Real Roots Research Articles

Helson zeta functions for characters with finitely many values

AbstractWe show that the analytic continuations of Helson zeta functions can have essentially arbitrary poles and zeroes in the strip (unconditionally), and in the whole critical strip under Riemann Hypothesis, with the function taking values in cubic roots of unity. If the sets of zeroes and poles are symmetric with respect to the real axis, the same can be achieved with taking values . The proof is constructive.

Symmetries of simple $A\mathbb{T}$-algebras

Let A be a unital simple A\mathbb{T} -algebra of real rank zero. Given an order two automorphism h: K_1(A)\to K_1(A) , we show that there is an order two automorphism \alpha : A\to A such that \alpha_{*0}=\mathrm {id} , \alpha_{*1}=h and the action of \mathbb{Z}_2 generated by \alpha has the tracial Rokhlin property. Consequently, C^*(A,\mathbb{Z}_2,\alpha) is a simple unital AH-algebra with no dimension growth, and with tracial rank zero. Thus our main result can be considered as the \mathbb{Z}_2 -action analogue of the Lin-Osaka theorem. As a consequence, a positive answer to a lifting problem of Blackadar is also given for certain split case.

On Solutions of the Bethe Ansatz for the Quantum KdV Model

We study the Bethe Ansatz Equations for the Quantum KdV model, which are also known to be solved by the spectral determinants of a specific family of anharmonic oscillators called monster potentials (ODE/IM correspondence). These Bethe Ansatz Equations depend on two parameters identified with the momentum and the degree at infinity of the anharmonic oscillators. We provide a complete classification of the solutions with only real and positive roots—when the degree is greater than 2—in terms of admissible sequences of holes. In particular, we prove that admissible sequences of holes are naturally parameterised by integer partitions, and we prove that they are in one-to-one correspondence with solutions of the Bethe Ansatz Equations, if the momentum is large enough. Consequently, we deduce that the monster potentials are complete, in the sense that every solution of the Bethe Ansatz Equations coincides with the spectrum of a unique monster potential. This essentially (i.e. up to gaps in the previous literature) proves the ODE/IM correspondence for the Quantum KdV model/monster potentials—which was conjectured by Dorey–Tateo and Bazhanov–Lukyanov–Zamolodchikov—when the degree is greater than 2. Our approach is based on the transformation of the Bethe Ansatz Equations into a free-boundary nonlinear integral equation—akin to the equations known in the physics literature as DDV or KBP or NLIE—of which we develop the mathematical theory from the beginning.

Permanence of real rank zero from a centrally large subalgebra to a C*-algebra

Let A A be an infinite dimensional simple unital C*-algebra and B B be a centrally large subalgebra of A A . Archey and Phillips [J. Operator Theory 83 (2020), pp. 353–389] showed that A A has real rank zero when B B has real rank zero and stable rank one. In this paper, we have removed the restriction of B B has stable rank one. Using matrix decomposition by some suitable projections, we have shown that A A has real rank zero if B B has real rank zero.

LiDAR-aid Inertial Poser: Large-scale Human Motion Capture by Sparse Inertial and LiDAR Sensors.

We propose a multi-sensor fusion method for capturing challenging 3D human motions with accurate consecutive local poses and global trajectories in large-scale scenarios, only using single LiDAR and 4 IMUs, which are set up conveniently and worn lightly. Specifically, to fully utilize the global geometry information captured by LiDAR and local dynamic motions captured by IMUs, we design a two-stage pose estimator in a coarse-to-fine manner, where point clouds provide the coarse body shape and IMU measurements optimize the local actions. Furthermore, considering the translation deviation caused by the view-dependent partial point cloud, we propose a pose-guided translation corrector. It predicts the offset between captured points and the real root locations, which makes the consecutive movements and trajectories more precise and natural. Moreover, we collect a LiDAR-IMU multi-modal mocap dataset, LIPD, with diverse human actions in long-range scenarios. Extensive quantitative and qualitative experiments on LIPD and other open datasets all demonstrate the capability of our approach for compelling motion capture in large-scale scenarios, which outperforms other methods by an obvious margin. We will release our code and captured dataset to stimulate future research.

