Parity Function Research Articles

Continuity risk evaluation of the Bayesian posterior integrity monitoring against multiple faults

Compared to the prior integrity monitoring of global navigation satellite system (GNSS), the posterior integrity monitoring incorporates the used GNSS measurements into the computation of integrity risk, thereby providing a more accurate representation of the reliability of position solution. However, the continuity risk evaluation in posterior integrity monitoring remains an unresolved issue. After formulating the posterior integrity risk of each event hypothesis as a parity function, we propose a method to evaluate the posterior continuity risk by computing the cumulative probability of false alarm within a sphere constructed in the parity space. This approach guarantees a conservative estimate of the probability of false alarm, thereby establishing an upper bound for posterior continuity risk. Experiment and analysis based on positioning cases involving GPS, Galileo, and BeiDou satellites suggest that the posterior integrity monitoring can achieve a lower probability of false alarm compared to prior integrity monitoring under certain integrity monitoring configurations. Furthermore, the evaluation accuracy of probability of false alarm inversely correlates with the severity of the differences in solution separation variances.

Associations of parity and cognitive decline, depression, and chronic comorbidity in West China: Results from WCHAT study.

To investigate the associations between parity (the number of offspring a female has borne) and cognitive function, depression, and chronic comorbidity in Western China. A total of 846 women aged 50-55 years were included in the current analysis. Cognitive status was measured using a 10-item short portable mental status questionnaire (SPMSQ). Depressive symptoms were assessed using the 15-item geriatric depression scale (GDS-15). Other characteristics were self-reported. The associations between parity and cognitive decline, depression, and chronic comorbidity were analyzed using univariable and multivariable models. Multivariable models were adjusted for age, ethnic group, occupation, marital status, educational level, lifestyle factors, and sleeping time. Among the enrolled women, 26.71% were either childless or had one child, 47.40% had two children, 18.32% had three children, and 7.57% had ≥4 children. Compared to women with low parity, women with two or more children exhibited a higher risk of cognitive decline. Moreover, having four or more children was significantly associated with depression and chronic comorbidity. After adjusting covariates, women with three or more children exhibited a higher risk of cognitive decline than those with low parity. However, high parity was not significantly associated with depression or chronic comorbidity after adjustment for covariates. Our study showed that ≥3 children was associated with cognitive decline in women. Longitudinal studies are needed to evaluate this conclusion and to investigate the mechanisms involved. More importantly, families and societies should pay more attention to women's long-term health outcomes related to fertility.

Congruences modulo powers of 5 for the crank parity function

In 1988, Andrews and Garvan introduced the partition statistic “crank” in order to give combinatorial interpretation for Ramanujan’s celebrated partition congruence modulo 11. In 2009, Choi, Kang and Lovejoy established congruences modulo powers of 5 for the crank parity function pC (n) by utilizing the theory of modular forms, where pC (n) denotes the difference between the number of partitions of n with even crank and the number of partitions of n with odd crank. In this paper, we provide a completely elementary proof of this congruence family. Moreover, we also prove several new congruences modulo small powers of 5.

Using Background Knowledge and Random Sampling in Genetic Programming: A Case Study in Learning Boolean Parity Functions

Using Background Knowledge and Random Sampling in Genetic Programming: A Case Study in Learning Boolean Parity Functions

Computational capabilities of a multicellular reservoir computing system.

The capacity of cells to process information is currently used to design cell-based tools for ecological, industrial, and biomedical applications such as detecting dangerous chemicals or for bioremediation. In most applications, individual cells are used as the information processing unit. However, single cell engineering is limited by the necessary molecular complexity and the accompanying metabolic burden of synthetic circuits. To overcome these limitations, synthetic biologists have begun engineering multicellular systems that combine cells with designed subfunctions. To further advance information processing in synthetic multicellular systems, we introduce the application of reservoir computing. Reservoir computers (RCs) approximate a temporal signal processing task via a fixed-rule dynamic network (the reservoir) with a regression-based readout. Importantly, RCs eliminate the need of network rewiring, as different tasks can be approximated with the same reservoir. Previous work has already demonstrated the capacity of single cells, as well as populations of neurons, to act as reservoirs. In this work, we extend reservoir computing in multicellular populations with the widespread mechanism of diffusion-based cell-to-cell signaling. As a proof-of-concept, we simulated a reservoir made of a 3D community of cells communicating via diffusible molecules and used it to approximate a range of binary signal processing tasks, focusing on two benchmark functions-computing median and parity functions from binary input signals. We demonstrate that a diffusion-based multicellular reservoir is a feasible synthetic framework for performing complex temporal computing tasks that provides a computational advantage over single cell reservoirs. We also identified a number of biological properties that can affect the computational performance of these processing systems.

