Residue Class Research Articles

On the existence of certain Lehmer numbers modulo a prime

A Lehmer number modulo an odd prime numberp is a residue class a∈Fp× whose multiplicative inverse ā has opposite parity. Lehmer numbers that are also primitive roots are called Lehmer primitive roots. Analogously, in this article we say that a residue class a∈Fp× is a Lehmer non-primitive root modulop if a is Lehmer number modulo p which is not a primitive root. We provide explicit estimates for the difference between the number of Lehmer non-primitive roots modulo a prime p and their “expected number”, which is p−1−ϕ(p−1)2. Similar explicit bounds are also provided for the number of k-consecutive Lehmer numbers modulo a prime, and k-consecutive Lehmer primitive roots We also prove that for any prime number p>3.05×1014, there exists a Lehmer non-primitive root modulo p. Moreover, we show that for any positive integer k≥2 (respectively, k≥5) and for all primes p≥exp(122k3) (respectively, p≥exp(6.87k)), there exist k consecutive Lehmer numbers modulo p (respectively, k consecutive Lehmer primitive roots modulo p). For large primes p, these theorems generalize two results which were proven in a paper by Cohen and Trudgian appeared in the Journal of Number Theory in 2019.

Relative perinormality and relative global perinormality in a new pullback construction and idealization

If (S, J) is a zero-dimensional local ring, K = S / J its residue class field and D an integral domain with quotient field K, we show that twin notions of relative perinormality (N. Epstein, J. Shapiro, Perinormality in pullbacks Journal of Commutative Algebra 11 (3):341–362, 2019) and relative global perinormality (H. Klawa, Globally perinormal domains and graded perinormality, Doctoral Dissertation, George Mason University, 2023) are preserved by a pullback construction R = D × K S . As a byproduct, we recover the preservation of these notions in a new pullback construction of Chang et al. and idealization.

Vitamin D receptor is associated with prognostic characteristics of breast cancer after neoadjuvant chemotherapy-an observational study.

Breast cancer (BC) is the most commonly diagnosed malignant tumor in women. The disease and its subsequent treatment pose a serious burden on the quality of life of patients. Neoadjuvant chemotherapy (NAC) has become one of the crucial strategies for the management of BC. Since the identification of the vitamin D receptor (VDR) in mammary tissues, extensive mechanistic research has been conducted on its function. The expression of VDR in BC cells and the tumor microenvironment could be a new prognostic factor for BC after NAC. This observational, single-center study compared data from clinical and histopathological records of 111 female subjects with the expression of VDR in different cellular and tissue components of breast specimens obtained from surgery after NAC. VDR expression was evaluated using an immunoreactive score assigned after immunohistochemistry. Intergroup comparisons and logistic regression were used to identify associations between VDR expression and clinicopathological features of BC. We found that the expression of VDR is associated with various clinical features (i.e., age, menopausal status, and NAC cycle number) and characteristics of prognostic significance, such as residual cancer burden class. Logistic regression analysis revealed that the expression of VDR in the nuclei and cytoplasm of surrounding normal mammary cells predicted vascular invasion and lymph node involvement. The expression of VDR in tumor cells and their microenvironment is related to the clinicopathological characteristics of BC after NAC.

Targeting the Epidermal Growth Factor Receptor Pathway in Chemotherapy-Resistant Triple-Negative Breast Cancer: A Phase II Study.

