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You are given an integer array nums of length n. An integer k is called sortable if k divides n and you can sort nums in non-decreasing order by sequentially performing the following operations: Partition nums into consecutive subarrays of length k. Cyclically rotate each subarray independently any number of times to the left or to the right. Return an integer denoting the sum of all possible sortable integers k.

You are given an integer array nums of length n. An integer k is called sortable if k divides n and you can sort nums in non-decreasing order by sequentially performing the following operations: Partition nums into consecutive subarrays of length k. Cyclically rotate each subarray independently any number of times to the left or to the right. Return an integer denoting the sum of all possible sortable integers k.

# CrossMemoryEnergy **Medication Safety Risk Stratification Based on Student Physical Fitness Monitoring and Medication Exposure Data** CrossMemoryEnergy is a multimodal learning framework designed for medication safety risk stratification by integrating physical fitness monitoring data with medication exposure information. The framework combines cross-modal representation alignment, temporal memory stabilization, and L2 regularization to estimate medication safety risk probabilities from participant-level data. ## Overview Medication safety risk prediction often relies on medication histories and clinical records, while physical fitness information may provide complementary physiological context. CrossMemoryEnergy integrates these heterogeneous data sources within a unified prediction framework. The model is designed to: * Integrate physical fitness and medication exposure features at the participant level. * Learn shared representations from heterogeneous data modalities. * Preserve historical information through exponential moving average updates. * Estimate medication safety risk probabilities using a regularized neural network. * Support risk stratification and longitudinal prediction evaluation. * Evaluate predictive performance across multiple real-world datasets. ## Key Features * **Multimodal representation learning:** Projects physical fitness and medication exposure features into a shared latent space. * **Temporal memory stabilization:** Uses an exponential moving average (EMA) to incorporate historical representations. * **Regularized optimization:** Applies L2 regularization to modality-specific projection parameters. * **Risk probability estimation:** Produces continuous medication safety risk scores using a sigmoid prediction head. * **Comprehensive evaluation:** Includes baseline comparisons, ablation studies, hyperparameter sensitivity analysis, interpretability analysis, and cross-dataset evaluation. ## Model Architecture CrossMemoryEnergy consists of three principal components. ### 1. Cross-Modal Alignment Mechanism Physical fitness features and medication exposure features are processed by separate projection layers and mapped into a shared 128-dimensional representation space. The aligned representation is defined as: $$ Z_t = H_{f,t} + H_{m,t} $$ where: * \(H_{f,t}\) represents the projected physical fitness features. * \(H_{m,t}\) represents the projected medication exposure features. * \(Z_t\) represents the fused multimodal representation. The model uses modality-specific linear projections followed by representation summation. It does not use an attention layer in the implemented architecture. ### 2. Memory Stabilization Framework For longitudinal observations, the model maintains historical information through an exponential moving average: $$ M_t = \gamma M_{t-1} + (1-\gamma)Z_t $$ where: * \(M_t\) is the stabilized representation at time \(t\). * \(Z_t\) is the current multimodal representation. * \(\gamma\) controls the contribution of historical information. The default memory coefficient is \(\gamma = 0.8\). The first valid representation initializes the memory state, and sequence masking excludes padded observations. ### 3. Energy-Based Regularization Strategy The model applies an L2 penalty to the modality-specific projection matrices: $$ E(\theta) = \|W_f\|_2^2 + \|W_m\|_2^2 $$ The training objective combines binary cross-entropy loss with the regularization penalty: $$ \mathcal{L} = \mathcal{L}_{BCE} + \lambda E(\theta) $$ The default regularization coefficient is \(\lambda = 0.001\). Here, "energy" refers to parameter regularization in the optimization objective, rather than physical energy consumption or a separate optimization algorithm. ### Prediction Head The stabilized representation is passed through a two-layer prediction head: `128 → 64 → 1` ReLU activation and dropout are used in the hidden layers, followed by