Complex Plane Research Articles

Exploration of treatment burden through examination of workload and patient capacity during transition onto kidney replacement therapy: a systematic review of qualitative research

BackgroundPatients with advanced chronic kidney disease requiring initiation of kidney replacement therapy (KRT) are frequently asked to enact complex management plans. Treatment burden has been defined as the effect of healthcare workload and the capacity a person has to manage this workload has on wellbeing. The aim of this review is to examine the experience of healthcare workload and the factors that affect capacity to meet that workload for people transitioning onto KRT for the first time, using a framework synthesis of published literature informed by normalisation process theory (NPT) and theory of patient capacity (TPC).MethodsMedline, Scopus and CINAHL were systematically searched with manual citation and reference searching. Studies were included if meeting the criteria of adults aged 18 or over transitioning for the first time onto any modality of KRT (haemodialysis, peritoneal dialysis or kidney transplantation), using qualitative methodologies to describe any aspect of experiences of healthcare workload or any factors that affect capacity to manage workload were included. s and full papers were independently screened by two reviewers and data extraction and quality appraisal were also independently conducted by two reviewers. Qualitative data were analysed using framework synthesis informed by NPT and TPC.ResultsA total of 24,380 studies were screened, 406 full texts were reviewed and 18 studies were included. There were four broad categories of workload described: making sense of KRT, working out what to do and how to do it, meeting the challenges of KRT, and reflecting on work done. Patient capacity influenced the experience of all types of workload and the treatment burden generated by the work.ConclusionsTransitioning onto KRT is a period of very high healthcare workload and potentially high treatment burden. The relationship between healthcare workload and capacity to handle workload is complex, multifactorial and changes over time. By better understanding workload, capacity and burden during transition, we can develop better ways of measuring these important aspects of care and develop interventions to reduce treatment burden in those transitioning onto KRT.

Abstract TP371: Effects of Empagliflozin on Ischemic Stroke in a Rat Model: Time-Frequency Electroencephalogram Features and Cerebral Infarction Size

Background: Emerging evidence suggests that empagliflozin (EMPA) may offer additional benefits beyond its primary use in diabetes. Ischemic stroke is known to cause distinctive alterations in EEG signals. However, the influences of EMPA on the stroke-induced EEG changes has not been investigated. Here, we study such influences via EEG time-frequency features using the Hilbert-Huang Transform (HHT) and examine how these features correlate with the cerebral infarction. Methods: n= 47 male SD rats were randomized into 3 groups: 1] Control (n = 16, regular diet); 2] Acute EMPA treatment (n = 16, EMPA 10 mg/kg, IV given at 10 mins prior to middle cerebral artery occlusion (MCAO) and 1 min prior to reperfusion); 3] Chronic EMPA treatment (n = 15, EMPA by food, 20 mg/kg for 7 days before MCAO). To induce ischemic stroke, standard MCAO was performed for 1 hour followed by 3 hours of reperfusion. Post-surgery, triphenyltetrazolium chloride (TTC) staining was used to measure volume of cerebral infarction. EEG signals were continuously recorded throughout the procedure. We applied the Hilbert-Huang Transform (HHT) to analyze the EEG signals. HHT includes 2 steps: 1] empirical mode decomposition (EMD) to decompose the EEG signal into a set of intrinsic mode functions (IMFs); 2] the Hilbert Transform is applied to each IMF to compute analytic signals. Such analytic signals are represented in the complex plane, where they often form circular or elliptical contours (Fig. 1). The area of these contours in the complex plane contains unique information about variations in the signal properties. We computed the HHT area metric for the first four IMFs by using EEG recordings at 3 hours post-reperfusion (duration: 2 minute). Results and Conclusions: Our results showed the control group demonstrates higher correlations between the HHT area metric and the cerebral infarction size, compared to acute or chronic EMPA treatments (Fig. 2), with an especially significant correlation for IMF1in controls (R= -0.76). No significant difference was found in the HHT area metric between control, acute, and chronic EMPA (Fig. 3). It revealed that while EMPA treatments do not alter the EEG time-frequency features compared to controls, control group exhibits a stronger correlation between EEG features and cerebral infarction size. This indicates that EMPA's neuroprotective effects might not be directly reflected in EEG changes, highlighting the need for further investigation of its mechanisms.

