Model: cristiano-sartori/stella_finetuned2_128d
Author: cristiano-sartori
Pipeline: sentence-similarity
Library: sentence-transformers
Tags: sentence-transformers, safetensors, new, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:1140, loss:MatryoshkaLoss, loss:MultipleNegativesRankingLoss, custom_code, en, arxiv:1908.10084, arxiv:2205.13147, arxiv:1705.00652, base_model:NovaSearch/stella_en_400M_v5, base_model:finetune:NovaSearch/stella_en_400M_v5, license:apache-2.0, model-index, text-embeddings-inference, endpoints_compatible, region:us
Base models (finetune): id: NovaSearch/stella_en_400M_v5
Card metadata: language: en
license: apache-2.0
tags: sentence-transformers, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:1140, loss:MatryoshkaLoss, loss:MultipleNegativesRankingLoss
base_model: NovaSearch/stella_en_400M_v5
widget: source_sentence: Give well chosen examples of applications that can be evaluated with the single metric derived from Precision/Recall and illustrate:
• a situation where more weight should be given to Precision;
• a situation where more weight should be given to Recall.
sentences: To understand the function of processors in a reorder buffer, let's break down the concept step by step.

### Step 1: Define the Reorder Buffer (ROB)
The reorder buffer is a hardware structure found in modern processors, particularly those that implement out-of-order execution. Its primary purpose is to manage instruction completion and ensure that results are committed in the correct order.

### Step 2: Explain Out-of-Order Execution
In out-of-order execution, instructions are processed as resources (like execution units) become available, rather than strictly in the order they appear in the program. This can enhance performance by allowing the processor to utilize idle resources effectively.

### Step 3: Identify the Role of the Reorder Buffer
The reorder buffer plays a crucial role in coordinating the results of out-of-order execution. It serves the following functions:

1. **Instruction Tracking**: The ROB keeps track of all instructions that have been issued and are either currently being executed or waiting to be executed. This includes both completed instructions and those that are still in the pipeline.

2. **Result Storage**: Once an instruction is executed, its result is temporarily stored in the ROB rather than being immediately written back to the register file or memory. This ensures that results can be held until it is safe to commit them.

3. **Ensuring In-Order Commitment**: The primary function of the ROB is to ensure that results are committed in program order. Even though instructions may execute out-of-order, they must be committed back to the architectural state (i.e., the register file and memory) in the original order they appeared in the program. The ROB checks for any potential hazards (like exceptions or mispredicted branches) before proceeding with the commitment.

### Step 4: Handling Exceptions and Branch Mispredictions
The reorder buffer also plays a vital role in handling exceptions and branch mispredictions:
- If an exception occurs in any instruction, the ROB will prevent the commitment of any subsequent instructions that depend on the results of the failed instruction.
- In the case of a branch misprediction, the ROB allows the processor to discard the results of incorrectly speculated instructions without affecting the correct program state.

### Step 5: Conclusion
In summary, the function of processors in a reorder buffer is to:
- Keep track of all instructions that are in-flight (executing or waiting).
- Store their results temporarily until it is safe to commit them.
- Ensure that results are committed in the order they were issued, while also handling exceptions and mispredictions effectively.

This organized structure contributes significantly to the efficiency and correctness of modern processors that leverage out-of-order execution., The transformation from total-order broadcast to consensus can be achieved by having each process propose a value and use the total-order broadcast to disseminate these proposals. Each process listens for all proposals and waits until it receives a predetermined number of messages (a quorum). After collecting the proposals, each process selects the first value that appears in the total order and is agreed upon by the quorum. This guarantees that all processes reach a consensus on the same value while utilizing the ordered nature of the broadcast to maintain consistency., Answer: 

1. **Applications Evaluated with Precision/Recall**: Common applications include medical diagnosis, spam detection, and information retrieval. 

2. **Weight on Precision**: In spam detection, a higher precision is crucial because misclassifying legitimate emails as spam (false positives) can lead to loss of important communication. Therefore, it is vital to ensure that the emails classified as spam are indeed spam.