sigmoid activation to generate a medication safety risk probability between 0 and 1. ## Input Features ### Physical Fitness Features The physical fitness modality includes measurements such as: * Body mass index (BMI) * Cardiovascular endurance * Muscular strength * Flexibility * Agility Continuous fitness variables are standardized within age and sex strata using statistics estimated from the training subset. ### Medication Exposure Features The medication exposure modality includes: * Medication class * Daily dose * Administration frequency * Exposure duration * Over-the-counter medication use * Concurrent medication use Medication classes are encoded categorically, and continuous exposure variables are normalized using training-subset statistics. ### Temporal Information For longitudinal modeling, observations are organized by their recorded timestamps. When a modality is not newly observed, the most recent valid measurement available before the prediction time may be retained together with a modality availability indicator. All observations from the same participant must remain within a single data partition to prevent participant-level information leakage. ## Experimental Setup The study evaluates CrossMemoryEnergy on four retrospective real-world datasets. | Dataset | Participants | Study Period | Outcome Prevalence | | ------------------------------------------- | -----------: | ------------ | -----------------: | | Student Physical Fitness Assessment | 3,842 | 2019–2023 | 8.4% | | Medication Exposure Tracking | 4,216 | 2018–2023 | 9.1% | | Pharmacoepidemiological Risk Stratification | 5,128 | 2017–2023 | 10.6% | | Real-World Medication Safety Monitoring | 6,034 | 2016–2023 | 11.2% | The study also uses a separate longitudinal fitness–medication monitoring cohort for temporal validation, comprising 2,486 participants and 12,938 repeated observations. The reported datasets were accessed through institutional research resources and authorized databases. They should not be assumed to be publicly downloadable. ### Data Partitioning Each primary dataset is partitioned at the participant level: * Training set: 70% * Validation set: 15% * Independent test set: 15% Preprocessing statistics, categorical mappings, and class weights are derived from the training subset. Model selection and classification threshold selection use the validation subset, while final evaluation uses the independent test subset. ## Implementation Details The experimental configuration reported in the paper is summarized below. | Parameter | Configuration | | ---------------------------------------- | ---------------------------------------------------- | | Framework | PyTorch | | Python version | 3.8 | | Operating system | Ubuntu 20.04 LTS | | Reported GPU | NVIDIA Tesla V100 | | Fitness projection dimension | 128 | | Medication projection dimension | 128 | | Shared representation dimension | 128 | | Prediction hidden dimension | 64 | | Dropout rate | 0.5 | | Memory coefficient (\(\gamma\)) | 0.8 | | Regularization coefficient (\(\lambda\)) | 0.001 | | Optimizer | Adam | | Initial learning rate | 0.001 | | Batch size | 64 | | Maximum training epochs | 50 | | Loss function | Weighted binary cross-entropy with L2 regularization | | Checkpoint selection | Highest validation ROC AUC | The paper reports ten independent experimental runs with different random seeds. **Note:** These are the experimental settings described in the manuscript. Executable implementation files, dependency versions, and exact reproduction commands must be provided separately before the repository can be considered fully reproducible. ## Results The reported mean ROC AUC values for CrossMemoryEnergy across the four primary datasets are shown below. | Dataset | Accuracy | ROC AUC | | ------------------------------------------- | -------: | ------: | | Student Physical Fitness Assessment | 89.90% | 88.77% | | Medication Exposure Tracking | 91.34% | 90.19% | | Pharmacoepidemiological Risk Stratification | 89.51% | 88.28% | | Real-World Medication Safety Monitoring | 90.98% | 89.98% | The results are reported as mean values across ten independent runs. According to the study, CrossMemoryEnergy achieved the highest mean ROC AUC among the evaluated baseline methods on all four datasets. ### Comparison with Baseline Methods The evaluated methods include: * Logistic Regression * Random Forest * XGBoost * LightGBM * Multilayer Perceptron (MLP) * Gated Recurrent Unit (GRU) * DistilBERT * RoBERTa * MobileBERT * ALBERT * XLNet * Longformer The strongest baseline varied by dataset. The reported mean ROC AUC improvements over the strongest baseline were 1.90, 1.44, 0.38, and 0.56 percentage points, respectively. ### Ablation and Additional Analyses The study also examines: * The contribution of cross-modal