Dimensional Complex Research Articles

The scalar curvature of CR submanifold of maximal CR dimension with a conformal change

In this paper, we investigate the codimension reduction theorem for an n-dimensional submanifold of an (n + p)-dimensional manifold (M, ~g) with a conformal change g = exp(–f)g where g denotes the Fubini-study metric on (n + p)-dimensional complex projective space P n+p/2 (C). Moreover, we tend to calculate the scalar curvature of an n-dimensional CR submanifold of maximal CR dimension of (M; g) and achieve the sufficient conditions for the existence of a totally geodesic submanifold M0 that includes M.

Inferring the interaction rules of complex systems with graph neural networks and approximate Bayesian computation.

Inferring the underlying processes that drive collective behaviour in biological and social systems is a significant statistical and computational challenge. While simulation models have been successful in qualitatively capturing many of the phenomena observed in these systems in a variety of domains, formally fitting these models to data remains intractable. Recently, approximate Bayesian computation (ABC) has been shown to be an effective approach to inference if the likelihood function for a model is unavailable. However, a key difficulty in successfully implementing ABC lies with the design, selection and weighting of appropriate summary statistics, a challenge that is especially acute when modelling high dimensional complex systems. In this work, we combine a Gaussian process accelerated ABC method with the automatic learning of summary statistics via graph neural networks. Our approach bypasses the need to design a model-specific set of summary statistics for inference. Instead, we encode relational inductive biases into a neural network using a graph embedding and then extract summary statistics automatically from simulation data. To evaluate our framework, we use a model of collective animal movement as a test bed and compare our method to a standard summary statistics approach and a linear regression-based algorithm.

Linear bounds for constants in Gromov’s systolic inequality and related results

Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $({r\over c(n)})^n$, then there exists a continuous map from $M^n$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$ in $M^n$. It was previously proven by Gromov that this result implies two by now famous Gromov's inequalities: $Fill Rad(M^n)\leq c(n)vol(M^n)^{1\over n}$ and, if $M^n$ is essential, then also $sys_1(M^n)\leq 6c(n)vol(M^n)^{1\over n}$ with the same constant $c(n)$. Here $sys_1(M^n)$ denotes the length of a shortest non-contractible closed curve in $M^n$. We prove that these results hold with $c(n)=({n!\over 2})^{1\over n}\leq {n\over 2}$. We demonstrate that for essential Riemannian manifolds $sys_1(M^n) \leq n\ vol^{1\over n}(M^n)$. All previously known upper bounds for $c(n)$ were exponential in $n$. Moreover, we present a qualitative improvement: In Guth's theorem the assumption that the volume of every metric ball of radius $r$ is less than $({r\over c(n)})^n$ can be replaced by a weaker assumption that for every point $x\in M^n$ there exists a positive $\rho(x)\leq r$ such that the volume of the metric ball of radius $\rho(x)$ centered at $x$ is less than $({\rho(x)\over c(n)})^n$ (for $c(n)=({n!\over 2})^{1\over n}$). Also, if $X$ is a boundedly compact metric space such that for some $r>0$ and an integer $n\geq 1$ the $n$-dimensional Hausdorff content of each metric ball of radius $r$ in $X$ is less than $({r\over 4n})^n$, then there exists a continuous map from $X$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$.

Codisk-cyclic Mixing Operators

Let be an infinite dimensional separable complex Hilbert space, and be a bounded linear operator. is called codisk mixing operator, - mixing operator, if for any non-empty open subsets of , there are and such that for all . In this paper, we studied a necessarily and sufficiently conditions of - mixing operators, and discused the direct sum of two - mixing operators.

The Polynomial Modules over Quantum Group Uq(sl3)

Let [Formula: see text] be a finite dimensional complex simple Lie algebra with Cartan subalgebra [Formula: see text]. Then [Formula: see text] has a [Formula: see text]-module structure if and only if [Formula: see text] is of type [Formula: see text] or of type [Formula: see text]; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over [Formula: see text] have been classified. In this paper, we prove that the quantum group [Formula: see text] has no rank one polynomial module.