Asymptotically holomorphic theory for symplectic orbifolds

We extend Donaldson’s asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a compact symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient orbifold satisfies those properties.

Relaxation of the greatest common divisor condition for the controllability of bilinear systems

When a bilinear system has an input matrix that has rank of one, bilinear controllability can be expressed in terms of the controllability and observability of an associated linear system, defined by the rank-one decomposition. Bilinear controllability also requires a greatest common divisor (GCD) condition to hold. We show in the absence of the GCD condition that the system is nearly controllable, i.e., controllable except for a set of Lebesgue measure zero. The presence of sparsity in the system dynamics structure then motivates a new definition of bilinear controllability in which initial and final conditions must satisfy certain combinations of non-zero state values.

Approximate solutions of the transcendental equation for the square quantum wells by finding the real root of the cubic equation

The energy eigenvalues E n in finite square quantum wells (SQW) cannot be found using an analytic expression. As a result, numerical methods are normally used to find the eigenvalues from a transcendental equation. In this report, it will be shown that the eigenvalue solution for a given state consists in finding the only real positive root of a depressed trinomial polynomial of third order, which is as easy to solve as a quadratic equation. The method proposed can also be applied for semi-infinite, finite, and asymmetric SQW, which are often presented in quantum mechanics textbooks at the undergraduate level. The proposed method can be applied during an exam when programmable calculators are not allowed as the real root of the trinomial polynomial can be found using the formula for the cubic equation found nearly 500 years ago.

Uncertainty quantification in the assessment of human exposure to pulsed or multi-frequency fields

Objective: pulsed fields or waveforms with multi-frequency content have to be assessed with suitable methods. This paper deals with the uncertainty quantification associated to these methods. Approach: among all possible approaches, the weighted peak method (WPM) is widely employed in standards and guidelines, therefore, in this paper, we consider its implementation both in time domain and frequency domain. For the uncertainty quantification the polynomial chaos expansion theory is used. By means of a sensitivity analysis, for several standard waveforms, the parameters with more influence on the exposure index are identified and their sensitivity indices are quantified. The output of the sensitivity analysis is used to set up a parametric analysis with the aim of evaluating the uncertainty propagation of the analyzed methods and, finally, also several measured waveforms generated by a welding gun are tested. Main results: it is shown that the time domain implementation of the weighted peak method provides results in agreement with the basilar mechanisms of electromagnetic induction and electrostimulation. On the opposite, the WPM in frequency domain is found to be too sensitive to parameters that should not influence the exposure index because its weight function includes sharp variations of the phase centered on real zeros and poles. To overcome this issue, a new definition for the phase of the weight function in frequency domain is proposed. Significance: it is shown that the time domain implementation of the WPM is the more accurate and precise. The standard WPM in frequency domain has some issues that can be avoided with the proposed modification of the phase definition of the weight function. Finally, all the codes used in this paper are hosted on a GitHub and can be freely accessed at https://github.com/giaccone/wpm_uncertainty.

A new framework integrating reinforcement learning, a rule-based expert system, and decision tree analysis to improve building energy flexibility

This study presents a new framework that integrates machine learning and a domain knowledge-based expert system to improve building energy flexibility. In this framework, a rule-based expert system was used to maximize the self-consumption of solar photovoltaics (PV) power, while a reinforcement learning (RL) agent was constructed to efficiently optimize the grid power import for battery charging and facilitate decision-making for battery discharging in response to the time of use electricity prices. Meanwhile, a Classification and Regression Tree (CART) model was developed to quantitatively analyze the relationships between building energy flexibility and external variables of interest to enhance the explainability of the framework. This work integrates the advantages of safety and simpleness of the domain knowledge-based expert system and the exploration and optimization capability of RL into building energy management. The performance of the proposed framework was evaluated using the four-year data collected from a real net zero energy office building. The system cost reduction ratio, self-consumption ratio, and self-sufficiency ratio were used as the energy flexibility indicators. The results showed that the integration of the RL and the rule-based expert system was able to reduce the electricity cost and grid power consumption by 7.0% and 10.6% respectively, and increase the self-consumption of PV power by 9.2% as compared with the use of the rule-based expert system only. The CART analysis also showed that external conditions can significantly influence the level of building energy flexibility. For instance, the average daily cost reduction ratio was 0.89 out of 1.0 when the daily maximum solar irradiance was above 717.5 W/m2, while it decreased to 0.28 when the daily mean solar irradiance was below 62.4 W/m2. This strategy can be used to facilitate building demand-side management and improve the design and control of building energy systems for enhanced demand flexibility.