3. **Weight on Recall**: In medical diagnosis for a serious disease, recall should be prioritized to ensure that as many actual positive cases (patients with the disease) are identified as possible. Missing a positive case (false negatives) can have severe consequences for patient health, so it is essential to catch as many true positives as possible, even at the expense of precision., source_sentence: Consider the following problem where we are given an edge-colored graph and we  wish to find a spanning tree that contains a specified number of edges of each color: \begin{description} \item[Input:] A connected undirected graph $G=(V,E)$ where the edges $E$ are partitioned into $k$ color classes $E_1, E_2, \dots, E_k$. In addition each color class $i$ has a target number $t_i \in \mathbb{N}$. \item[Output:] If possible, a spanning tree $T \subseteq E$ of the graph satisfying the color requirements: \begin{align*} |T \cap E_i| = t_i \qquad  \mbox{ for $i=1,\dots, k$.} \end{align*} Otherwise, i.e., if no such spanning tree $T$ exists, output that no solution exists. \end{description} \noindent {Design} a polynomial time algorithm for the above problem. You should analyze the correctness of your algorithm, i.e.,  why it finds a solution if possible. To do so, you are allowed to use algorithms and results seen in class without reexplaining them.
sentences: To solve the problem of finding a spanning tree in an edge-colored graph with specified color constraints, we can utilize a combination of graph theory concepts and flow networks. Hereâs a step-by-step guide to designing a polynomial-time algorithm for this problem.

### Step 1: Understanding the Problem

We have a connected undirected graph \( G = (V, E) \) with edges partitioned into \( k \) color classes \( E_1, E_2, \ldots, E_k \). For each color class \( i \), we have a target number \( t_i \) which specifies how many edges of that color must be included in the spanning tree \( T \).

### Step 2: Formulating the Problem as a Flow Network

1. **Graph Construction**: We will construct a flow network based on the input graph \( G \).
   - Create a source node \( s \) and a sink node \( t \).
   - For each color class \( i \):
     - Create two nodes \( u_i \) (representing the start of color class \( i \)) and \( v_i \) (representing the end of color class \( i \)).
     - Connect \( s \) to \( u_i \) with an edge that has a capacity equal to \( t_i \) (the target number of edges of color \( i \)).
     - Connect \( v_i \) to \( t \) with an edge that has infinite capacity.
     - For each edge \( e \in E_i \) (color class \( i \)), create an edge from \( u_i \) to \( v_i \) with capacity 1. This represents that we can use at most one edge of color \( i \) in the spanning tree.

2. **Connecting Color Classes**: For each vertex \( v \in V \):
   - Connect \( v \) to the respective \( v_i \) nodes for all color classes \( i \) where there are edges incident to \( v \).

### Step 3: Flow Network Algorithm

We can now apply a max-flow algorithm (e.g., Edmonds-Karp or Dinic's algorithm) to find the maximum flow in this network:

- **Max-Flow Calculation**: The maximum flow from \( s \) to \( t \) must equal the sum of the targets \( \sum_{i=1}^k t_i \) in order to satisfy the constraints for the spanning tree.
- **Feasibility Check**: If the maximum flow equals \( \sum_{i=1}^k t_i \), then there exists a way to select the edges such that the spanning tree conditions are met. Otherwise, it's impossible to construct such a spanning tree.

### Step 4: Extracting the Spanning Tree

If the flow is equal to \( \sum_{i=1}^k t_i \):
- We can reconstruct the spanning tree \( T \) by tracking which edges were used in the flow from \( s \) to \( t \). For every edge that contributes to the flow, we include it in \( T \).

### Step 5: Correctness and Complexity Analysis

- **Correctness**: The algorithm guarantees that if the maximum flow is equal to \( \sum_{i=1}^k t_i \), we can find a spanning tree that satisfies the color constraints. This is because the flow model directly enforces the number of edges taken from each color class.
- **Complexity**: The max-flow algorithm runs in polynomial time (specifically \( O(VE^2) \) for Edmonds-Karp), which is efficient for our purpose given that \( |E| \) is the number of edges in the graph.

### Conclusion

To summarize, we can find a spanning tree with specified color constraints by modeling the problem as a flow network and applying a max-flow algorithm. If the flow meets the required total, we can reconstruct the spanning tree; otherwise, we declare that no solution exists. This approach is both systematic and efficient, adhering to the requirements laid out in the problem statement., The Spectre attack exploits speculative execution in modern processors to access sensitive data. To determine if it can be adapted to Itanium architecture, we need to analyze its features:

1. **Speculative Execution**: Itanium does support speculative execution, but its architecture is significantly different from x86 and other common architectures where Spectre was first demonstrated.

2. **Instruction Set**: Itanium uses a Very Long Instruction Word (VLIW) architecture, which changes how instructions are issued and executed compared to traditional architectures. The way speculative execution is performed may not directly translate to Itanium's model.

3. **Branch Prediction**: Spectre attacks often rely on branch prediction mechanisms to influence the speculative execution path. Itanium employs a different branch prediction strategy, which could alter the effectiveness of the attack.

