Mixing Sum Research Articles

Neutrino Mixing Sum Rules and the Littlest Seesaw

In this work, we study the neutrino mixing sum rules arising from discrete symmetries and the class of Littlest Seesaw (LS) neutrino models. These symmetry-based approaches all offer predictions for the cosine of the leptonic CP phase cosδ in terms of the mixing angles, θ13, θ12, θ23, while the LS models also predict the sine of the leptonic CP phase sinδ, as well as making other predictions. In particular, we study the solar neutrino mixing sum rules, arising from charged lepton corrections to tri-bimaximal (TB), bimaximal (BM), golden ratio (GR) and hexagonal (HEX) neutrino mixing, and the atmospheric neutrino mixing sum rules, arising from preserving one of the columns of these types of mixing—for example, the first or second column of the TB mixing matrix (TM1 or TM2)—and we confront them with an up-to-date global fit of the neutrino oscillation data. We show that some mixing sum rules, such as an atmospheric neutrino mixing sum rule arising from a version of neutrino golden ratio mixing (GRa1), are already excluded at 3σ, and we determine the remaining models allowed by the data. We also consider the more predictive LS models (which obey the TM1 sum rules and offer further predictions) based on constrained sequential dominance CSD(n) with n≈3. We compare for the first time the three cases n=2.5, n=3 and n=1+6≈3.45, which are favored by theoretical models, using a new type of analysis to accurately predict the observables θ12, θ23 and δ. We study all the above approaches, solar and atmospheric mixing sum rules and LS models, together so that they may be compared and to give an up-to-date analysis of the predictions of all of these possibilities, when confronted with the most recent global fits.

Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions

Analytic and asymptotic properties of the generalized Student and generalized Lomax distributions are discussed, with the main focus on the representation of these distributions as scale mixtures of the laws that appear as limit distributions in classical limit theorems of probability theory, such as the normal, folded normal, exponential, Weibull, and Fréchet distributions. These representations result in the possibility of proving some limit theorems for statistics constructed from samples with random sizes in which the generalized Student and generalized Lomax distributions are limit laws. An overview of known properties of the generalized Student distribution is given, and some simple bounds for its tail probabilities are presented. An analog of the ‘multiplication theorem’ is proved, and the identifiability of scale mixtures of generalized Student distributions is considered. The normal scale mixture representation for the generalized Student distribution is discussed, and the properties of the mixing distribution in this representation are studied. Some simple general inequalities are proved that relate the tails of the scale mixture with that of the mixing distribution. It is proved that for some values of the parameters, the generalized Student distribution is infinitely divisible and admits a representation as a scale mixture of Laplace distributions. Necessary and sufficient conditions are presented that provide the convergence of the distributions of sums of a random number of independent random variables with finite variances and other statistics constructed from samples with random sizes to the generalized Student distribution. As an example, the convergence of the distributions of sample quantiles in samples with random sizes is considered. The generalized Lomax distribution is defined as the distribution of the absolute value of the random variable with the generalized Student distribution. It is shown that the generalized Lomax distribution can be represented as a scale mixture of folded normal distributions. The convergence of the distributions of maximum and minimum random sums to the generalized Lomax distribution is considered. It is demonstrated that the generalized Lomax distribution can be represented as a scale mixture of Weibull distributions or that of Fréchet distributions. As a consequence, it is demonstrated that the generalized Lomax distribution can be limiting for extreme statistics in samples with random size. The convergence of the distributions of mixed geometric random sums to the generalized Lomax distribution is considered, and the corresponding extension of the famous Rényi theorem is proved. The law of large numbers for mixed Poisson random sums is presented, in which the limit random variable has a generalized Lomax distribution.

Moments of mixed Kloosterman sums, supercharacters, and elliptic curves

Moments of mixed Kloosterman sums, supercharacters, and elliptic curves

New Designing Mixed Double Moving Average-Cumulative Sum Control Chart for Detecting Mean Shifts with Symmetrically and Asymmetrically Distributed Observations

Herein, we present a new control chart called the mixed double moving average-cumulative sum control chart (DMA-CUSUM: MDC) used for detecting shifts in the process mean when symmetrically and asymmetrically distributed. The performance of the MDC chart is compared with shewhart, cumulative sum (CUSUM), double moving average (DMA) and mixed cumulative sum-double moving average (CUSUM-DMA: MCD) control charts by using average run length (ARL) and median run length (MRL) with the monte carlo simulation (MC). The research results show that the proposed (MDC) control chart was more efficient than the Shewhart, CUSUM, DMA and MCD charts for all distributions tested. We apply the MDC chart to real sets of data: I) the tensile data of single carbon fiber and II) the survival times of guinea pigs infected with virulent bacilli.