Experimental simulation of volumetric compacts formation from spherical waxy elements

The growth in metal intensity of industrial production and the volume of consumption of finished metal products determine the relevance of development and research of energy efficient technological processes aimed at reducing costs by reducing the number of operations while maintaining product performance. In mechanical engineering, the problem of obtaining blanks with increased dimensional and geometric accuracy and complex configuration is solved by using a common method of investment casting. Expansion of the use of such technological approach to produce blanks in mechanical engineering is hindered by a number of physical phenomena associated with the thermal expansion of investment and ceramic materials, which leads to an increase in the product final cost. A significant number of defect-forming factors can be eliminated by applying an innovative solution consisting in the formation of porous removable models by compacting mixtures based on waxy materials. This solves the problem of material shrinkage and increases the crack resistance of ceramic molds, which significantly reduces the share of machining in the overall volume of technological operations. Technical tests of the new method have revealed the reason why the machining of castings cannot be completely eliminated at present. The problem mainly lies in elastic response of compacted material of the model mixture, which, in some cases, affects the increase in the compacts size. This paper considers the effect of initial packing of spherical-shaped elements simulating one- and two-component model mixtures on the stress-strain state of a powder body subjected to unilateral compaction in a rigid cylindrical matrix to technologically justified density values. The results of the experiment are presented in the form of stress-strain relations. Preferable conditions of compact formation with minimal values of elastic response of the compacted material are considered.

On the classification of 3-dimensional complex hom-Lie algebras

Let hom-Ln(C) be the algebraic set of complex n-dimensional hom-Lie algebras. The group GL(n,C) acts on it via change of basis. We classify hom-Lie structures with nilpotent twisting map on 3-dimensional complex Lie algebras, up to isomorphism, and we give the classification of orbit closures in such family. For this purpose, we introduce some invariants of hom-Lie algebras to study both problems. The ideas and techniques presented here can be easily extrapolated to study similar problems on other linear algebraic structures.

Bayesian inference of axisymmetric plasma equilibrium

We present a Bayesian method for inferring axisymmetric plasma equilibria from the magnetic field and plasma pressure measurements. The method calculates all possible solutions for plasma current and pressure distributions consistent with the measurements and magnetohydrodynamic (MHD) force balance. Toroidal plasma current and magnetic field coils are modelled as a set of axisymmetric current-carrying solid beams. The other parameters such as plasma pressure and poloidal current flux are given as a function of poloidal magnetic flux, which is determined given a 2D current distribution. Plasma pressure and poloidal current flux profiles are modelled as Gaussian processes whose smoothness is optimally chosen based on the principle of Occam’s razor. To find equilibrium solutions, we introduce an MHD force balance constraint at every plasma current beam as a part of the prior knowledge. Given all these physical quantities, predictions calculated by the predictive (forward) models for diagnostics are compared to the observations. The high dimensional complex posterior probability distribution is explored by a new algorithm based on the Gibbs sampling scheme.

Mathematical model development and simplification for multi-route complex chemical mechanism

A complete inventory and a clearly defined structure of any complex reaction mechanism are difficult to obtain. This paper addresses a complex catalytic cycle of dehydrogenation of butane in theoretical and computational beliefs. Mathematically, simplification is done by “model reduction,” that is, the rigorous way of approximating and representing a complex model in simplified form. Firstly, a simple single-route mechanism has been considered then a complex system is discussed in detail. The most practical and helpful way to represent a complicated system is using a simple model so that the model may be improved and transformed into a more accurate one that can be solved numerically. To reduce the complexity of a higher dimensional complex system, need to be a split reaction mechanism in different available routes. With the help of the concept of the multi-route model, we can reduce a complicated higher-dimensional system to a low-dimensional manifold and investigate each route independently. For model development and standardization, sensitivity analysis is a powerful tool used in this paper. The influence of parameters on target output is discussed through graphical results depicted in the symbology toolbox in MATLAB.

Rogue waves, modulation instability of the (2+1)-dimensional complex modified Korteweg-de Vries equation on the periodic background

In this paper, we construct rogue periodic wave solutions (rogue wave solutions under the Jacobian elliptic dn- and cn-periodic waves background) and the modulation instability of a (2+1)-dimensional complex modified Korteweg–de Vries (CMKdV) equation. Under the linear stability analysis, we reveal that modulation instability is the condition for the existence of rogue waves. Next, we derive the expression of eigenvalues and corresponding squared periodic eigenfunctions by the method of nonlinearization of the Lax pair. Then, we construct the periodic and non-periodic solution of the Lax pair. At last, we present the rogue periodic wave solutions of the (2+1)-dimensional CMKdV equation by the Darboux transformation (DT) method, and analyze the non-linear dynamics in some figures.