Efficient deterministic MapReduce algorithms for parallelizable problems

The MapReduce framework has firmly established itself as one of the most widely used parallel computing platforms for processing big data on tera- and peta-byte scale. Approaching it from a theoretical standpoint has proved to be notoriously difficult, however. In continuation of Goodrich et al.'s early efforts, explicitly espousing the goal of putting the MapReduce framework on footing equal to that of long-established models such as the PRAM, we investigate the obvious complexity question of how the computational power of MapReduce algorithms compares to that of combinational Boolean circuits commonly used for parallel computations. Relying on the standard MapReduce model introduced by Karloff et al. a decade ago, we develop an intricate simulation technique to show that any problem in NC (i.e., a problem solved by a logspace-uniform family of Boolean circuits of polynomial size and a depth polylogarithmic in the input size) can be solved by a MapReduce computation in O(T(n)/log⁡n) rounds, where n is the input size and T(n) is the depth of the witnessing circuit family. Thus, we are able to closely relate the standard, uniform NC hierarchy modeling parallel computations to the deterministic MapReduce hierarchy DMRC by proving that NCi+1⊆DMRCi for all i∈N. Besides the theoretical significance, this result has important applied aspects as well. In particular, we show for all problems in NC1—many practically relevant ones, such as integer multiplication and division, the parity function, and recognizing balanced strings of parentheses being among these—how to solve them in a constant number of deterministic MapReduce rounds.

Gender differences in the association between parity and cognitive function: Evidence from the UK biobank

While much previous work linking fertility history with late-life cognition has focused on a narrow set of cognitive measures and/or has used modest sample sizes in the analysis, our paper expands the size and scope of these linkages by analyzing cognitive function across five domains and precisely estimating gendered patterns between men and women. Results point to important gendered associations between parity and cognition: having children is likely associated with better cognitive function for fathers in all five domains. However, mothers show worse cognitive function in some domains (i.e., numeric memory, prospective memory, and fluid intelligence) than childless women. We explore the possibility of confounding in these associations and rule out the effects of genetic cognitive ability on fertility. We also find that adding controls for educational attainment differ by gender—strengthening associations between parity and cognition for men and largely eliminating them for women. The findings support previous work done on how life course contexts may link to the risk of dementia or cognitive impairment, highlighting parity as potential protective or risk factors to parents’ cognitive health. The use of five cognitive domains yields variations in results, giving implications on measure selection of cognitive function and calling for replicated work covering more cognitive domains.

Batch Gradient Learning Algorithm with Smoothing L1 Regularization for Feedforward Neural Networks

Regularization techniques are critical in the development of machine learning models. Complex models, such as neural networks, are particularly prone to overfitting and to performing poorly on the training data. L1 regularization is the most extreme way to enforce sparsity, but, regrettably, it does not result in an NP-hard problem due to the non-differentiability of the 1-norm. However, the L1 regularization term achieved convergence speed and efficiency optimization solution through a proximal method. In this paper, we propose a batch gradient learning algorithm with smoothing L1 regularization (BGSL1) for learning and pruning a feedforward neural network with hidden nodes. To achieve our study purpose, we propose a smoothing (differentiable) function in order to address the non-differentiability of L1 regularization at the origin, make the convergence speed faster, improve the network structure ability, and build stronger mapping. Under this condition, the strong and weak convergence theorems are provided. We used N-dimensional parity problems and function approximation problems in our experiments. Preliminary findings indicate that the BGSL1 has convergence faster and good generalization abilities when compared with BGL1/2, BGL1, BGL2, and BGSL1/2. As a result, we demonstrate that the error function decreases monotonically and that the norm of the gradient of the error function approaches zero, thereby validating the theoretical finding and the supremacy of the suggested technique.