Bounds for the Rate of Convergence in the Generalized Rényi Theorem

In the paper, an overview is presented of the results on the convergence rate bounds in limit theorems concerning geometric random sums and their generalizations to mixed Poisson random sums, including the case where the mixing law is itself a mixed exponential distribution. The main focus is on the upper bounds for the Zolotarev ζ-metric as the distance between the pre-limit and limit laws. New results are presented that extend existing estimates of the rate of convergence of geometric random sums (in the well-known Rényi theorem) to a considerably more general class of random indices whose distributions are mixed Poisson, including generalized negative binomial (e.g., Weibull-mixed Poisson), Pareto-type (Lomax)-mixed Poisson, exponential power-mixed Poisson, Mittag-Leffler-mixed Poisson, and one-sided Linnik-mixed Poisson distributions. A transfer theorem is proven that makes it possible to obtain upper bounds for the rate of convergence in the law of large numbers for mixed Poisson random sums with mixed exponential mixing distribution from those for geometric random sums (that is, from the convergence rate estimates in the Rényi theorem). Simple explicit bounds are obtained for ζ-metrics of the first and second orders. An estimate is obtained for the stability of representation of the Mittag-Leffler distribution as a geometric convolution (that is, as the distribution of a geometric random sum).

Implications of the zero 1-3 flavour mixing hypothesis: predictions for {theta}_{23}^{mathrm{PMNS}} and δPMNS

We revisit mixing sum rule relations in the lepton and quark sectors under the assumption that the 1-3 elements of the flavour mixing matrices ( {V}_L^u , {V}_L^d , {V}_L^e , {V}_L^{nu } ) are zero in the flavour basis. We consider the exact relations resulting from the validity of this “zero 1-3 flavour mixing hypothesis” and analyse their implications based on the current experimental data, including effects from RG running. In particular, we analyse how the existing precise measurement of {theta}_{13}^{mathrm{PMNS}} allows to derive predictions for {theta}_{23}^{mathrm{PMNS}} in models with constrained {theta}_{12}^{mathrm{e}} . As examples, we calculate the predictions for {theta}_{23}^{mathrm{PMNS}} which arise in 12 classes of Pati-Salam models and SU(5) GUTs that relate {theta}_{12}^{mathrm{e}} to {theta}_{12}^{mathrm{d}} . We also derive a novel “lepton phase sum rule”, valid under the additional assumption of small charged lepton mixing contributions. We furthermore point out that, in the context of GUT flavour models, the quark and lepton CP violating phases δCKM and δPMNS can both be predicted from a single imaginary element in the mass matrices.

A Patient Selection Approach Based on NTCP Models and DVH Parameters for Definitive Proton Therapy in Locally Advanced Sinonasal Cancer Patients.

Simple SummaryThe role of proton therapy as a radiation treatment option for locally advanced sinonasal cancer patients has increased in the last years, showing promising results in terms of clinical outcomes. Definition of strategies to identify patients who would benefit the most from proton therapy in terms of reduced toxicity is highly desirable, due to limited availability and higher costs of this treatment option. The novelty of our in silico study relies on assessing the suitability of a mixed dose volume histograms and normal tissue complication probability model-based approach, aiming at providing, to the scientific community, a patient selection criteria possibly leading the therapeutic choice between proton therapy and advanced photon techniques.(1) Background: In this work, we aim to provide selection criteria based on normal tissue complication probability (NTCP) models and additional explanatory dose-volume histogram parameters suitable for identifying locally advanced sinonasal cancer patients with orbital invasion benefitting from proton therapy. (2) Methods: Twenty-two patients were enrolled, and two advanced radiation techniques were compared: intensity modulated proton therapy (IMPT) and photon volumetric modulated arc therapy (VMAT). Plans were optimized with a simultaneous integrated boost modality: 70 and 56 Gy(RBE) in 35 fractions were prescribed to the high risk/low risk CTV. Several endpoints were investigated, classified for their severity and used as discriminating paradigms. In particular, when NTCP models were already available, a first selection criterion based on the delta-NTCP was adopted. Additionally, an overall analysis in terms of DVH parameters was performed. Furthermore, a second selection criterion based on a weighted sum of the NTCP and DVH was adopted. (3) Results: Four patients out of 22 (18.2%) were suitable for IMPT due to NTCP > 3% for at least one severe toxicity, 4 (18.2%) due to NTCP > 20% for at least three concurrent intermediate toxicities and 16 (72.7%) due to the mixed sum of NTCP and DVH criterion. Since, for some cases, both criteria were contemporary fulfilled, globally 17/22 patients (77.3%) would benefit from IMPT. (4) Conclusions: For this rare clinical scenario, the use of a strategy including DVH parameters and NTCPs when comparing VMAT and IMPT is feasible. We showed that patients affected by sinonasal cancer could profit from IMPT compared to VMAT in terms of optical and central nervous system organs at risk sparing.