Epidermal growth factor receptor (EGFR) pathway activation causes chemotherapy resistance, and inhibition of the EGFR pathway sensitizes triple-negative breast cancer (TNBC) cells to chemotherapy in preclinical models. Given the high prevalence of EGFR overexpression in TNBC, we conducted a single-arm phase II study of panitumumab (anti-EGFR monoclonal antibody), carboplatin, and paclitaxel as the second phase of neoadjuvant therapy (NAT) in patients with doxorubicin and cyclophosphamide (AC)-resistant TNBC (NCT02593175). Patients with early-stage, AC-resistant TNBC, defined as disease progression or ≤80% reduction in tumor volume after four cycles of AC, were eligible for this study and received panitumumab (2.5 mg/kg i.v., every week × 13), paclitaxel (80 mg/m2 i.v. every week × 12), and carboplatin (AUC = 4 i.v., every 3 weeks × 4) as the second phase of NAT. A two-stage Gehan-type design was used to detect an improvement in the pathological complete response (pCR)/residual cancer burden class I (RCB-I) rate from 5% to 20%. Whole-exome sequencing was performed on diagnostic tumor biospecimens, where available. From November 3, 2016, through August 23, 2021, 43 patients with AC-resistant TNBC were enrolled. The combined pCR/RCB-I rate was 30.2%. The most common treatment-related adverse events were neutropenia (72%) and anemia (61%), with 7 (16%), 16 (37%), and 8 (19%) patients experiencing grade 4 neutropenia, grade 3 neutropenia, and grade 3 anemia, respectively. No new safety signals were observed. This study met its primary endpoint (pCR/RCB-I = 30.2% vs. 5% in historical controls), suggesting that panitumumab should be evaluated as a component of NAT in patients with chemotherapy-resistant TNBC in a larger, randomized clinical trial. The epidermal growth factor receptor (EGFR) pathway has been implicated as a driver of chemotherapy resistance in triple-negative breast cancer (TNBC). Here, we evaluate the combination of panitumumab, carboplatin, and paclitaxel as the second phase of neoadjuvant therapy (NAT) in patients with AC-resistant TNBC. This study met its primary efficacy endpoint, and molecular alterations in EGFR pathway genes did not seem to influence response to the study regimen.

S-слівна арифметика та високоточні обчислення

The intricacies of using S-word arithmetic, the influence of the value of the parameter S on the estimation of the rounding error are analyzed; what are high-precision calculations and where they are used. The problems of two-key cryptography, computer steganography and the problem of transcomputational complexity are considered as areas of application of S-word arithmetic. For the development of S-word arithmetic algorithms, sequential, parallel, quantum computing models are used, and systems of residual classes are used. The architectural features of the computer system for the implementation of an effective algorithm in various models of calculations are considered. For the parallel computing model, the importance of reducing the connected steps is indicated, which can increase the amount of processed data, but allows to involve a larger number of parallel processors. This approach is in conflict with a method that reduces the amount of processed data, and there is a need to maintain a balance between these two methods in a parallel computing model. For the quantum computing model, the connection of qubits is a key factor in determining the quantum volume. The physical scheme determines which pairs of qubits can be entangled in a quantum computer. Peculiarities of transferring algorithms to another computing model are considered. An analysis of the complexity of implementing S-word arithmetic operations in sequential, parallel, and quantum computing models is carried out. For the parallel computing model, the importance of reducing the connected steps is indicated, which can increase the amount of processed data, but allows to involve a larger number of parallel processors. This approach is in conflict with a method that reduces the amount of processed data, and there is a need to maintain a balance between these two methods in a parallel computing model. For the quantum computing model, the connection of qubits is a key factor in determining the quantum volume. The physical scheme determines which pairs of qubits can be entangled in a quantum computer. Information is provided about the ongoing scientific forum «Calculation optimization issues», the subject of which is closely related to the topic (1969-2023)

The Prime Geodesic Theorem in Arithmetic Progressions

Abstract We address the prime geodesic theorem in arithmetic progressions and resolve conjectures of Golovchanskiĭ–Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.

DNA-directed immobilization fluorescent immunoarray for multiplexed antibiotic residue determination in milk.