Note on boundedness of the $L$-index in the direction of the composition of slice entire functions

We study a composition of two functions belonging to a class of slice holomorphic functions in the whole $n$-dimensional complex space. The slice holomorphy in the space means that for some fixed direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ and for every point $z^0\in\mathbb{C}^n$ the function is holomorphic on its restriction on the slice $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}.$ An additional assumption on joint continuity for these functions allows to construct an analog of theory of entire functions having bounded index. The analog is applicable to study properties of slice holomorphic solutions of directional differential equations, describe local behavior and value distribution.In particular, we found conditions providing boundedness of $L$-index in the direction $\mathbf{b}$ for a function $f(\underbrace{\Phi(z),\ldots,\Phi(z)}_{m\text{ times}}),$where $f: \mathbb{C}^n\to\mathbb{C}$ is a slice entire function, $\Phi: \mathbb{C}^n\to\mathbb{C}$ is a slice entire function,${L}: \mathbb{C}^n\to\mathbb{R}_+$ is a continuous function.The obtained results are also new in one-dimensional case, i.e. for $n=1,$ $m=1.$ They are deduced using new approach in this area analog of logarithmic criterion.For a class of nonvanishing outer functions in the composition the sufficient conditions obtained by logarithmic criterion are weaker than the conditions by the Hayman theorem.

Positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg–de Vries equations

Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of (2+1)-dimensional equations are decomposed into multi-solitons as well as the (1+1)-dimensions. Moreover, the interactions between positon and soliton are analyzed. In addition, the hybrid solutions of b-positon and breather are obtained using the plane wave seed solution, and their evolutions with time are discussed.

Targeted Treatment of Inappropriate Sinoatrial Node Tachycardia Based on Electrophysiological and Structural Mechanisms

The purpose of this review is to determine the causal mechanisms and treatment of inappropriate sinoatrial tachycardia (IST), defined as a non-physiological elevation in resting heart rate. IST is defined as a resting daytime sinus rate >100 beats/minute and an average 24-hour heart rate >90 beats/minute. Potential causal mechanisms include sympathetic receptor hypersensitivity, blunted parasympathetic tone, or enhanced intrinsic automaticity within the sinoatrial node (SAN) pacemaker-conduction complex. These anomalies may coexist in the same patient. Recent ex-vivo near-infrared transmural optical imaging of the SAN in human and animal hearts provides important insights into the functional and molecular features of this complex structure. In particular, it reveals the existence of preferential sinoatrial conduction pathways that ensure robust SAN activation with electrical conduction. The mechanism of IST is debated because even high-resolution electroanatomical mapping approaches cannot reveal intramural conduction in the 3-dimensional SAN complex. It may be secondary to enhanced automaticity, intranodal re-entry, or sinoatrial conduction pathway re-entry. Different pharmacological approaches can target these mechanisms. Long-acting β blockers in IST can act on both primarily increased automaticity and dysregulated autonomic system. Ivabradine targets sources of increased SAN automaticity. Conventional or hybrid ablation may target all the described abnormalities. This review provides a state-of-the-art overview of putative IST mechanisms. In conclusion, based on current knowledge, pharmacological and ablation approaches for IST, including the novel hybrid SAN sparing ablation, are discussed.

On Frobenius-Perron dimension

We propose a notion of Frobenius-Perron dimension for certain free Z \mathbb {Z} -modules of infinite rank and compute it for the Z \mathbb {Z} -modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. The definition of Frobenius-Perron dimension that we are introducing naturally generalizes the well-known Frobenius-Perron dimension on the category of finite dimensional complex representations of a finite group.

Generalized Perfect Codes for Symmetric Classical-Quantum Channels

We define a new family of codes for symmetric classical-quantum channels and establish their optimality. To this end, we extend the classical notion of generalized perfect and quasi-perfect codes to channels defined over some finite dimensional complex Hilbert output space. The resulting optimality conditions depend on the channel considered and on an auxiliary state defined on the output space of the channel. For certain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> -qubit classical-quantum channels, we show that codes based on a generalization of Bell states are quasi-perfect and, therefore, they feature the smallest error probability among all codes of the same blocklength and cardinality.