Coupled dynamic analysis of moored floating structures by a hybrid Laplace-time domain method

The coupled dynamic analysis of moored floating structures has often been conducted in the time domain by iteratively solving the Cummins equation, treating the mooring effects as an additional nonlinear load. However, the time domain (TD) method requires performing a convolution integral at each time step, which is costly in computational time. This paper innovatively develops an efficient hybrid Laplace-time domain method (HLTD) on implementing the coupled dynamic analysis of moored floating structures. The proposed method divides the external load into a number of segments and models the mooring system by the catenary theory. Under each segment, while the iterative operations used in the TD method is borrowed to handle the coupled behavior, the HLTD method computes the motions of the floating structure under each iteration by the pole-residue operations in the Laplace domain (LD). As the complicated convolutional integral computation required in the TD method is replaced by simple algebraic pole-residue calculations in the complex plane, the HLTD method is more efficient. Additionally, the HLTD method derives analytical response solutions for the moored floating system. Its efficiency and accuracy are demonstrated by comparing with the TD method through a moored float-over barge to irregular waves.

A New Fast Root-finder for Black Box Polynomials

Univariate polynomial root-finding has been studied for four millennia and very intensively in the last decades. Our black box root-finder involves no coefficients and works for a black box polynomial, defined by an oracle (that is, black box subroutine) for its evaluation. Such root-finders have various benefits, e.g., are particularly efficient where a polynomial can be evaluated fast, say, is a sum of a small number of shifted monomials (x−c)a. With incorporation of a fast algorithm by the first author for compression of a disc on the complex plane without losing roots, our root-finder approximates all d complex zeros of a dth degree polynomial p(x) (aka roots of equation p(x)=0) by using near-optimal Las Vegas expected number of bit-operations,1 that is, the root-finder is expected to run almost as fast as one accesses the coefficients with a precision required for the solution within a prescribed error bound. The only other known near-optimal polynomial root-finder was presented by the first author at ACM STOC 1995. It is quite involved and has never been implemented, while already in its initial implementation our new root-finder competed with user's choice package of root-finding subroutines MPSolve, according to extensive numerical experiments with standard test polynomials. Furthermore we readily extend our black box root-finder to approximation of the eigenvalues of a matrix in record expected bit operation time, while the root-finder of STOC 1995, using the coefficients of p(x), does not support such extension.

Code Provisions to Limit Plan Area Projections in RC Buildings with Complex Plan Geometry

Code Provisions to Limit Plan Area Projections in RC Buildings with Complex Plan Geometry

Disentangling the component processes in complex planning impairments following ventromedial prefrontal lesions.

Damage to ventromedial prefrontal cortex (vmPFC) in humans disrupts planning abilities in naturalistic settings. However, it is unknown which components of planning are affected in these patients, including selecting the relevant information, simulating future states, or evaluating between these states. To address this question, we leveraged computational paradigms to investigate the role of vmPFC in planning, using the board game task 'Four-in-a-Row' (18 lesion patients, 9 female; 30 healthy control participants, 16 female), and the simpler 'Two-Step' task measuring model-based reasoning (49 lesion patients, 27 female; 20 healthy control participants, 13 female). Damage to vmPFC disrupted performance in Four-in-a-Row compared to both control lesion patients and healthy age-matched controls. We leveraged a computational framework to assess different component processes of planning in Four-in-a-Row and found that impairments following vmPFC damage included shallower planning depth, and a tendency to overlook game-relevant features. In the 'Two-Step' task, which involves binary choices across a short future horizon, we found little evidence of planning in all groups, and no behavioural differences between groups. Complex yet computationally tractable tasks such as 'Four-in-a-row' offer novel opportunities for characterising neuropsychological planning impairments, which in vmPFC patients we find are associated with oversights and reduced planning depth.Significance Statement The ability to plan in real-world settings is often disrupted after damage to ventromedial prefrontal cortex (vmPFC). However, real-world planning consists of many different cognitive processes, and it is uncertain which processes are disturbed by these lesions. Here we use rich computational models of planning to characterise behaviour in two planning tasks performed by patients with vmPFC damage and controls. VmPFC damage only affected behaviour in the more complex planning task, and behavioural modelling revealed this was associated with planning less far into the future and overlooking important features. These findings shed light on the neural mechanisms supporting complex planning, demonstrating how novel computational methods can strike the balance between task complexity and interpretability.