4. **Memory Access Patterns**: The way Itanium handles memory access and caching may also differ, affecting how side-channel information can be leaked or exploited.

5. **Mitigation Strategies**: To protect sensitive parts of code on Itanium, developers can implement the following strategies:
   - **Avoid Speculative Execution**: Use compiler options or code patterns that minimize speculative execution paths.
   - **Data Isolation**: Separate sensitive operations from less sensitive ones, ensuring that speculative execution cannot influence access to sensitive data.
   - **Control Flow Integrity**: Implement control flow integrity checks to ensure that the execution flow does not deviate unexpectedly.

In conclusion, while it is theoretically possible to adapt Spectre attacks to Itanium, the differences in architecture, instruction handling, and execution models would make direct adaptation complex and potentially less effective. Therefore, while adaptations may be possible, they would require a tailored approach taking into account the unique characteristics of the Itanium architecture., <think>
The recall at k metric is commonly used in information retrieval and recommendation systems to evaluate how well a system retrieves relevant items. It measures the proportion of relevant items that are retrieved in the top k results. However, a common mistake is to assume that simply counting relevant items is sufficient; context matters significantly, such as the definition of "relevant" and the way items are ranked. It's crucial to ensure that the items considered relevant are correctly identified, as this can skew results. Additionally, one must consider edge cases where k may be larger than the total number of relevant items available, which could lead to misleading interpretations of recall performance. Therefore, caution should be exercised in both implementation and interpretation.
</think>

<answer>
def recall_at_k(relevant_items, retrieved_items, k):
    """
    Calculate the recall at k metric.
    
    Parameters:
    relevant_items (set): A set of relevant items.
    retrieved_items (list): A list of retrieved items in ranked order.
    k (int): The number of top results to consider for recall calculation.
    
    Returns:
    float: Recall at k value.
    """
    if k <= 0:
        raise ValueError("k must be greater than 0")
    
    # Take only the top k retrieved items
    retrieved_at_k = set(retrieved_items[:k])
    
    # Calculate the number of relevant items retrieved
    true_positives = len(relevant_items.intersection(retrieved_at_k))
    
    # Calculate the recall
    if len(relevant_items) == 0:
        return 0.0  # Avoid division by zero if no relevant items exist
    
    recall = true_positives / len(relevant_items)
    return recall
</answer>, source_sentence: You are given three classes (Student, Exam and Course which are defined below) and the method generatePassedExams, which from a given list of students and a list of courses, generates a list of students and all their successfully passed courses together with the corresponding grade. A course is considered as successfully passed if the grade for that course is greater than 2. case class Student(name: String, exams: List[Exam]) case class Exam(courseId: String, grade: Double) case class Course(id: String, name: String) def generatePassedExams( students: List[Student], courses: List[Course]): List[(String, String, Double)] = { for { s <- students e <- s.exams if e.grade > 2 c <- courses if e.courseId == c.id } yield (s.name, c.name, e.grade) } Your task is to rewrite the method generatePassedExams to use map, flatMap and filter instead of the for-comprehension. The resulting method should of course have the same result as the for-comprehension above.
sentences: You can rewrite the `generatePassedExams` method using `map`, `flatMap`, and `filter` instead of a for-comprehension. Here’s how you can do that:

```scala
def generatePassedExams(students: List[Student], courses: List[Course]): List[(String, String, Double)] = {
  students.flatMap { student =>
    student.exams.filter(_.grade > 2).flatMap { exam =>
      courses.filter(_.id == exam.courseId).map { course =>
        (student.name, course.name, exam.grade)
      }
    }
  }
}
```

### Explanation:
1. **`flatMap` on `students`:** This iterates over each `Student` and allows us to flatten the results.
2. **`filter` on `exams`:** This filters the exams of the student to keep only those with a grade greater than 2.
3. **`flatMap` on filtered `exams`:** For each exam that passed the filter, we again use `flatMap` to find the corresponding `Course`.
4. **`filter` on `courses`:** This checks which course matches the `courseId` of the exam.
5. **`map` to create tuples:** Finally, we map the filtered courses to create a tuple containing the student's name, course name, and the grade of the exam.