Mixed correlator dispersive CFT sum rules

Conformal field theory (CFT) dispersion relations reconstruct correlators in terms of their double discontinuity. When applied to the crossing equation, such dispersive transforms lead to sum rules that suppress the double-twist sector of the spectrum and enjoy positivity properties at large twist. In this paper, we construct dispersive CFT functionals for correlators of unequal scalar operators in position- and Mellin-space. We then evaluate these functionals in the Regge limit to construct mixed correlator holographic CFT functionals which probe scalar particle scattering in Anti-de Sitter spacetime. Finally, we test properties of these dispersive sum rules when applied to the 3D Ising model, and we use truncated sum rules to find approximate solutions to the crossing equation.

Wargaming with Quadratic Forms and Brauer Configuration Algebras

Recently, Postnikov introduced Bert Kostant’s game to build the maximal positive root associated with the quadratic form of a simple graph. This result, and some other games based on Cartan matrices, give a new version of Gabriel’s theorem regarding algebras classification. In this paper, as a variation of Bert Kostant’s game, we introduce a wargame based on a missile defense system (MDS). In this case, missile trajectories are interpreted as suitable paths of a quiver (directed graph). The MDS protects a region of the Euclidean plane by firing missiles from a ground-based interceptor (GBI) located at the point (0,0). In this case, a missile success interception occurs if a suitable positive number associated with the launches of the enemy army can be written as a mixed sum of triangular and square numbers.

Binary sequences and lattices constructed by discrete logarithms

<abstract><p>In 1997, Mauduit and Sárközy first introduced the measures of pseudorandomness for binary sequences. Since then, many pseudorandom binary sequences have been constructed and studied. In particular, Gyarmati presented a large family of pseudorandom binary sequences using the discrete logarithms. Ten years later, to satisfy the requirement from many applications in cryptography (e.g., in encrypting "bit-maps'' and watermarking), the definition of binary sequences is extended from one dimension to several dimensions by Hubert, Mauduit and Sárközy. They introduced the measure of pseudorandomness for this kind of several-dimension binary sequence which is called binary lattices. In this paper, large families of pseudorandom binary sequences and binary lattices are constructed by both discrete logarithms and multiplicative inverse modulo $ p $. The upper estimates of their pseudorandom measures are based on estimates of either character sums or mixed exponential sums.</p></abstract>

Investigating Thin Silicone Oil Films Using Four-Wave Mixing Spectroscopy and Sum Frequency Generation Vibrational Spectroscopy.

This article applies four-wave mixing (FWM) spectroscopy, a third-order nonlinear optical spectroscopic technique which is not intrinsically surface- or interface-sensitive, to study silicone oil thin films, supplemented by second-order nonlinear-optical sum frequency generation (SFG) vibrational spectroscopy. Although studies of thin organic films using coherent antistokes Raman spectroscopy (CARS), a special case of FWM, have been reported previously, in this study we demonstrate the feasibility of using a more general FWM process which involves three independent excitation laser beams to investigate silicone oil thin films. The results show that the FWM method has the potential to detect and provide molecular-level information on ultrathin silicone oil layers, down to a film thickness of 1 nm. This developed FWM methodology is widely applicable and can be utilized to study important issues in the biopharmaceutical field, e.g., to examine the distribution of silicone oil on syringe glass surfaces with subnanometer sensitivity. It can also be used to study the potentially slow reactions between silicone oil and glass surfaces as proposed in the literature but without direct molecular-level information.

Navier–Stokes regularity criteria in sum spaces

In this paper, we will consider regularity criteria for the Navier--Stokes equation in mixed Lebesgue sum spaces. In particular, we will prove regularity criteria that only require control of the velocity, vorticity, or the positive part of the second eigenvalue of the strain matrix, in the sum space of two scale critical spaces. This represents a significant step forward, because each sum space regularity criterion covers a whole family of scale critical regularity criteria in a single estimate. In order to show this, we will also prove a new inclusion and inequality for sum spaces in families of mixed Lebesgue spaces with a scale invariance that is also of independent interest.

Blowup criteria of a dissipative system modeling electrohydrodynamics in sum spaces

In this paper, we consider some sufficient conditions for the breakdown of local smooth solutions to the Cauchy problem of the 3D Navier–Stokes/Poisson–Nernst–Planck system modeling electro-diffusion in terms of the horizontal velocity (or vorticity, or the positive part of the second eigenvalue of the strain matrix, or pressure) in scale-critical mixed Lebesgue sum spaces.

