absolute value of 〈 v H , v K 〉 π. This is the correlation constant c (π ; H , K) defined by Gross [3]. In this paper, we study this question for G = GL 2 (F q) , where F q is the finite field of q = p m elements of odd characteristic p , H is its split torus and K is a non-split torus. The first main theorem of this paper gives an explicit formula for | 〈 v H , v K 〉 π | 2 modulo p. The key idea here is to analyse the mod p reduction of π. The second main theorem relates the behaviour of 〈 v H , v K 〉 π under Shintani base change and gives a sufficient condition for 〈 v H , v K 〉 π to vanish for an irreducible representation π = BC (τ) of PGL 2 (E) , in terms of the epsilon factor of the base changing representation τ of PGL 2 (F) , where E / F is a finite extension of finite fields. [ABSTRACT FROM AUTHOR]

While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space ℋ k , g parametrizing smooth degree k, genus g covers of ℙ 1 . Let k = 3 , 4 , 5 . We prove that the rational Chow rings of ℋ k , g stabilize in a suitable sense as g tends to infinity. In the case k = 3 , we completely determine the Chow rings for all g. We also prove that the rational Chow groups of the simply branched Hurwitz space ℋ k , g s ⊂ ℋ k , g are zero in codimension up to roughly g k . In [S. Canning and H. Larson, The Chow rings of the moduli spaces of curves of genus 7 , 8 and 9, preprint 2021, arXiv:2104.05820], results developed in this paper are used to prove that the Chow rings of ℳ 7 , ℳ 8 , and ℳ 9 are tautological. [ABSTRACT FROM AUTHOR]

In order to solve the problem that it is difficult to get out of the local optimum to achieve further optimization in large-scale dense Distributed Constraint Optimization Problems (DCOPs). This paper proposed a local parallel search framework (LPOS) for solving algorithms of DCOPs. In LPOS, the Agent searches all the value assignments of its local neighbors in parallel according to its current value assignment to further expand the search of solution space, so as to avoid the algorithm falling into local optimum prematurely. In order to ensure the convergence of the algorithm, this paper designed an adaptive equilibrium factor K to balance the exploration and exploitation ability of the DCOPs solving algorithms. At the theoretical analysis, it proved that the LPOS can expand the search of solution space and the K can achieve the balance. The experimental results demonstrate that the performance of the proposed LPOS is better than the-state-of-the-art algorithms in both low density and high density. Specifically, LPOS significantly superior to the competitors when solving high density DCOPs. [ABSTRACT FROM AUTHOR]

Two–dimensional (2D) transition metal dichalcogenides endow individually addressable valleys in momentum space at the K and K' points in the first Brillouin zone due to the breaking of inversion symmetry and the effect of spin–orbit coupling. However, the application of 2H–VS2 monolayer in valleytronics is limited due to the valence band maximum (VBM) located at the Γ point. Here, by involving the 2D ferroelectric (FE) CuInP2Se6 (CIPSe), the ferrovalley polarization, electronic structure, and magnetic properties of 2D 2H–VS2/CIPSe heterostructures with different stacking patterns and FE polarizations have been investigated by using first–principles calculations. It is found that, for the energetically favorable AB–stacking pattern, the valley polarization is preserved when the FE polarization of CIPSe is upwards (CIPSe↑) or downwards (CIPSe↓) with the splitting energies slightly larger or smaller compared with that of the pure 2H–VS2. It is intriguing that, for the FE CIPSe↑ case, the VBM is expected to pass through the Fermi energy level, which can be eventually achieved by applying biaxial strain and thus the valleytronic nature is turned off; however, for the CIPSe↓ situation, the heterostructure basically remains semiconducting even under biaxial strains. Therefore, with the influence of proper strains, the FE polar reversal of CIPSe can be used as a switchable on/off to regulate the valley polarization in VS2. These results not only demonstrate that 2H–VS2/CIPSe heterostructures are promising potential candidates in valleytronics, but also shed some light on developing practical applications of valleytronic technology. [ABSTRACT FROM AUTHOR]