This approach maintains the same logic and returns the same results as the original for-comprehension implementation., In analyzing the competitive ratio of any deterministic algorithm for exchanging 1 Euro to Swiss francs over two days, we consider the exchange rates \( x_1 \) and \( x_2 \) within the range \([1, W^2]\). By choosing \( x_1 = W \), we evaluate two scenarios:

1. **Trading on Day 1**: If we trade at day 1, we receive \( W \) Swiss francs. In the worst-case scenario, if \( x_2 \) reaches its maximum value of \( W^2 \), the ratio becomes:
   \[
   \frac{W}{\max\{W, W^2\}} = \frac{W}{W^2} = \frac{1}{W}.
   \]

2. **Waiting for Day 2**: If we wait until day 2, the potential payout depends on \( x_2 \). The best outcome we can achieve is \( x_2 \), but since we can only guarantee at most \( W^2 \), we again find that our algorithm will yield:
   \[
   \frac{W}{W^2} = \frac{1}{W}.
   \]

In both cases, the competitive ratio does not exceed \( \frac{1}{W} \). Thus, we conclude that any deterministic algorithm has a competitive ratio of at most \( \frac{1}{W} \), confirming that the choice of \( x_1 = W \) effectively bounds the performance across all scenarios., To address the question of common types of exceptions that could possibly be implemented imprecisely, let's break it down step by step:

### Step 1: Understanding Exceptions
Exceptions are events that disrupt the normal flow of a program's execution. They can arise from various issues, such as errors in the code, invalid input, or external factors like network failures. Imprecise implementation of exceptions can lead to misleading error messages, unhandled errors, or poor user experience.

### Step 2: Identifying Common Types of Exceptions
Two common types of exceptions that are frequently encountered in programming are:

1. **Null Pointer Exception (NPE)**
2. **Index Out of Bounds Exception (IOBE)**

### Step 3: Analyzing Null Pointer Exception (NPE)
- **Description**: A Null Pointer Exception occurs when a program attempts to use an object reference that has not been initialized (i.e., is null).
- **Imprecise Implementation**: 
  - NPEs can be imprecisely implemented if the error handling does not provide clear information about where the null reference originated. 
  - For instance, if a method catches an NPE and simply logs it without specifying which object was null or what operation caused it, it can lead to confusion and difficulty in debugging.
- **Reasoning**: The lack of specific context makes it hard for developers to trace back to the source of the problem, leading to wasted time and potential bugs in the future.

### Step 4: Analyzing Index Out of Bounds Exception (IOBE)
- **Description**: An Index Out of Bounds Exception occurs when a program tries to access an index of an array or collection that does not exist (either negative or exceeds the size).
- **Imprecise Implementation**: 
  - Similar to NPEs, if an IOBE is caught and handled without providing detailed information about which index was out of bounds or the size of the collection being accessed, it can be problematic.
  - For example, if an exception handler only states "Index out of bounds" without additional details, it leaves the developer with little information to diagnose the issue.
- **Reasoning**: This lack of detail can lead to difficulty in identifying logical errors in the code, especially in larger applications where multiple indices may be used.

### Step 5: Conclusion
In summary, both Null Pointer Exceptions and Index Out of Bounds Exceptions are common exceptions that can be implemented imprecisely. When these exceptions are not handled with clear and specific information, it can create confusion and hinder the debugging process, ultimately affecting the reliability of the software.

### Final Answer
Two common types of exceptions that could possibly be implemented imprecisely are Null Pointer Exceptions (NPE) and Index Out of Bounds Exceptions (IOBE). Both can lead to confusion and difficulty in debugging when not handled with detailed information about the cause and context of the error., source_sentence: The company in which you work has just hired a new CTO, freshly graduated from a theoretical university. The CTO decides that in order to minimize bugs in the product, all new code must now be covered at least 80% in terms of paths in the code. Is this a good idea, and why?
Can you suggest something better, given the CTO's objective? (the answer must be based on the definition of path coverage.)
sentences: To calculate the raw agreement between the two annotators, we can use the formula for raw agreement, which is defined as the number of items (in this case, movie reviews) that both annotators agreed on, divided by the total number of items that were annotated.

Given:
- Total number of reviews = 1000
- Reviews rated as positive by both annotators = 80
- Reviews rated as negative by both annotators = (total reviews) - (positive reviews by first annotator + positive reviews by second annotator - both positive reviews)

Let's denote:
- \( A_p \) = number of reviews rated positive by the first annotator = \( a \)
- \( A_n \) = number of reviews rated negative by the first annotator = \( 1000 - a \)
- \( B_p \) = number of reviews rated positive by the second annotator = \( b \)
- \( B_n \) = number of reviews rated negative by the second annotator = \( 1000 - b \)

The total number of agreements is the sum of the number of reviews both annotators rated positive and the number they both rated negative. 