The spectrum properties of an integrable G2 invariant vertex model

This paper is concerned with the study of properties of the exact solution of the fundamental integrable G2 vertex model. The model R-matrix and respective spin chain are presented in terms of the basis generators of the G2 Lie algebra. This formulation permits us to relate the number of the Bethe roots of the respective Bethe equations with the eigenvalues of the U(1) conserved charges from the Cartan subalgebra of G2. The Bethe equations are solved by a peculiar string structure which combines complex three-strings with real roots allowing us to determine the bulk properties in the thermodynamic limit. We argue that G2 spin chain is gapless but the low-lying excitations have two different speeds of sound and the underlying continuum limit is therefore not strictly Lorentz invariant. We have investigated the finite-size corrections to the ground state energy and proposed that the critical properties of the system should be governed by the product of two c=1 conformal field theories. By combining numerical and analytical methods we have computed the bulk free-energy of the G2 vertex model. We found that there are three regimes in the spectral parameter in which the free-energy is limited and continuous. There exists however at least two sharp corner points in which the bulk-free energy is not differentiable.

On symmetric closed subsets of real affine root systems

Any symmetric closed subset of a finite crystallographic root system must be a closed subroot system. This is not, in general, true for real affine root systems. In this paper, we determine when this is true and also give a very explicit description of symmetric closed subsets of real affine root systems. At the end, using our results, we study the correspondence between symmetric closed subsets of real affine root systems and the regular subalgebras generated by them.

Density questions on arithmetic equivalence

It is a classic result that two number fields have equal Dedekind zeta functions if and only if the arithmetic type of a prime p p is the same in both fields for almost all prime p p . Here, almost all means with the possible exception of a set of Dirichlet density zero. One of the results of this paper shows that the condition density zero can be improved to a specific positive density that depends solely on the degree of the fields. More specifically, for every positive n n we exhibit a positive constant c n c_{n} such that any two degree n n number fields K K and L L are arithmetically equivalent if and only if the set of primes p p such that the arithmetic type of p p in K K and L L is not the same has Dirichlet density at most c n c_n . We in fact show that c n = 1 4 n 2 \displaystyle c_n=\frac {1}{4n^2} works and give a heuristic evidence that points to the fact that this value might be improved to 2 n 2 \displaystyle \frac {2}{n^2} . We also show that to check whether or not two number fields are arithmetically equivalent it is enough to check equality between finitely many coefficients of their zeta functions, and we give an upper bound for such number.

On a variant of Pillai’s problem with binary recurrences and S-units

Let [Formula: see text] be a fixed binary recurrence with real characteristic roots [Formula: see text] satisfying [Formula: see text] and let [Formula: see text] be fixed distinct prime numbers. In this paper, we show that there exist effectively computable, positive constants [Formula: see text] and [Formula: see text] such that the Diophantine equation [Formula: see text] has at most [Formula: see text] solutions [Formula: see text] if [Formula: see text] and at most [Formula: see text] solutions [Formula: see text] if [Formula: see text]. In order to demonstrate the strength of our method we show that for the binary recurrence sequence [Formula: see text], with [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text], the Diophantine equation [Formula: see text] has at most six solutions [Formula: see text] unless [Formula: see text] in which case it has exactly seven solutions.

On the Probability of Real Roots of a Quadratic Equation when the Coefficients are iid Uniform (α, β) Variates

On the Probability of Real Roots of a Quadratic Equation when the Coefficients are iid Uniform (α, β) Variates

Voltage Decay of Li‐Rich Layered Oxides: Mechanism, Modification Strategies, and Perspectives

AbstractLi‐rich layered oxides (LLOs) have been considered as the most promising cathode materials for achieving high energy density Li‐ion batteries. However, they suffer from continuous voltage decay during cycling, which seriously shortens the lifespan of the battery in practical applications. This review comprehensively elaborates and summarizes the state‐of‐the‐art of the research in this field. It is started from the proposed mechanism of voltage decay that refers to the phase transition, microscopic defects, and oxygen redox or release. Furthermore, several strategies to mitigate the voltage decay of LLOs from different scales, such as surface modification, elemental doping, regulation of components, control of defect, and morphology design are summarized. Finally, a systematic outlook on the real root of voltage decay is provided, and more importantly, a potential solution to voltage recovery from electrochemistry. Based on this progress, some effective strategies with multiple scales will be feasible to create the conditions for their commercialization in the future.