The presence of antibiotic residues in cow's milk entails high risk for consumers, the dairy industry, and the environment. Therefore, the development of highly specific and sensitive screening tools for the rapid and cost-effective identification of traces of these compounds is urgently needed. A multiplexed screening platform utilizing DNA-directed immobilization (DDI) was developed aiming to detect three classes of antibiotic residues (fluoroquinolones, sulfonamides, and tylosin) prevalently found in milk. Throughout this work, each oligonucleotide sequence was conjugated to a different hapten molecule, while the three complementary strands were immobilized in 24 independent microarray chips on a single glass slide. First, the array was incubated with the pool of hapten-oligonucleotide conjugate site encoded the signal through DNA hybridization. Next, commercial milk samples were incubated with the cocktail of monoclonal antibodies following a secondary fluorophore-labeled antibody which was required for fluorescent readout. Direct sample detection was achieved in milk diluting 20 times in assay buffer. The limits of detection (LODs) reached were 1.43µgkg-1, 1.67µgkg-1, and 0.89µgkg-1 for TYLA, STZ, and CIP, respectively, which represented in raw milk 7.15µgkg-1, 8.35µgkg-1, and 4.45µgkg-1 for TYLA, STZ, and CIP, respectively, that are below the EU regulatory limits. Cross-reactivity profiles were evaluated against the family of structurally related antibiotics in order to demonstrate the capability to detect antibiotics from the same family of compounds. A pre-validation study was performed by spiking 20 blind samples above and below the maximum residue limits established by the EU guidelines. The system was successfully implemented towards randomized sample classification as compliant or non-compliant. The proposed DDI-based immunoarray provides a fast and cost-effective alternative to obtain semi-quantitative information about the presence of three veterinary residues simultaneously in milk samples.

Synthesis of Self-Checking Circuits for Train Route Traffic Control at Intermediate Stations with Control of Calculations Based on Weight-Based Sum Codes

When synthesizing systems for railway interlocking, it is recommended to use automated models to implement the logic of railway automation and remote control units. Finite-state machines (FSMs) can be implemented on any hardware component. When using relay technology, the functional safety of electrical interlocking is achieved by using uncontrolled (safety) relays with a high coefficient of asymmetry of failures in types 1 → 0 and 0 → 1. When using programmable components, the use of backup and diverse protection methods is required. This paper presents a flexible approach to synthesizing FSMs for railway automation and remote control units that offer both individual and route-based control. Unlike existing solutions, this proposal considers the pre-failure states of railway automation and remote control units during the finite-state machine synthesis stage. This enables the implementation of self-checking and self-diagnostic modules to manage automation units. By increasing the number of states for individual devices and considering the states of interconnected objects, the transition graphs can be expanded. This expansion allows for the synthesis of the transition graph of the control subsystem and other systems. The authors used a field-programmable gate array (FPGA) to implement a finite-state machine. In this case, the proposal is to encode the states of a finite-state machine using weight-based sum codes in the residue class ring based on a given modulus. The best coverage of errors occurring at the outputs of the logic converter in the structure of the FSM can be ensured by selecting the weighting coefficients and the value of the module. This paper presents an example of synthesizing an FPGA-based FSM using state encoding through modular weight-based sum codes. The operation of the synthesized device was modeled. It was found to operate according to the same algorithm as the real devices. When synthesizing self-checking and self-controlled train control devices, it is recommended to consider the solutions proposed in this paper.

Sensing noise exposure and its inequality based on noise complaint data through vision-language hybrid method

This study seeks to reveal urban noise exposure patterns and inequalities using noise complaint data and vision-language hybrid method. By applying a natural language processing model to 17,243 noise complaint records, we uncovered distinct patterns of traffic, industrial, and living noise exposures across residential communities. Our analysis of street view images near complaint locations, utilizing a Residual Network (ResNet) model and Class Activation Mapping (CAM), identified the key environmental elements of different noise sources. Notably, our assessment of noise exposure inequality across 9791 communities yielded a counterintuitive finding: contrary to previous studies in Western contexts, rich communities in China experience higher and more unequal noise exposure compared to average communities, with Gini coefficients exceeding 0.8. This unexpected result likely stems from China's unique rapid urbanization process. Our use of crowdsourced complaint data aligns more closely with human subjective perceptions of noise, offering a novel perspective on noise exposure inequality. These findings challenge existing assumptions about the relationship between socioeconomic status and environmental quality in urban China, and have significant implications for urban planning and noise management strategies in rapidly developing cities.

Infinitude of Palindromic Almost-Prime Numbers

Abstract It is proven that, in any given base, there are infinitely many palindromic numbers having at most six prime divisors, each relatively large. The work involves equidistribution estimates for the palindromes in residue classes to large moduli, offering upper bounds for moments and averages of certain products closely related to exponential sums over palindromes.