To find the number of reviews rated negative by both annotators, we can calculate it as follows:

1. Reviews rated negative by both = \( (1000 - a) + (1000 - b) - 80 \)
2. This simplifies to \( 2000 - a - b - 80 = 1920 - a - b \)

Now, the total number of agreements is:
- Total agreements = reviews both rated positive + reviews both rated negative
- Total agreements = \( 80 + (1920 - a - b) \)

Thus, the total agreements can be expressed as:
\[ 
\text{Total agreements} = 2000 - a - b 
\]

Now we can find the raw agreement:
\[
\text{Raw Agreement} = \frac{\text{Total agreements}}{\text{Total reviews}} = \frac{2000 - a - b}{1000}
\]

The raw agreement simplifies to:
\[
\text{Raw Agreement} = 2 - \frac{a + b}{1000}
\]

To express this in decimal value to three decimal places, we need the values of \( a \) and \( b \) to compute the final agreement. Without specific values for \( a \) and \( b \), we cannot compute a numerical answer.

However, if we assume \( a + b \) can be determined or set to certain values, you could substitute into the equation to find the raw agreement.

For example, if \( a = 500 \) and \( b = 600 \):
\[
\text{Raw Agreement} = 2 - \frac{500 + 600}{1000} = 2 - 1.1 = 0.9
\]

In this case, the raw agreement would be \( 0.900 \). 

Since we do not have specific values for \( a \) and \( b \), we cannot provide a definitive numerical answer, but the formula derived can be used to compute the raw agreement once those values are known., To determine if there is a general "merging" procedure for checking the satisfaction of a SLOP/x clause, we need to analyze the merging of two term-offset inverted lists based on the specifications of the SLOP operator.

### Understanding the SLOP/x Operator
The SLOP operator allows for a flexible positioning of two terms within a specified distance (x). For example:
- SLOP/1 requires the two terms to be adjacent.
- SLOP/2 allows for one word between the terms.
- And so on, with higher values of x allowing for greater distances.

### Term-Offset Lists
Each term in the inverted list comes with its document occurrences and specific offsets. For example, if the term "Obama" appears at offsets `[3, 10]` in a document and "Governor" appears at offsets `[4, 12]`, we need to check if any occurrence of "Obama" is within x words of any occurrence of "Governor".

### Merging Process Strategy
We can implement a two-pointer technique to traverse the lists of offsets for the two terms:
1. **Initialization**: Start with two pointers, one for each term's offset list (let's call them `pointer1` for QueryTerm1 and `pointer2` for QueryTerm2).
2. **Comparison**:
   - If the offset at `pointer1` is less than the offset at `pointer2` minus x, increment `pointer1` (the offset for QueryTerm1 is too far behind).
   - If the offset at `pointer1` is greater than the offset at `pointer2` plus x, increment `pointer2` (the offset for QueryTerm2 is too far behind).
   - If the offset at `pointer1` is within the range defined by QueryTerm2 (i.e., between the offset at `pointer2` and `pointer2 + x`), we have a match and can record this occurrence.
3. **Termination**: The process continues until one of the pointers reaches the end of its respective list.

### Analysis of Options
1. **Option 1**: The merging can be accomplished in a number of steps linear in L regardless of x, and we can ensure that each pointer moves only to the right (i.e., forward).
   - This is **true**. The two-pointer approach guarantees that each pointer moves forward, resulting in a linear time complexity of O(L), where L is the total number of occurrences of both terms.

2. **Option 2**: The merging can be accomplished in a number of steps linear in L, but a pointer may be forced to move to the left (i.e., backwards).
   - This is **false**. In the merging strategy described, both pointers only move forward, ensuring that the process remains linear without any backward movements.

3. **Option 3**: The merge can require x × L steps in some cases.
   - This is **false**. The merging strategy's efficiency does not depend on x multiplying the number of steps. The approach remains linear, as the pointers will traverse the lists without needing to perform additional iterations based on the distance x.

### Conclusion
The correct answer is **1**: The merging can be accomplished in a number of steps linear in L regardless of x, and we can ensure that each pointer moves only to the right (i.e., forward). This merging strategy provides an efficient way to evaluate the SLOP/x operator by leveraging the sorted nature of the offset lists and maintaining a linear traversal of the data., To evaluate the CTO's decision to mandate at least 80% path coverage for new code, we first need to understand what path coverage entails. Path coverage is a testing criterion that requires that all possible paths through a program's control flow graph be executed at least once during testing. It is a more stringent measure than line or statement coverage, as it considers the various combinations of control flow that can occur based on the program's logic.