Hadron-quark hybrid model, modular transformation, and Roberge-Weiss transition

In the framework of modular transformations, we reformulate the recently proposed hadron-quark hybrid model when the imaginary baryonic chemical potential is introduced. In this case, the number density of the hybrid model is obtained by the modular transformation of the complex number densities of the baryons (antibaryons) and the quarks (antiquarks). We can regard these number densities as the basis in the complex plane. As a result, we can consider the torus, which is characterized by the basis. Since the complex structure of the torus is invariant under the modular transformation, we can extract the topological property of the hadron-quark system using the untransformed baryon (antibaryon) and the quark (antiquark) number densities. We apply this model to analyze the Roberge-Weiss transition. It is shown that the torus vanishes at the baryonic chemical potential where the Roberge-Weiss transition appears because the number density of baryons (antibaryons) is not linearly independent of the number density of quarks (antiquarks). When the temperature T is lower than the Roberge-Weiss transition temperature TRW, the torus shrinks smoothly to the one-dimensional object at the Roberge-Weiss transition point, but the discontinuity does not appear. On the other hand, the discontinuity of the geometrical object appears when T>TRW. We also calculate the modulus of the torus and transform it into the fundamental region. The transformed moduli are symmetric below TRW, but the symmetry is broken above TRW. Published by the American Physical Society 2025

Neural Density Functional Theory of Liquid-Gas Phase Coexistence

We use supervised machine learning together with the concepts of classical density functional theory to investigate the effects of interparticle attraction on the pair structure, thermodynamics, bulk liquid-gas coexistence, and associated interfacial phenomena in many-body systems. Local learning of the one-body direct correlation functional is based on Monte Carlo simulations of inhomogeneous systems with randomized thermodynamic conditions, randomized planar shapes of the external potential, and randomized box sizes. Focusing on the prototypical Lennard-Jones system, we test predictions of the resulting neural attractive density functional across a broad spectrum of physical behavior associated with liquid-gas phase coexistence in bulk and at interfaces. We analyze the bulk radial distribution function g(r) obtained from automatic differentiation and the Ornstein-Zernike route and determine (i) the Fisher-Widom line, i.e., the crossover of the asymptotic (large distance) decay of g(r) from monotonic to oscillatory, (ii) the (Widom) line of maximal correlation length, (iii) the line of maximal isothermal compressibility, and (iv) the spinodal by calculating the poles of the structure factor in the complex plane. The bulk binodal and the density profile of the free liquid-gas interface are obtained from density functional minimization and the corresponding surface tension from functional line integration. We also show that the neural functional describes accurately the phenomena of drying at a hard wall and of capillary evaporation for a liquid confined in a slit pore. Our neural framework yields results that improve significantly upon standard mean-field treatments of interparticle attraction. Comparison with independent simulation results demonstrates a consistent picture of phase separation even when restricting the training to supercritical states only. We argue that phase coexistence and its associated signatures can be discovered as emerging phenomena via functional mappings and educated extrapolation. Published by the American Physical Society 2025

Zero exclusion sectors for polynomials whose coefficients exhibit a +/-/+ sign pattern

The zeros of polynomials whose coefficients exhibit a +/-/+ sign pattern are sometimes excluded from sectors in the complex plane. It is shown how to obtain such sectors, as well as the number of zeros that lie between the sectors, along with bounds on their magnitudes. The results are illustrated with an application to polynomial eigenvalue problems.