Estimations on some hybrid exponential sums related to Kloosterman sums

In this paper, we revisit the bounds of the mixed exponential sums introduced by Lv and Zhang (2020). Moreover, we give some estimations for some new hybrid exponential sums related to Kloosterman sums over finite fields of odd characteristic by using the properties of Jacobi sums and Gaussian sums.

Leptonic sum rules from flavour models with modular symmetries

Sum rules in the lepton sector provide an extremely valuable tool to classify flavour models in terms of relations between neutrino masses and mixing parameters testable in a plethora of experiments. In this manuscript we identify new leptonic sum rules arising in models with modular symmetries with residual symmetries. These models simultaneously present neutrino mass sum rules, involving masses and Majorana phases, and mixing sum rules, connecting the mixing angles and the Dirac CP-violating phase. The simultaneous appearance of both types of sum rules leads to some non-trivial interplay, for instance, the allowed absolute neutrino mass scale exhibits a dependence on the Dirac CP-violating phase. We derive analytical expressions for these novel sum rules and present their allowed parameter ranges as well as their predictions at upcoming neutrino experiments.

The extended homogeneously weighted moving average control chart

AbstractControl charts are widely applied in many manufacturing processes to monitor the quality characteristic of interest. Recently, a homogeneously weighted moving average (HWMA) control chart was proposed as an improvement of the exponentially weighted moving average (EWMA) chart for efficiently monitoring of small shifts in the process mean. In the present article, we extend the HWMA chart by imitating exactly the double EWMA (DEWMA) technique. The proposed scheme is regarded as double HWMA (DHWMA) control chart. The run‐length characteristics of the proposed chart are evaluated by performing Monte Carlo simulations. A comparison study versus the EWMA, DEWMA, HWMA, mixed EWMA cumulative sum (CUSUM), CUSUM, and GWMA charts indicates that the DHWMA chart is more effective in detecting small to moderate shifts, while it performs similarly with its competitors for large shifts. We also study the robustness of the proposed chart under several nonnormal distributions, and it is shown that the DHWMA chart is in‐control robust for small values of the smoothing parameters. Finally, two examples are given to demonstrate the implementation of the proposed chart.

A mixed cumulative sum homogeneously weighted moving average control chart for monitoring process mean

AbstractThe homogeneously weighted moving average (HWMA) control chart is famous to identify small deviations in the process mean. The plotting statistic of the HWMA chart assigns equal weight among the previous samples as compared to the plotting statistic of the exponentially weighted moving average chart. We propose a new HWMA chart that uses the plotting statistic of the cumulative sum chart. The run length performance of the proposed chart is measured in terms of the average, the standard deviation, some percentile points, and compared with some existing counterparts' charts. The comparison shows that the proposed chart performs superior to their existing counterparts. An application based on a real‐life dataset is also presented.

Bounds for convergence rate in laws of large numbers for mixed Poisson random sums

In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev ζ-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous Rényi theorem) to a considerably wider class of random indices with mixed Poisson distributions including, e.g., those with the (generalized) negative binomial distribution.

HOMOGENEOUS STRUCTURES WITH NONUNIVERSAL AUTOMORPHISM GROUPS

AbstractWe present three examples of countable homogeneous structures (also called Fraïssé limits) whose automorphism groups are not universal, namely, fail to contain isomorphic copies of all automorphism groups of their substructures.Our first example is a particular case of a rather general construction on Fraïssé classes, which we call diversification, leading to automorphism groups containing copies of all finite groups. Our second example is a special case of another general construction on Fraïssé classes, the mixed sums, leading to a Fraïssé class with all finite symmetric groups appearing as automorphism groups and at the same time with a torsion-free automorphism group of its Fraïssé limit. Our last example is a Fraïssé class of finite models with arbitrarily large finite abelian automorphism groups, such that the automorphism group of its Fraïssé limit is again torsion-free.

Prediction of Genotypic Values and Selection Gains in Peanut Lines Derived from a Synthetic Amphidiploid

Genotypic values and selection gains were estimated in peanut lines derived from backcrossing with synthetic amphidiploid (A. batizocoi x A. duranensis)4x and a commercial cultivar, based on agronomical traits. Thirteen BC1F3 lines were grown in the field and submitted to 21 d of water suppression. Two earliness and drought tolerant cultivars (BR 1 and Senegal 55 437) were used as controls. Eight agronomic traits were adopted to characterize the lines. The estimates were based on mixed models (REML/BLUP) and Sum of Ranks Selection Index. We found that residual variance was greater than genotypic one, indicating the existence of variability within the families. Most traits showed high accuracy values. Three families showed an increased number of pods/plant, pod length and several seeds/pod. Negative gains were achieved for time for blooming and full pod maturation. As these traits are used as criteria for selection for earliness, these results favour the identification of short-cycle plants and prone to tolerate dry environments.