### Pros and Cons of 80% Path Coverage

**Pros:**
1. **Thoroughness:** Achieving high path coverage can lead to the discovery of bugs that might not be found with lower coverage criteria. It ensures that more logical paths through the code are tested.
2. **Quality Assurance:** High path coverage can enhance the quality of the software by ensuring that edge cases and less frequently executed paths are evaluated.

**Cons:**
1. **Complexity and Cost:** Achieving 80% path coverage can be very complex and time-consuming, particularly in large applications with numerous conditional statements and loops. The effort required for testing can increase significantly, which might not always yield proportional benefits in terms of bug reduction.
2. **Misleading Metric:** Path coverage does not guarantee that the tested paths are free from bugs. It is possible to achieve high path coverage while still having undetected defects in the code. Coverage metrics can sometimes give a false sense of security.
3. **Maintenance Overhead:** As code evolves, maintaining high path coverage may add overhead, requiring frequent updates to tests and potentially leading to higher technical debt.

### A Better Approach

Given the CTO's objective to minimize bugs, a more balanced and effective strategy would be a combination of path coverage with other testing practices. Here are some suggestions:

1. **Risk-Based Testing:** Prioritize testing efforts based on the risk and impact of different components of the code. Focus on critical paths and areas with a higher likelihood of failure rather than aiming for a universal percentage of path coverage.

2. **Use of Unit Tests and Integration Tests:** Implement a robust suite of unit tests that focus on individual components and integration tests that assess how those components work together. These tests can catch many bugs before they reach the production stage.

3. **Code Reviews and Static Analysis:** Incorporate formal code reviews and use static analysis tools to identify potential bugs before the code goes into testing. This can complement path coverage by catching issues at an early stage.

4. **Mutation Testing:** Utilize mutation testing, which involves making small changes to the code and checking if existing tests can catch these changes. This can help assess the effectiveness of the current test suite beyond just coverage metrics.

5. **Continuous Integration/Continuous Deployment (CI/CD):** Implement CI/CD practices to ensure that tests are run automatically with every code change. This promotes consistent testing and quicker identification of bugs.

### Conclusion

While aiming for 80% path coverage might seem like a step toward improving code quality, it is essential to recognize the limitations of this approach. A more holistic testing strategy that includes various methodologies and focuses on high-risk areas is likely to be more effective in minimizing bugs and enhancing overall software quality., source_sentence: In this problem we are going to investigate the linear programming relaxation of a classical scheduling problem. In the considered problem, we are given a set $M$ of $m$ machines and a set $J$ of $n$ jobs. Each job $j\in J$ has a processing time $p_j > 0$ and can be processed on a subset $N(j) \subseteq M$ of the machines.  The goal is to assign each job $j$ to a machine in $N(j)$ so as to complete all the jobs by a given deadline $T$. (Each machine can only process one job at a time.) If we, for $j\in J$ and $i\in N(j)$, let $x_{ij}$ denote the indicator variable indicating that $j$ was assigned to $i$, then we can formulate the scheduling problem as the following integer linear program: \begin{align*} \sum_{i\in N(j)} x_{ij} & = 1 \qquad \mbox{for all }  j\in J & \hspace{-3em} \mbox{\small \emph{(Each job $j$ should be assigned to a machine $i\in N(j)$)}} \\ \sum_{j\in J: i \in N(j)} x_{ij} p_j & \leq T \qquad \mbox{for all } i \in M & \hspace{-3em} \mbox{\small \emph{(Time needed to process jobs assigned to $i$ should be $\leq T$)}} \\ x_{ij} &\in \{0,1\}   \  \mbox{for all } j\in J, \ i \in N(j) \end{align*} The above integer linear program  is NP-hard to solve, but we can obtain a linear programming relaxation by relaxing the constraints $x_{ij} \in \{0,1\}$ to $x_{ij} \in [0,1]$. The obtained linear program can be solved in polynomial time using e.g. the ellipsoid method. \\[2mm] \emph{Example.} An example is as follows. We have two machines $M = \{m_1, m_2\}$ and three jobs $J= \{j_1, j_2, j_3\}$. Job $j_1$ has processing time $1/2$ and can only be assigned to $m_1$;  job $j_2$ has processing time $1/2$ and can only be assigned to $m_2$; and job $j_3$ has processing time $1$ and can be assigned to either machine. Finally, we have the ``deadline'' $T=1$. An extreme point solution to the linear programming relaxation is $x^*_{11} = 1, x^*_{22} =1, x^*_{13} = 1/2$ and $x^*_{23} = 1/2$. The associated graph $H$ (defined in subproblem~\textbf{a}) can be illustrated as follows: \begin{tikzpicture} \node[vertex] (a1) at (0,1.7) {$a_1$}; \node[vertex] (a2) at (0,0.3) {$a_2$}; \node[vertex] (b1) at (3,2.5) {$b_1$}; \node[vertex] (b2) at (3,1) {$b_2$}; \node[vertex] (b3) at (3,-0.5) {$b_3$}; \draw (a1) edge (b3); \draw (a2) edge (b3); \end{tikzpicture} Use the structural result proved in the first subproblem to devise an efficient rounding algorithm that, given an instance and a feasible extreme point $x^*$ in the linear programming relaxation corresponding to the instance, returns a schedule that completes all jobs by deadline $T + \max_{j\in J} p_j$. In other words, you wish to assign jobs to machines so that the total processing time of the jobs a machine receives is at most $T + \max_{j\in J} p_j$.
sentences: To determine whether the expression `e2` is referentially transparent, we need to analyze the behavior of the code involved, particularly focusing on the definition of `Student2`, the `count2` function, and how the objects are being compared.