Estimating the association between parity and mid‐ to later‐life cognitive function in rural South Africa: Evidence from “Health and Aging in Africa: A Longitudinal Study of an INDEPTH Community in South Africa” (HAALSI)

AbstractBackgroundThe burden of cognitive decline is projected to rise over the next decades, particularly in low‐ and middle‐income countries, contributing to economic hardship, morbidity, and mortality. Reproductive life‐history, including parity, may be associated with later‐life cognitive ability. Using data from the population‐representative “Health and Aging in Africa: A Longitudinal Study of an INDEPTH Community in South Africa” (HAALSI) cohort, we estimated the association between parity and mid‐ to later‐life global cognitive function.MethodData were from in‐person interviews with 2242 men and 2604 women aged ≥40 in the rural Agincourt sub‐district, Mpumalanga province, South Africa in 2014/15. Cognitive function was assessed as time orientation and immediate and delayed recall of a 10‐word list. Number of living children was self‐reported. We used multivariable linear regression with parity (number of children: 0, 1‐2, 3‐4, 5‐9, 10+) as the categorical predictor and z‐standardized cognitive function as the outcome. Analyses were stratified by sex/gender, controlling for age, education, and literacy.ResultAmong men, there was a U‐shaped relationship between parity and cognitive function (p<0.001), whereby men with 3‐4 children had the highest mean z‐standardized cognitive scores (mean: 0.244, SD: 0.984, Table 1, Figure 1). Women also had a U‐shaped relationship between parity and cognitive function (p = 0.0254), whereby women with 5‐9 children had the highest mean z‐standardized cognitive scores (mean: ‐0.017, SD: 0.935, Table 2, Figure 1). Among men, adjusted regression models showed that parity was associated with higher cognitive function, across all categories compared to no children (Estimates [95% CI]: 1‐2: 0.32 [0.17‐0.48], 3‐4: 0.50 [0.35‐0.65], 5‐9: 0.50 [0.36‐0.74], 10+: 0.55 [0.36‐0.74]; Table 3). Among women, only the 5‐9 children group (Estimate [95% CI]: 0.16 [0.00‐0.31]) had cognitive scores that were significantly different from having no children (Table 3).ConclusionData from this representative sample in rural South Africa indicate that the relationship between parity and later life cognition varies by sex/gender, with having children being associated with greater cognitive performance in men, compared to women. This sex/gender difference may be due to sex and gender differences in biological and social roles involved in childbearing and rearing.

The Sample Complexity of Distribution-Free Parity Learning in the Robust Shuffle Model


 
 
 We provide a lowerbound on the sample complexity of distribution-free parity learning in the realizable case in the shuffle model of differential privacy. Namely, we show that the sample complexity of learning d-bit parity functions is Ω(2d/2). Our result extends a recent similar lowerbound on the sample complexity of private agnostic learning of parity functions in the shuffle model by Cheu and Ullman . We also sketch a simple shuffle model protocol demon- strating that our results are tight up to poly(d) factors.
 
 

Нижняя оценка схемной сложности линейной функции в одном бесконечном базисе

The paper is devoted to realization of parity functions by circuits consisting of generalized conjunctors of infinite fan-in. We present a new circuit complexity lower bound for parity function in this model.

Lower Bound of Circuit Complexity of Parity Function in a Basis of Unbounded Fan-In

Lower Bound of Circuit Complexity of Parity Function in a Basis of Unbounded Fan-In

Parallel Quantum Computation Approach for Quantum Deep Learning and Classical-Quantum Models

The paradigm of Quantum computing and artificial intelligence has been growing steadily in recent years and given the potential of this technology by recognizing the computer as a physical system that can take advantage of quantum mechanics for solving problems faster, more efficiently, and accurately. We suggest experimentation of this potential through an architecture of different quantum models computed in parallel. In this work, we present encouraging results of how it is possible to use Quantum Processing Units analogically to Graphics Processing Units to accelerate algorithms and improve the performance of machine learning models through three experiments. The first experiment was a reproduction of a parity function, allowing us to see how the convergence of a given Quantum model is influenced significantly by computing it in parallel. For the second and third experiments, we implemented an image classification problem by training quantum neural networks and using pre-trained models to compare their performances with the same experiments carried out with parallel quantum computations. We obtained very similar results in the accuracies, which were close to 100% and significantly improved the execution time, approximately 15 times faster in the best-case scenario. We also propose an alternative as a proof of concept to address emotion recognition problems using optimization algorithms and how execution times can be positively affected by parallel quantum computation. To do this, we use tools such as the cross-platform software library PennyLane and Amazon Web Services to access high-end simulators with Amazon Braket and IBM quantum experience.