BFT2: a general class of 2d mathcal{N} = (0, 2) theories, 3-manifolds and toric geometry

We introduce and initiate the study of a general class of 2d mathcal{N} = (0, 2) quiver gauge theories, defined in terms of certain 2-dimensional CW complexes on oriented 3-manifolds. We refer to this class of theories as BFT2’s. They are natural generalizations of Brane Brick Models, which capture the gauge theories on D1-branes probing toric Calabi-Yau 4-folds. The dynamics and triality of the gauge theories translate into simple transformations of the underlying CW complexes. We introduce various combinatorial tools for analyzing these theories and investigate their connections to toric Calabi-Yau manifolds, which arise as their master and moduli spaces. Invariance of the moduli space is indeed a powerful criterion for identifying theories in the same triality class. We also investigate the reducibility of these theories.

Topological classification of complex vector bundles over 8-dimensional spin$$^{c}$$ manifolds

In this paper, complex vector bundles of rank r over 8-dimensional spin\(^{c}\) manifolds are classified in terms of the Chern classes of the complex vector bundles, where \(r = 3\) or 4. As an application, we see that two rank 3 complex vector bundles over 4-dimensional complex projective space \(\mathbb {C}P^{4}\) are isomorphic if and only if they have the same Chern classes. Moreover, the Chern classes of rank 3 complex vector bundles over \(\mathbb {C}P^{4}\) are determined. Together with results of Thomas and Switzer, this completes the classification of complex vector bundles of any rank over \(\mathbb {C}P^4\).

Classification of incompatibility for two orthonormal bases

For two orthonormal bases of a $d$-dimensional complex Hilbert space, the notion of complete incompatibility was introduced recently by De Bi\`{e}vre [Phys. Rev. Lett. 127, 190404 (2021)]. In this work, we introduce the notion of $s$-order incompatibility with positive integer $s$ satisfying $2\leq s\leq d+1.$ In particular, $(d+1)$-order incompatibility just coincides with the complete incompatibility. We establish some relations between $s$-order incompatibility, minimal support uncertainty and rank deficiency of the transition matrix. As an example, we determine the incompatibility order of the discrete Fourier transform with any finite dimension.

Syntheses, Characterization, and X-Ray Crystal Structure of a 1:1 Co-Crystal of bis{bis[((2-(1H-imidazol-2-yl)methylidene)amino)phenolato-κ3N,N’,O]nickel(II)} and bis{bis[((2-(1H-imidazol-2-yl)methylidene) amino)phenol-κ3N,N’,O]nickel(II)} tetra(chloride)

The present investigation describes the synthesis and structural study of the ligand 2-((1H-imidazol-2-yl)methyleneamino)phenol, which was used to generate three dimensio-nal supramolecular complex formulated as {[Ni(HL)2]2.[Ni(H2L)2]2.(Cl)4.(H2O)5}. The title compound crystallizes in the orthorhombic space group Fddd with the following unit cell parameters: a = 13.9269(4) Å, b = 29.5295(16) Å, c = 43.1739(15) Å, V = 17755.5(12) Å3, Z = 8, R1 = 0.043 and wR2 = 0.129. For this compound, the structure reveals that one neutral mononuclear unit [Ni(HL)2] and one cationic mononuclear unit [Ni(H2L)2]2+. In the neutral unit, the organic molecule acts as a tridentate through one imino nitrogen atom, one nitrogen atom from the imidazole ring and one phenolate oxygen atom. In the cationic unit, it acts as tridentate through one imino nitrogen atom, one nitrogen atom from the imidazole ring and one oxygen atom of non-deprotonated phenol group. In both units the nickel(II) ion is hexacoordinated and the coordination environment can be described as distorted octahedral geometry. Numerous hydrogen bonds link the molecules into three dimensional network.

Dynamics of charge imbalance resolved negativity after a global quench in free scalar field theory

In this paper, we consider the time evolution of charge imbalance resolved negativity after a global quench in the 1+1 dimensional complex Klein-Gordon theory. We focus on two types of global quenches which are called boundary state quench and mass quench respectively. We first study the boundary state quench where the post-quench dynamic is governed by a massless Hamiltonian. In this case, the temporal evolution of charged imbalance resolved negativity can be obtained first by evaluating the correlators of the fluxed twist field in the upper half plane and then applying Fourier transformation. We test our analytical formulas in the underlying lattice model numerically. We also study the mass quench in the complex harmonic chain where the system evolves according to a massive Hamiltonian after the quench. We argue that our results can be understood in the framework of quasi-particle picture.