1. **Referential Transparency Definition**: An expression is referentially transparent if it always evaluates to the same result, regardless of the program's state. This means that if you replace the expression with its evaluated value, the program's behavior should remain unchanged.

2. **Class Definition**: The class `Student2` is defined with mutable attributes (`var gpa: Double` and `var name: String`). This mutability is crucial. If an instance of `Student2` is modified after its creation, its value can change, leading to different outcomes when the same instance is used in comparisons.

3. **Count Function**: The `count2` function takes a list of `Student2` objects and a `Student2` object (the student to count). Inside `count2`, the expression `course.map(s => new Student2(student.gpa, student.name))` creates a new list of `Student2` objects based on the `student` parameter. However, since `Student2` has mutable fields, the new instances created in this mapping will not retain the same identity or state as the original `student` object passed in.

4. **Equality Check**: The equality check (`s == student`) in the `filter` method relies on how equality is defined for `Student2` (assuming default reference equality). Two different instances of `Student2` (even if they have the same `gpa` and `name`) are not considered equal because they do not reference the same object in memory, unless an appropriate `equals` method is overridden.

5. **Expression Evaluation**: When evaluating `count2(students2, Student2(6.0, "*"))`, a new `Student2` instance is created. If the properties of this instance were to change (due to mutability), then subsequent calls to `count2` could yield different results based on those changes even if the parameters appear the same.

### Conclusion:
The expression `e2` is **not referentially transparent** due to the mutability of the `Student2` class. The outcome of the expression can vary based on the state of its object instances, particularly because equal comparisons depend on the mutable state of those instances. Given that the results could change if the state of the `Student2` instances were modified, we cannot confidently replace `e2` with its evaluated value without potentially changing the program's behavior., To devise an efficient rounding algorithm based on the linear programming relaxation for the scheduling problem described, we need to follow these steps:

### Step 1: Understand the Linear Programming Relaxation

Given the linear programming relaxation of the scheduling problem, we have:
- Variables \(x_{ij}\) representing the fraction of job \(j\) assigned to machine \(i\).
- Constraints ensuring each job is assigned exactly once and that the total processing time assigned to each machine does not exceed the deadline \(T\).

### Step 2: Identify the Maximum Processing Time

Calculate \(P = \max_{j \in J} p_j\), which is the maximum processing time of all jobs. This will be used to adjust the deadline in the rounding algorithm.

### Step 3: Define the New Deadline

The new effective deadline for scheduling jobs becomes \(T' = T + P\).

### Step 4: Create a Rounding Scheme

1. **Initialization**: Start by initializing an empty schedule for each machine.
2. **Job Assignment**:
   - For each job \(j\):
     - If \(x_{ij} > 0\) for a machine \(i\), assign job \(j\) to machine \(i\) with a specific probability based on the value of \(x_{ij}\):
       - Assign job \(j\) to machine \(i\) with probability \(x_{ij}\).
       - Use a random number generator to determine if the job is assigned based on the value of \(x_{ij}\).
       - If \(x_{ij} = 1\), assign job \(j\) to machine \(i\) deterministically.
3. **Check Capacity**: After assigning jobs to machines, check if the total processing time assigned to each machine exceeds \(T'\):
   - If the total exceeds \(T'\), adjust the assignments. This can be done by removing the job with the largest processing time until the total processing time is within limits.
4. **Output the Schedule**: The output schedule contains jobs assigned to each machine, ensuring that the total processing time does not exceed \(T'\).