Impact of parity on cardiac structure and function in apparently healthy pregnant women in Nigeria

Abstract Background There are a few and conflicting data in the literature about parity and maternal cardiac function. Purpose The purpose of this study was to assess the impact of parity on cardiac structure and function in apparently healthy pregnant women in Nigeria. Methods This was a cross-sectional study carried out in 4 tertiary centers in Kano, and 1 in Ile- Ife, Nigeria. 112 apparently healthy pregnant women were consecutively recruited between 28thand 38th weeks of gestation. Left ventricular (LV)systolic dysfunction was defined as LV ejection fraction of below 50%, and diastolic dysfunction was graded using mitral filling pattern and tissue velocities. Results LV systolic dysfunction and diastolic dysfunction were found in 6 (5.4%) subjects and 20 (17.9%) subjects respectively. Age (p= <0.0001), left atrial (LA)size (P= <0.0001), interventricular septal thickness at end diastole (IVSD) (p=0.005) and posterior wall thickness at end diastole (PWTD) (p=0.004) increased progressively with parity (Table 1). There was significant positive correlation between parity and age (r=0.475, p= <0.0001), LA size (r=0.332, p= <0.0001), IVSD (r=0.264, p=0.005) and PWD (r=0.343, p= <0.0001). LV systolic function was not significantly associated with parity. Conclusion Our findings suggests that parity causes significant myocardial remodeling in apparently healthy pregnant women that should be taken in to consideration while interpreting their echocardiograms. Funding Acknowledgement Type of funding sources: None.

Congruences modulo powers of 5 for the rank parity function

It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for the rank parity function is f (q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.

252 Degree of proteinuria and duration of expectant management in women with severe preeclampsia

To evaluate the association between the degree of proteinuria and duration of expectant management (EM) and other perinatal outcomes in women with severe preeclampsia (SPE). Retrospective cohort study of women with SPE delivering live, non-anomalous singletons at 23-34 weeks gestational age (GA) at a single tertiary center (2016-2018). Women with proteinuria assessment (24-hour total urine protein or urine protein:creatinine ratio extrapolation) within 3 days of SPE diagnosis were included. Women delivered for an indication other than SPE, i.e., preterm labor, were excluded. Women were categorized by degree of proteinuria (mg): none (less than or equal to 300), mild (301-1000), moderate (1001-3000), and massive (greater than or equal to 3001). The primary outcome was the proportion of potential EM time achieved (%EM), i.e., days of EM achieved divided by maximum potential EM time (days from SPE diagnosis to 34 weeks). Secondary outcomes included delivery GA, days of EM, and perinatal outcomes. Bivariable and multivariable analyses compared outcomes across groups. Of 285 women included, 21% had no proteinuria, 33% mild, 19% moderate, and 27% massive. Demographics differed between groups with respect to GA at diagnosis, age, parity, chronic hypertension, and kidney function (Table 1). %EM was not significantly different between groups (Table 2; adjusted β coefficient 0.5 (95% CI -0.5, 1.4) for mild proteinuria vs none, -0.5 (95% CI -1.6, 0.6) for moderate proteinuria vs none, and -0.6 (95% CI -1.6, 0.5) for massive proteinuria vs none). Increasing proteinuria was associated with earlier delivery GA but not days of EM (Table 2). However, massive proteinuria was associated with an increased odds of maternal (aOR 2.3, 95% CI 1.0, 5.3) and neonatal (aOR 2.5, 95% CI 1.2, 5.4) composite morbidities, compared to no proteinuria (Table 2). Among women with SPE, the degree of proteinuria was not associated with duration of EM. However, massive proteinuria, compared to no proteinuria, was associated with higher odds of maternal and neonatal adverse outcomes.View Large Image Figure ViewerDownload Hi-res image Download (PPT)

Parity, education, and postmenopausal cognitive function.

Parity, education, and postmenopausal cognitive function.