Some Properties of Subspaces over Residue Class Rings

Some Properties of Subspaces over Residue Class Rings

USING IFS TO REVEAL BIASES IN THE DISTRIBUTION OF PRIME NUMBERS

It was long assumed that the pseudorandom distribution of prime numbers was free of biases. Specifically, while the prime number theorem gives an asymptotic measure of the probability of finding a prime number and Dirichlet’s theorem on arithmetic progressions tells us about the distribution of primes across residue classes, there was no reason to believe that consecutive primes might “know” anything about each other — that they might, for example, tend to avoid ending in the same digit. Here, we show that the Iterated Function System method (IFS) can be a surprisingly useful tool for revealing such unintuitive results and for more generally studying structure in number theory. Our experimental findings from a study in 2013 include fractal patterns that reveal “repulsive” phenomena among primes in a wide range of classes having specific congruence properties. Some of the phenomena shown in our computations and interpretation relate to more recent work by Lemke Oliver and Soundararajan on biases between consecutive primes. Here, we explore and extend those results by demonstrating how IFS points to the precise manner in which such biases behave from a dynamic standpoint. We also show that, surprisingly, composite numbers can exhibit a notably similar bias.

Interval Valued Anti-Fuzzy R -Subgroups Weak BiIdeals in Gamma near Rings

This research was aimed to establish the cogitation of Gammanear-rings (N-Rings) residue class and Interval Valued Anti Fuzzy R -Sub-groups. In Gamma -N-rings, weak bi-ideals, homomorphism, and endomorphism of interval-valued anti fuzzy R sub-groups week bi-ideals are studied.

Subspace Sum Graphs over Residue Class Rings

Subspace Sum Graphs over Residue Class Rings

Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series

Let $s_d(n)$ be the number of distinct decompositions of the $d$-dimensional hypercube with $n$ rectangular regions that can be obtained via a sequence of splitting operations. We prove that the generating series $y = \sum_{n \geq 1} s_d(n)x^n$ satisfies the functional equation $x = \sum_{n\geq 1} \mu_d(n)y^n$, where $\mu_d(n)$ is the $d$-fold Dirichlet convolution of the Möbius function. This generalizes a recent result by Goulden et al., and shows that $s_1(n)$ also gives the number of natural exact covering systems of $\mathbb{Z}$ with $n$ residual classes. We also prove an asymptotic formula for $s_d(n)$ and describe a bijection between $1$-dimensional decompositions and natural exact covering systems.

Abstract PO1-28-07: Neoadjuvant docetaxel plus cisplatin versus docetaxel plus doxorubicin and cyclophosphamide in early-stage triple-negative breast cancer (HNBC-001): results from a multicenter, randomized controlled, open-label phase II trial