### Step 5: Ensure Feasibility of the Schedule

The rounding algorithm is designed to ensure that:
- Each job is assigned to a machine in its feasible set \(N(j)\).
- The total processing time for each machine does not exceed \(T + P\).

### Example Application

Let’s illustrate the rounding algorithm with the provided example:

- Machines: \(M = \{m_1, m_2\}\)
- Jobs: \(J = \{j_1, j_2, j_3\}\)
- Processing times: \(p_1 = 1/2\), \(p_2 = 1/2\), \(p_3 = 1\)
- Assignments: \(x^*_{11} = 1\), \(x^*_{22} = 1\), \(x^*_{13} = 1/2\), \(x^*_{23} = 1/2\)
- Deadline: \(T = 1\)
- Maximum processing time: \(P = 1\)
- New deadline: \(T' = 2\)

**Rounding Steps**:
1. Assign \(j_1\) to \(m_1\) (since \(x_{11} = 1\)).
2. Assign \(j_2\) to \(m_2\) (since \(x_{22} = 1\)).
3. For job \(j_3\):
   - Assign with probability \(1/2\) to either machine. Let’s say it’s assigned to \(m_1\).
4. Check machine \(m_1\):
   - Total processing time = \(p_1 + p_3 = 1/2 + 1 = 1.5 \leq 2\) (valid).
5. Check machine \(m_2\):
   - Total processing time = \(p_2 = 1/2 \leq 2\) (valid).

### Conclusion

The proposed rounding algorithm yields a feasible schedule that respects the new deadline \(T'\) while ensuring that jobs are assigned appropriately to machines. This approach leverages the structural properties of the linear programming solution, providing a practical method for achieving a valid job schedule., It's best to use branching strategies instead of maintaining separate repositories for old versions. Create a branch for each version in the main repository, allowing you to easily apply bug fixes and features across versions. This way, you can maintain a clean history, facilitate code reviews, and simplify merging changes. Additionally, using tags to mark releases can help track versions effectively while keeping the main repository organized. This approach enhances collaboration and minimizes the risk of discrepancies between versions.
pipeline_tag: sentence-similarity
library_name: sentence-transformers
metrics: cosine_accuracy@1, cosine_accuracy@3, cosine_accuracy@5, cosine_accuracy@10, cosine_precision@1, cosine_precision@3, cosine_precision@5, cosine_precision@10, cosine_recall@1, cosine_recall@3, cosine_recall@5, cosine_recall@10, cosine_ndcg@10, cosine_mrr@10, cosine_map@100
model-index: name: BGE base Financial Matryoshka
results: task: type: information-retrieval
name: Information Retrieval
dataset: name: dim 768
type: dim_768
metrics: type: cosine_accuracy@1
value: 0.2912280701754386
name: Cosine Accuracy@1
verified: no, type: cosine_accuracy@3
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name: Cosine Accuracy@3
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verified: no, type: cosine_accuracy@10
value: 0.9719298245614035
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verified: no, type: cosine_precision@5
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name: Cosine Precision@5
verified: no, type: cosine_precision@10
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name: Cosine Precision@10
verified: no, type: cosine_recall@1
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name: Cosine Recall@1
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name: Cosine Ndcg@10
verified: no, type: cosine_mrr@10
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name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.5626716230567321
name: Cosine Map@100
verified: no, task: type: information-retrieval
name: Information Retrieval
dataset: name: dim 512
type: dim_512
metrics: type: cosine_accuracy@1
value: 0.3017543859649123
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verified: no, type: cosine_accuracy@3
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value: 0.9403508771929825
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verified: no, type: cosine_recall@10
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name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.568853410172187
name: Cosine Map@100
verified: no, task: type: information-retrieval
name: Information Retrieval
dataset: name: dim 256
type: dim_256
metrics: type: cosine_accuracy@1
value: 0.2982456140350877
name: Cosine Accuracy@1
verified: no, type: cosine_accuracy@3
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name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.564921899494268
name: Cosine Map@100
verified: no, task: type: information-retrieval
name: Information Retrieval
dataset: name: dim 128
type: dim_128
metrics: type: cosine_accuracy@1
value: 0.2771929824561403
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verified: no, type: cosine_accuracy@3
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verified: no, type: cosine_map@100
value: 0.5515041358901008
name: Cosine Map@100
verified: no, task: type: information-retrieval
name: Information Retrieval
dataset: name: dim 64
type: dim_64
metrics: type: cosine_accuracy@1
value: 0.3087719298245614
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verified: no, type: cosine_accuracy@3
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name: Cosine Accuracy@3
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verified: no
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Created: 2025-06-06 08:49:27