Power of uninitialized qubits in shallow quantum circuits

We study uninitialized qubits, whose initial state is arbitrary and unknown, in relation to the computational power of shallow quantum circuits. To do this, we consider uniform families of shallow quantum circuits with n input qubits, O(log⁡n) initialized ancillary qubits, and nO(1) uninitialized ancillary qubits, where the input qubits only act as control qubits. We show that such a circuit with depth O((log⁡n)2) can compute any symmetric Boolean function on n bits that is computable by a uniform family of polynomial-size classical circuits. Since it is unlikely that this can be done with only O(log⁡n) initialized ancillary qubits, our result provides evidence that the presence of uninitialized ancillary qubits increases the computational power of shallow quantum circuits with only O(log⁡n) initialized ancillary qubits. On the other hand, to understand the limitations of uninitialized qubits, we focus on sub-logarithmic-depth quantum circuits and show the impossibility of computing the parity function on n bits.

Covariance and Spinorial Statistical Description of Simple Relativistic Stochastic Kinematics

It is shown that Generalized Poisson–Kac processes are closed with respect to Lorentz transformations, providing a class of covariant kinematic processes. The transformation properties of the associated partial probability densities (waves) display spinorial character in a probability space, and their spinorial character is intrinsically related to the parametrization of the internal degrees of freedom of the process. Parity function analysis associated with the bias induced in the partial-wave recombination process by a Lorentz boost, indicates a symmetry breaking in the recombination dynamics. In an inertial reference frame moving with constant velocity [Formula: see text] with respect to the rest frame of the process, stochastic fluctuations are progressively damped out till complete suppression in the limit for [Formula: see text].

Sensitivity, affine transforms and quantum communication complexity

In this paper, we study the Boolean function parameters sensitivity (s), block sensitivity (bs), and alternation (alt) under specially designed affine transforms and show several applications. For a function f:F2n→{−1,1}, and A=Mx+b for M∈F2n×n and b∈F2n, the result of the transformation g is defined as ∀x∈F2n,g(x)=f(Mx+b).As a warm up, we study alternation under linear shifts (when M is restricted to be the identity matrix) called the shift invariant alternation (the smallest alternation that can be achieved for the Boolean function f by shifts, denoted by salt(f)). By a result of Lin and Zhang (2017) [7], it follows that bs(f)≤O(salt(f)2s(f)). Thus, to settle the Sensitivity Conjecture (∀f,bs(f)≤poly(s(f))), it suffices to argue that ∀f,salt(f)≤poly(s(f)). However, we exhibit an explicit family of Boolean functions for which salt(f) is 2Ω(s(f)).Going further, we use an affine transform A, such that the corresponding function g satisfies bs(f,0n)≤s(g). We apply this in the setting of quantum communication complexity to prove that for F(x,y)=deff(x∧y), the bounded error quantum communication complexity of F with prior entanglement, Q1/3⁎(F) is Ω(bs(f,0n)). Our proof builds on ideas from Sherstov (2010) [17] where we use specific properties of the above affine transformation. Using this, we show the following.(a)For a fixed prime p and an ϵ, 0<ϵ<1, any Boolean function f that depends on all its inputs with degp(f)≤(1−ϵ)log⁡n must satisfy Q1/3⁎(F)=Ω(nϵ/2log⁡n). Here, degp(f) denotes the degree of the multilinear polynomial over Fp which agrees with f on Boolean inputs.(b)For Boolean function f such that there exists primes p and q with degq(f)≥Ω(degp(f)δ) for δ>2, the deterministic communication complexity - D(F) and Q1/3⁎(F) are polynomially related. In particular, this holds when degp(f)=O(1). Thus, for this class of functions, this answers an open question (see Buhrman and de Wolf (2001) [15]) about the relation between the two measures. Restricting back to the linear setting, we construct linear transformation A, such that the corresponding function g satisfies, alt(f)≤2s(g)+1. Using this new relation, we exhibit Boolean functions f (other than the parity function) such that s(f) is Ω(sparsity(f)) where sparsity(f) is the number of non-zero coefficients in the Fourier representation of f. This family of Boolean functions also rule out a potential approach to settle the XOR Log-Rank conjecture via the recently settled Sensitivity conjecture (Hao Huang (2019) [5]).