The spectral ζ-function for quasi-regular Sturm–Liouville operators

In this work, we analyze the spectral ζ-function associated with the self-adjoint extensions, TA,B, of quasi-regular Sturm–Liouville operators that are bounded from below. By utilizing the Green’s function formalism, we find the characteristic function, which implicitly provides the eigenvalues associated with a given self-adjoint extension TA,B. The characteristic function is then employed to construct a contour integral representation for the spectral ζ-function of TA,B. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the ζ-function to a larger region of the complex plane. We also present a method for computing the value of the spectral ζ-function of TA,B at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral ζ-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of s.

Dynamic Stress Analysis of a Strip Plate with Elliptical Holes Subjected to Incident Shear Horizontal Waves

The dynamic stress analysis of a strip plate with elliptical holes under the action of an incident SH wave was performed using a complex function method and a successive mirror method. Firstly, a complex plane coordinate system of elliptic holes was established by using the complex variable function method and integral transformation method. The elliptic hole wave field and stress were established using the wave function expansion method. Then, the relation between the argument angle of any point on the edge of the ellipse hole and the angle between the vertical line and the coordinate axis was established. Using boundary conditions to solve the unknown coefficients in the equation, finally, the integral equation was simplified to a linear equation by means of the effective truncation method, and the steady-state response of dynamic stress under different parameters was analyzed. In addition, by comparing the finite element solution with the numerical solution, the accuracy of the results was effectively verified. The results show that the studied geometric model can provide solid theoretical support for the inspection of plate and shell structures, which is of great significance in practical engineering.

Orthogonal Polynomials on Radial Rays in the Complex Plane: Construction, Properties and Applications

Orthogonal polynomials on radial rays in the complex plane were introduced and studied intensively in several papers almost three decades ago. This paper presents an account of such kinds of orthogonality in the complex plane, as well as a number of new results and examples. In addition to several types of standard orthogonality, the concept of orthogonality on arbitrary radial rays is introduced, some or all of which may be infinite. A general method for numerical constructing, the so-called discretized Stieltjes–Gautschi procedure, is described and several interesting examples are presented. The main properties, zero distribution and some applications are also given. Special attention is paid to completely symmetric cases. Recurrence relations for such kinds of orthogonal polynomials and their zero distribution, as well as a connection with the standard polynomials orthogonal on the real line, are derived, including the corresponding linear differential equation of the second order. Finally, some applications in physics and electrostatics are mentioned.

Quantum–Fractal–Fractional Operator in a Complex Domain

In this effort, we extend the fractal–fractional operators into the complex plane together with the quantum calculus derivative to obtain a quantum–fractal–fractional operators (QFFOs). Using this newly created operator, we create an entirely novel subclass of analytical functions in the unit disk. Motivated by the concept of differential subordination, we explore the most important geometric properties of this novel operator. This leads to a study on a set of differential inequalities in the open unit disk. We focus on the conditions to obtain a bounded turning function of QFFOs. Some examples are considered, involving special functions like Bessel and generalized hypergeometric functions.

Multi-Criteria Assessment of Urban Thermal Hotspots: A GIS-Based Remote Sensing Approach in a Mediterranean Climate City