alignment. * The contribution of temporal memory stabilization. * The effect of L2 regularization. * Sensitivity to the memory and regularization coefficients. * Potential confounding by demographic and health-related factors. * Feature contributions using SHAP. * Rare-outcome classification and probability calibration. * Generalization across different datasets. * The predictive contribution of combining both input modalities. The reported findings support the predictive value of integrating physical fitness and medication exposure information within the evaluated populations. ## Risk Stratification The model produces a continuous predicted probability of a medication safety event. For the risk-stratification analysis described in the paper, predictions are grouped into three categories: | Risk Category | Predicted Probability | | ----------------- | ---------------------- | | Low Risk | \(p < 0.30\) | | Intermediate Risk | \(0.30 \leq p < 0.60\) | | High Risk | \(p \geq 0.60\) | These categories are intended for research evaluation and are not validated clinical decision thresholds. The study reports that observed medication safety event rates increased across the three risk strata. However, the calibration analysis also identified discrepancies between predicted probabilities and observed event frequencies. ## Repository Structure The following is a suggested repository layout for organizing the implementation and experimental materials. It is a proposed structure, not a claim about files already available. ```text CrossMemoryEnergy/ ├── README.md ├── requirements.txt ├── configs/ │ └── default.yaml ├── data/ │ └── README.md ├── models/ │ ├── cross_modal_alignment.py │ ├── memory_stabilization.py │ └── cross_memory_energy.py ├── scripts/ │ ├── preprocess.py │ ├── train.py │ └── evaluate.py ├── experiments/ │ ├── ablation.py │ └── cross_dataset_evaluation.py ├── tests/ └── LICENSE ``` ## Getting Started The manuscript describes a PyTorch implementation, but a complete runnable implementation and its dependency manifest have not been established by the supplied document alone. Before running experiments, prepare the source code, install the pinned dependencies, configure the dataset paths, and implement the preprocessing and participant-level splitting procedures described above. Training and evaluation commands should be documented here once the corresponding scripts are available. ## Data Availability and Privacy The study uses de-identified physical fitness records, medication exposure records, and medication safety outcomes obtained through institutional research resources and authorized databases. The source data should not be redistributed without appropriate authorization. Any implementation using similar information must follow applicable institutional requirements, data access agreements, privacy protections, and ethical approvals. No individual-level health or medication records should be committed to this repository. ## Limitations * The reported findings are based on retrospective observational data. * Predictive associations do not establish causal medication effects. * External and prospective validation is needed to assess generalizability. * The reported calibration results indicate that predicted probabilities require further evaluation before potential clinical use. * The model's risk categories should not be interpreted as treatment recommendations or validated clinical thresholds. * Reproduction of the reported results requires access to suitable data, implementation details, and the corresponding experimental pipeline. ## Citation If you use this framework or build upon the associated research, please cite the original work once its publication details are verified. **Manuscript title:** *Medication Safety Risk Stratification Based on Student Physical Fitness Monitoring and Medication Exposure Data: A Real-World Pharmacoepidemiological Study* **Authors:** Fei Zhou, Donghui Wang, and Chongmin Li. A DOI, publication year, and final publication URL should be added after they are verified. ## License A license has not yet been specified. Add a `LICENSE` file after deciding how the source code may be used, modified, and redistributed. ## Disclaimer CrossMemoryEnergy is a research framework for medication safety risk prediction. It is not a medical device, does not establish causal relationships, and should not be used as a substitute for professional medical judgment or clinical evaluation.

You are given a 0-indexed integer array nums and a target element target. A target index is an index i such that nums[i] == target. Return a list of the target indices of nums after sorting nums in non-decreasing order. If there are no target indices, return an empty list. The returned list must be sorted in increasing order.