Cubic equation and topology of complex numbers

The study of complex analysis is an intriguing and technical topic. Over the past few centuries, mathematicians have produced ground-breaking work in this field. Mathematicians initially defined complex numbers, and then they gradually investigated their algebraic and geometric properties. They stumbled upon and researched intricate calculations hundreds of years later. The solution of the cubic equation has motivated mathematicians to explore the study of complex numbers, but it has also advanced mathematics in many other ways by exposing previously undiscovered issues. First off, when there are multiple real roots to the equation, it is impossible to obtain all of them using Cardano's formula at that time without understanding the fundamentals of complex numbers. Therefore, when they test various examples, mathematicians occasionally discover that there are other roots that are quite clear but are impossible to pass via Cardano's formula. In this paper, it start from the research of cubic equaiton and then discuss in detail topology of complex numbers.

On the Practicality of the Analytical Solutions for all Third- and Fourth-Degree Algebraic Equations with Real Coefficients

In order to propose a deeper analysis of the general quartic equation with real coefficients, the analytical solutions for all cubic and quartic equations were reviewed here; then, it was found that there can only be one form of the resolvent cubic that satisfies the following two conditions at the same time: (1) Its discriminant is identical to the discriminant of the general quartic equation. (2) It has at least one positive real root whenever the general quartic equation is non-biquadratic. This unique special form of the resolvent cubic is defined here as the “Standard Form of the Resolvent Cubic”, which becomes relevant since it allows us to reveal the relationship between the nature of the roots of the general quartic equation and the nature of the roots of all the forms of the resolvent cubic. Finally, this new analysis is the basis for designing and programming efficient algorithms that analytically solve all algebraic equations of the fourth and lower degree with real coefficients, always avoiding the application of complex arithmetic operations, even when these equations have non-real complex roots.

Chain enumeration, partition lattices and polynomials with only real roots

The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. The chain polynomials of the partition lattices and their standard type \(B\) analogues are shown to have only real roots. The real-rootedness of the chain polynomial is conjectured for all geometric lattices and is shown to be preserved by the pyramid and the prism operations on Cohen-Macaulay posets. As a result, new families of convex polytopes whose barycentric subdivisions have real-rooted \(f\)-polynomials are presented. An application to the face enumeration of the second barycentric subdivision of the boundary complex of the simplex is also included.Mathematics Subject Classifications: 05A05, 05A18, 05E45, 06A07, 26C10Keywords: Chain polynomial, geometric lattice, partition lattice, real-rooted polynomial, flag \(h\)-vector, convex polytope, barycentric subdivision

An Investigation in to the Properties of Functions Defining Distinguished Varieties

An inner toral polynomial is a polynomial in \(\mathbb{C}\) [z,w] such that its zero set is contained in \(\mathbb{D}\)2 \(\cup\) \(\mathbb{T}\)2 \(\cup\) \(\mathbb{E}\)2 where \(\mathbb{D}\) is the open unit disc, \(\mathbb{T}\) is the unit circle and \(\mathbb{E}\) is the exterior of the closed unit disc in \(\mathbb{C}\). Given such a polynomial p, it's zero set that lies inside \(\mathbb{D}\)2 , i.e V = Z (p) \(\cap\) \(\mathbb{D}\)2 is called a distinguished variety, and p is called a polynomial defining the distinguished variety V . An inner toral polynomial always gives a distinguished variety and vice versa. Finite Blaschke products generate inner toral polynomials such a way that, given a finite Blaschke product B(z) the numerator of wm - B(z) is an inner toral polynomial. In this paper, we investigate the conditions that make the sum and the composition of inner toral polynomials generated by finite Blaschke products, inner toral.