Abstract Background: Adding platinum to anthracycline- and taxane-based neoadjuvant chemotherapy has improved pathological complete response (pCR) in patients with triple-negative breast cancer (TNBC). However, the efficacy of combining docetaxel and platinum without anthracycline remains controversial. Methods: The HNBC-001 trial was a randomized, phase 2 controlled, and open-label investigation carried out in China at 6 hospitals. The participants who were aged 18–70 years, were histologically confirmed for TNBC clinical stage II–III, suitable for potentially curative surgery, and had an Eastern Cooperative Oncology Group performance status (ECOG-PS) of 0 or 1 were selected for this trial. Participants were randomly categorized into two equal groups; those who received docetaxel plus cisplatin (75 mg/m2, respectively) and those who received docetaxel plus doxorubicin and cyclophosphamide (docetaxel 75 mg/m2, doxorubicin 50 mg/m2, and cyclophosphamide 500 mg/m2). These regimens were given every 3 weeks for 6 cycles. Randomization was stratified by tumor size and nodal status. The primary endpoint was the number of individuals achieving a pCR (ypT0/is ypN0). The trial was registered with chictr.org (number ChiCTR-1800019501). Results: Between November 28, 2018, and June 11, 2022, 212 patients were selected (n=106/treatment). The number of individuals who achieved pCR after docetaxel plus cisplatin treatment was 51.9%, and that of those who attained pCR after docetaxel plus doxorubicin and cyclophosphamide was 35.8% (P=0.019). After 21 months of median follow-up [interquartile range (IQR), 13 to 33], 11 of 106 patients (10.4%) in the docetaxel plus cisplatin group and 14 of 106 patients (13.2%) in the docetaxel plus doxorubicin and cyclophosphamide group had event-free survival (EFS) events [95% confidence interval (CI) =0.360 to 1.783, hazard ratio (HR) =0.801, P=0.585]. The incidence of grade 3 or 4 events was similar in both groups [57 (54%) vs. 51 (48%)]. However, no treatment-associated deaths were identified. Conclusions: In stage II to III TNBC, the docetaxel plus cisplatin regimen achieved higher pCR rates than docetaxel plus doxorubicin and cyclophosphamide, with a manageable toxicity profile. Consistent with the literature, the docetaxel plus cisplatin regimen demonstrated a favorable risk-to-benefit profile and could serve as an optimal neoadjuvant chemotherapy option for patients with high-risk TNBC. table 1: Patient characteristics figure 1: Study endpoints Error bars denote 95% CIs based on normal approximation. p values were calculated from the Chi-square test. (A) Frequency of patients who achieved a pathological complete response per treatment group (primary endpoint). (B) Frequency of patients who achieved a clinical breast tumour response per treatment group, assessed by serial MRI scans after completion of neoadjuvant treatment. (C) Frequency of patients who achieved a minimal residual disease (residual cancer burden class 0 or 1). (D) Frequency of patients who received breast-conservation surgery. table 2: Treatment-emergent adverse events Citation Format: Dechuang Jiao, Jianghua Qiao, Chengzheng Wang, Juntao Li, Xianfu Sun, Zhenduo Lu, Chongjian Zhang, Lianfang Li, Min Yan, Yueqing Feng, Yong Zhou, Miao Deng, Xinlan Liu, Mingde Ma, Haiquan Jia, Qingxin Xia, Xiuchun Chen, Zhenzhen Liu. Neoadjuvant docetaxel plus cisplatin versus docetaxel plus doxorubicin and cyclophosphamide in early-stage triple-negative breast cancer (HNBC-001): results from a multicenter, randomized controlled, open-label phase II trial [abstract]. In: Proceedings of the 2023 San Antonio Breast Cancer Symposium; 2023 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2024;84(9 Suppl):Abstract nr PO1-28-07.

Reprint of: Robust inference on correlation under general heterogeneity

Considerable evidence in past research shows size distortion in standard tests for zero autocorrelation or zero cross-correlation when time series are not independent identically distributed random variables, pointing to the need for more robust procedures. Recent tests for serial correlation and cross-correlation in Dalla, Giraitis, and Phillips (2022) provide a more robust approach, allowing for heteroskedasticity and dependence in uncorrelated data under restrictions that require a smooth, slowly-evolving deterministic heteroskedasticity process. The present work removes those restrictions and validates the robust testing methodology for a wider class of innovations and regression residuals allowing for heteroscedastic uncorrelated and non-stationary data settings. The updated analysis given here enables more extensive use of the methodology in practical applications. Monte Carlo experiments confirm excellent finite sample performance of the robust test procedures even for extremely complex white noise processes. The empirical examples show that use of robust testing methods can materially reduce spurious evidence of correlations found by standard testing procedures.

Fragmentation characteristics-based nontargeted screening method of exogenous chemical residues in animal-derived foods using reversed-phase and hydrophilic interaction liquid chromatography–high-resolution mass spectrometry

Fragmentation characteristics are crucial for nontargeted screening to discover and identify unknown exogenous chemical residues in animal-derived foods. In this study, first, fragmentation characteristics of 51 classes of exogenous chemical residues were summarized based on experimental mass spectra of standards in reversed-phase and hydrophilic interaction liquid chromatography–high-resolution mass spectrometry (MS) and mass spectra from the MassBank of North America (MoNA) library. According to the proportion of fragmentation characteristics to the total number of chemical residues in each class, four screening levels were defined to classify 51 classes of chemical residues. Then, a nontargeted screening method was developed based on the fragmentation characteristics. The evaluation results of 82 standards indicated that more than 90 % of the chemical residues with MS/MS spectra can be identified at concentrations of 100 and 500 μg/kg, and about 80 % can be identified at 10 μg/kg. Finally, the nontargeted screening method was applied to 16 meat samples and 21 egg samples as examples. As a result, eight chemical residues and transformation products (TPs) of 5 classes in the exemplary samples were found and identified, in which 3 TPs of azithromycin were identified by fragmentation characteristics-assisted structure interpretation. The results demonstrated the practicability of the nontargeted screening method for routine risk screening of food safety.