One of the most significant urban challenges focuses on addressing the effects of urban overheating as a consequence of climate change. Several methods have been developed to characterize urban heat islands (UHIs); however, the most widely used involve complex planning, huge time consumption, and substantial human and technical resources on field monitoring campaigns. Therefore, this study aims to provide an easily accessible and affordable remote sensing method for locating urban hotspots and addresses a multi-criteria assessment of urban heat-related parameters, allowing for a comprehensive city-wide evaluation. The novelty is based on leveraging the potential of the last Landsat 9 satellite, the application of kernel spatial interpolation, and GIS open access data, providing very high-resolution land surface temperature images over urban spaces. Within GIS workflow, the city is divided into LCZs, thermal hotspots are detected, and finally, it is analyzed to understand how urban factors, such as urban boundaries, building density, and vegetation, affect urban scale LST, all using graphical and analytical cross-assessment. The methodology has been tested in Seville, a representative warm Mediterranean city, where variations of up to 10 °C have been found between homogeneous residential areas. Thermal hotspots have been located, representing 11% of the total residential fabric, while results indicate a clear connection between the urban factors studied and overheating. The conclusions support the possibility of generating a powerful affordable tool for future research and the design of public policy renewal actions in vulnerable areas.

Starlikeness associated with certain strongly functions.

Let κ be an analytic function in the open unit disc in the complex plane with . Then if and only if , where ≺ is a subordination relation and the function , for , maps to a symmetric domain about the real axis in the right-half plane. This article finds the sharp radii of starlikeness for three classes of normalized analytic functions in the open unit disc. It also finds the solutions to first-order differential equations and utilizes these results together with the differential subordination developed by Miller and Mocanu to obtain the conditions so that some first-order differential relations associated with are subordinate to certain Ma and Minda functions. Consequently, it provides sufficient conditions for the normalized analytic functions to belong to some established subclasses of starlike functions.

Effects of local mutations in quadratic iterations.

We introduce mutations in the process of discrete iterations of complex quadratic maps in the family fc(z)=z2+c. More specifically, we consider a "correct" function fc1 acting on the complex plane. A "mutation" fc0 is a different ("erroneous") map acting on a locus of given radius r around a mutation focal point ξ∗. The effect of the mutation is interpolated radially to eventually recover the original map fc1 when reaching an outer radius R. We call the resulting map a "mutated" map. In the theoretical framework of mutated iterations, we study how a mutation affects the temporal evolution of the system and the asymptotic behavior of its orbits. We use the prisoner set of the system to quantify simultaneously the long-term behavior of the entire space under mutated maps. We analyze how the position, timing, and size of the mutation can alter the system's long-term evolution (as encoded in the topology of its prisoner set). The framework is then discussed as a metaphoric model for studying the impact of copying errors in natural replication systems.

Levi-umbilical weakly 2-Hopf hypersurface in nonflat complex planes

Levi-umbilical weakly 2-Hopf hypersurface in nonflat complex planes

GUI of Complex plane on Excel spreadsheets

In this paper we provide two spreadsheets in Excel for university students and teachers. These spreadsheets contains the geometrical representation of complex numbers, conversion in polar and rectangular forms and four basic math operation that are addition, subtraction, multiplication and division. Further we also established that complex number satisfies the parallelogram law

Study of the Relationship Between Two Types of Mixed-Order Evolution Equations and the Existence of Their Solutions

The purpose of this article is to study the relationship between two types of mixed-order evolution equations and the existence of their solutions. Initially, the mild solutions of the two system are obtained by Laplace transform. Then, we use semigroup theorem and sector operator theorem to get the norm estimation results of under- standing operator through special paths in the complex plane. Further, the sufficient conditions for existence and uniqueness of mild solution of the proposed system are verified by applying fixed point theorems. Finally, examples are provided to illustrate the main results.

Interconnection Between Schur Stability and Structured Singular Values

In this paper, we present new results on the interconnections between Schur stability and structured singular values of real-valued matrices, denoted as Rn×n . Most new findings are obtained for n = 2 and n = 3. These novel insights into the relationship between Schur stability and structured singular values are developed by applying various tools from linear algebra, system theory, and matrix analysis. Schur stability ensures that all eigenvalues lie within the unit circle in the complex plane, which is fundamental for the boundedness and stability of system responses. Structured singular values, on the other hand, provide a measure of robustness, stability, and performance against structured perturbations in system parameters, offering valuable insights into the stability margins and performance limits under such uncertainties.