# OptimalFutureConfidence — reconstructed research implementation This is a **new implementation reconstructed from the supplied 62-page manuscript**, not the original research software. It includes runnable core components and research-workflow scaffolding. It **has not reproduced** any reported PISA/C19 numerical results because the raw analytic datasets, exact validated feature lists, and historical experimental artifacts were not supplied. Do not describe it as a verified reproduction until the procedures are executed and outputs checked. ## Manuscript mapping - Sec. 3.3/4.4: encoder 256→128, entropic OT + barycentric projection, one-step latent transformation (stochastic dimension 64), confidence gate. - Sec. 3.4/4.4.5: BCE + OT + representation preservation + Brier + hinge constraint. - Sec. 4.1: country-specific, weighted ESCS first quartile; each of ten math plausible values compared with the corresponding weighted 75th percentile of **all eligible** students in that country; majority vote >=5. - Sec. 4.5.3: BRS six items, reverse 2/4/6, mean score >4.30 high resilience; **independent** C19 fitting. - Sec. 4.3: AUROC, AUPRC, F1, balanced accuracy, Brier, 10-bin ECE, NLL; paired t tests, 95% t CI, Holm adjustment. - Sec. 4.4: batch 256, AdamW, cosine annealing, patience 20, seeds 42/52/62/72/82. ## Installation ```bash python -m venv .venv source .venv/bin/activate pip install -r requirements.txt python -m pytest -q ``` ## Prepare legal public-use data 1. Download the original PISA 2018 / PISA 2022 files from official OECD sources, and C19 ISWS from https://doi.org/10.5281/zenodo.5946964, observing their data-use conditions. 2. Decode original questionnaire data using the relevant source codebook. Merge PISA student and school files **by country and school identifier** if using school-level predictors; retain unique student identifiers. Resolve all year-specific column names and answer-code differences using the official year-specific dictionaries. 3. Inspect `configs/pisa_template.json` or `configs/c19_template.json`. Replace each example field with actual column names; **set a fully audited set of non-achievement predictors** and, only after validating the mappings, switch `verified_mapping` to `true`. 4. Inputs: CSV, Parquet, Stata DTA or SPSS SAV. The full manuscript harmonized feature inventory and exact cross-cycle category recodes were **not contained in the supplied PDF**, so they cannot be reconstructed without extra documentation. ## Run a verified dataset ```bash python scripts/run_experiment.py --config configs/pisa_template.json --model OptimalFutureConfidence --seed 42 --out outputs/pisa18_ofc_42 python scripts/run_experiment.py --config configs/pisa_template.json --model XGBoost --seed 42 --out outputs/pisa18_xgb_42 python scripts/run_experiment.py --config configs/pisa_template.json --model OptimalFutureConfidence --seed 42 --group-by-school --out outputs/pisa18_school_42 python scripts/run_experiment.py --config configs/pisa_template.json --model OptimalFutureConfidence --seed 42 --remove future --out outputs/pisa18_no_future_42 python scripts/run_experiment.py --config configs/c19_template.json --model OptimalFutureConfidence --seed 42 --out outputs/c19_ofc_42 python scripts/aggregate.py outputs/*/metrics.json --output outputs/summary.json ``` Templates deliberately fail before real data/feature mappings are supplied. This prevents pretending that an arbitrary feature list reproduces the study. Run each model with seeds 42,52,62,72,82 to obtain paired comparisons. `--remove` supports OT, future, calibration, constraint, and parameters include `--epsilon`, `--delta`, `--stochastic-dim`. ## Scope and limitations **Implemented:** core network; balanced minibatch Sinkhorn; sample-wise out-of-sample transport with frozen training-derived target duals; major loss terms; training/validation/test partitioning, group-disjoint school split; train-only imputation/scaling/one-hot; PISA and BRS label construction; four conventional baselines; calibration metrics; significance aggregation; basic tests. **Not complete:** source data inventory and true cross-cycle harmonization mappings; missing PISA design nuances / plausible-value sampling uncertainty and replicate-weight inference; full hyperparameter search; true TabNet and FT-Transformer comparisons; per-table plotting, source-matched model complexity/FLOPs; end-to-end replication of Tables 2–12; direct PISA 2018→2022 input alignment/transfer. A group shuffle split maintains school separation but does **not** guarantee the paper's claimed close preservation of country/class distributions. Per-model baseline hyperparameter tuning is simplified. ### Implementation choices that cannot be verified against the manuscript The paper does not specify exact reference bank size or target-potential maintenance procedure. The reconstruction uses a capped 512-point training-only bank (configurable), refreshed after each epoch, and Sinkhorn target dual values fitted on a training-only subset at each validation checkpoint. This is a practical approximation, not proof of algorithmic equivalence. For inference, a deterministic zero-noise latent realization is used; the paper gives a conceptual stochastic expectation but no Monte Carlo sample count. The training objective and formulas are structurally similar, but historical results should not be expected to match. The code should be audited against the full original source and official datasets before public claims of reproducibility. ## Data governance No participant data or results are bundled. Only use authorized, de-identified releases. Do not publicly upload identifying or restricted records.

Given an array nums that represents a permutation of integers from 1 to n. We are going to construct a binary search tree (BST) by inserting the elements of nums in order into an initially empty BST. Find the number of different ways to reorder nums so that the constructed BST is identical to that formed from the original array nums. For example, given nums = [2,1,3], we will have 2 as the root, 1 as a left child, and 3 as a right child. The array [2,3,1] also yields the same BST but [3,2,1] yields a different BST. Return the number of ways to reorder nums such that the BST formed is identical to the original BST formed from nums. Since the answer may be very large, return it modulo 109 + 7.