Safety and efficacy of anlotinib combined with taxane and lobaplatin in neoadjuvant treatment of clinical stage II/III triple-negative breast cancer in China (the neoALTAL trial): a single-arm, phase 2 trial

Anlotinib is a new type of tyrosine kinase inhibitor that targets vascular endothelial growth factor receptors 1/2/3, platelet-derived growth factor receptors α/β, and fibroblast growth factor receptors 1-4 and c-Kit, with a broad spectrum of inhibitory effects on tumor angiogenesis and growth. It has been proven effective in HER2-negative metastatic breast cancer, but its efficacy in early-stage triple-negative breast cancer (TNBC) is unknown. This phase 2 study aims to evaluate the efficacy and safety of adding anlotinib to neoadjuvant chemotherapy in patients with TNBC. Patients with clinical stage II/III TNBC were treated with 5 cycles of anlotinib (12mg, d1-14, q3w) plus 6 cycles of taxanes (docetaxel 75 mg/m2 ,d1, q3w or nab-paclitaxel 125 mg/m2, d1 and d8, q3w) and lobaplatin (30 mg/m2, d1, q3w), followed by surgery. The primary endpoint was pathological complete response (pCR; ypT0/is ypN0) and the secondary endpoints include breast pCR (bpCR), axillary pCR (apCR), residual cancer burden (RCB), objective response rate (ORR), survival, and safety. Exploratory endpoints were efficacy biomarkers based on Fudan University Shanghai Cancer Center Immunohistochemical (FUSCC IHC) classification for TNBC and next-generation sequencing (NGS) of DNA from tumor tissue and blood samples of patients with 425-gene panel. This trial is registered with www.chictr.org.cn (ChiCTR2100043027). From Jan 2021 to Aug 2022, 48patients were assessed and 45 were enrolled. All patients received at least one dose of study treatment and underwent surgery. The median age was 48.5 years (SD: 8.7), 71% were nodal involved, and 20% had stage III. In the intention-to-treat population, 26 out of 45patients achieved pCR (57.8%; 90% CI, 44.5%-70.3%), and 39 achieved residual cancer burden class 0-I (86.7%; 95% CI, 73.2%-94.9%). The bpCR and apCR rate were 64.4% (29/45) and 71.9% (23/32), respectively. No recurrence or metastasis occurred during the short-term follow-up. Based on the FUSCC IHC-based subtypes, the pCR rates were 68.8% (11/16) for immunomodulatory subtype, 58.3% (7/12) for basal-like immune-suppressed subtype and 33.3% (4/12) for luminal androgen receptor subtype, respectively. NGS revealed that the pCR were 77% (10/13) and 50% (14/28) in MYC-amplified and wild-type patients, respectively, and 78% (7/9) and 53% (17/32) in gBRCA1/2-mutated and wild-type patients, respectively. The median follow-up time of the study was 14.9 months (95% CI: 13.5-16.3 months). There was no disease progression or death during neoadjuvant therapy. No deaths occurred during postoperative follow-up. In the safety population (N=45), Grade 3 or 4 treatment emergent adverse events occurred in 29patients (64%), and the most common events were neutropenia (38%), leukopenia (27%), thrombocytopenia (25%), anemia (13%), and hypertension (13%), respectively. The addition of anlotinib to neoadjuvant chemotherapy showed manageable toxicity and encouraging antitumor activity for patients with clinical stage II/III TNBC. Chongqing Talents Project, Chongqing Key Project of Technology Innovation and Application Development and Chongqing Outstanding Youth Natural Science Foundation.

The distribution of the multiplicative index of algebraic numbers over residue classes

The distribution of the multiplicative index of algebraic numbers over residue classes