User: 

You are given access to make searches on the Internet. The format for making a search is: 1. The user sends a message. 2. You will decide whether to make a search. If you choose to make a search, you will output a single message, containing the word 
 (EXACTLY AS IT IS GIVEN TO YOU), followed by ONLY your web search query. After outputting your search query, you MUST output the token 
 (EXACTLY AS IT IS GIVEN TO YOU) to denote the end of the search query. Keep your search query concise as there is a 400 characters limit.
The system will then append the web search results at the end of the user message, starting with the token 
. Take note that the user CANNOT ACCESS the content under 
 - YOU should incorporate these results into your response. Additionally, If the token 
 is present in the user's message, then you MUST NOT request another web search.

IMPORTANT! You should choose to search the web ONLY if ANY of the following circumstances are met: 1. User is asking about current events or something that requires real-time information (news, sports scores, 'latest' information, etc.)
2. User is asking about some term you are unfamiliar with (e.g. if it's a little-known term, if it's a person you don't know well, etc.)
3. User is asking about anything that involves numerical facts (e.g. distances between places, population count, sizes, date and time, etc.)
4. User explicitly asks you to search or provide links to references.
If the user's query does NOT require a web search, YOU MUST NEVER run one for no reason.

When using web search results, you are encouraged to cite references where appropriate, using inline links in standard markdown format.
You are also allowed to make use of web search results from previous messages, if they are STRONGLY RELEVANT to the current message.

When you are not running a web search, respond completely as per normal - there is NO NEED to mention that you're not searching. 
Do your best to be informative. However, you MUST NOT make up any information. If you think you might not know something, search for it.

Assistant: Understood. I will strictly follow these instructions in this conversation.
User: what's the branching factor formula in chess engine?
Assistant: The branching factor in chess engine refers to the average number of possible moves that a player can make at each turn. The formula to calculate the branching factor in chess engine is typically the number of legal moves available at a given position. This can vary depending on the position on the board, as well as other factors such as piece activity and board control.  
  
If you are looking for a more precise formula or specific details on how the branching factor is determined in chess engines, please let me know if you would like me to search for more information on this topic.
User: search for it please


 for "branching factor formula in chess engine calculations":

1.
URL: https://www.nature.com/articles/s41598-024-59689-x
Title: An exact algorithm to find a maximum weight clique in a weighted undirected graph | Scientific Reports - Nature.com
Content: In each step, below the graph are the current values of important variables as they are sampled at the beginning of the Expand procedure (before line 2 in Table 4) for steps 1–4 and 6–9 or at the update of Qmax in the Expand procedure (after line 13 in Table 4) for steps 5 and 10: R—graph vertices that remain to be explored; wcolors—upper bounds to the weight of a clique, where each wcolor corresponds to the vertex in R at the same index in the list; C—color classes determined in the ColorSortWeight procedure, where each color class corresponds to the vertex in R at the same index in the list; Q—vertices of the current growing clique; Qmax—vertices of the current maximum weight clique.  At each step, Expand selects the vertex p ∈ R with the highest cumulative weight (wcolor) (line 3 in Table 4), which is the last vertex in set R, and removes this vertex p from the set R. If the weight of Q, which is the sum of the weights of all its constituting vertices plus the weight of the selected vertex p is greater than the weight of Qmax, the vertex p is added to Q (line 4 in Table 4).  The value of min_k_w is found iteratively by starting with min_k_w = 1 and increasing it to maxno; the search is stopped when the sum of the weights of the k smallest color classes is greater than the difference between the weight of the maximum clique found so far Qmax and the weight of the currently growing clique Q.  In Step 7, the Expand procedure backtracks to the recursion level of Step 1, and vertex no. 3 is added to the set Q. In Step 8 vertex no. 1 is added to the set Q, as the weight of Q (5), which now includes vertex no. 3, plus the wcolor of vertex no. 1 (15), exceeds the weight of Qmax (18). In the next step, the ColorSortWeight procedure determines the color class (min_k_w) below which vertices cannot be used to extend the growing weighted clique based on the difference between the weight of the currently growing clique Q and the weight of the maximum weight clique found so far Qmax (line 18 in Table 3).

2.
URL: https://www.nature.com/articles/s41598-024-58203-7
Title: Response times are affected by mispredictions in a stochastic game | Scientific Reports - Nature.com
Content: If the Kolmogorov–Smirnov test rejects the equality of distributions, then the sequences in F(w) are added to \({\hat{\tau }}\) and deleted from the List C. Otherwise, the sequences in F(w) are deleted from the list C and the sequence \((w_{-l(w)+1},\cdots ,w_{-1})\) is added to the end of C.  Given the sequence \(w = (w_{-k}, \ldots , w_{-k})\), let \(n_{(1,w)}, \ldots , n_{(2,w)}, n_{(3,w)}, \ldots\) be the successive steps ending in the occurrence of w. Namely, and for \(j>1\) Let \(n_{(1,w \rightarrow a)}, \ldots , n_{(2,w \rightarrow a)}, n_{(3,w \rightarrow a)}, \ldots ,\) be the successive steps ending in a occurrence of w with correctly predicted transitions from w to a. That is, Also, let \(N_{ (w \rightarrow a ) }\) be the total number of correctly predicted transitions from w to a.  Advertisement Explore content About the journal Publish with us Search Quick links Scientific Reports (Sci Rep) ISSN 2045-2322 (online) nature.com sitemap About Nature Portfolio Discover content Publishing policies Author & Researcher services Libraries & institutions Advertising & partnerships Professional development Regional websites  Since the sequence of kicks consists in a repetition of \(0 * 1\) with \(*\) taking the value of 2 with probability \(p = 0.7\) and 1 with probability \(1-p\), we reasoned that the contexts 01, 11 and 21 might be affected by the congruence between the participants choices and those of the penalty taker.  More precisely, in the second epoch, the mode context tree deduced from the response times of a large majority of participants (15 out of 22) coincided with the context tree used to generate the sequence of choices of the penalty taker.

3.
URL: https://www.nature.com/articles/s41598-024-57778-5
Title: Innovative dead-time correction and background subtraction for neutron multiplicity measurements using neural ... - Nature.com
Content: If \(F'_{l}\) is the probability per reaction of recording l pulses during the neutron counting gate, \(B'_x\) the probability of recording x background pulses during the background counting gate, \(D_x\) and \(B_x\) are the real probabilities of occurrence of x neutron pulses during the neutron counting gate and x background pulses during the background counting gate respectively, the following equations may be written as where \(k_{nn}\) is the probability that two neutron pulses, occurring during the neutron counting gate, overlap and appear as one pulse; \(k_{nb}\) is the probability that one neutron and one background pulse overlap; \(k_{bb}\) is the probability that two background pulses overlap; and, \(^{X} \, C_Y\) is the combination probability of selecting Y items out of X2,10, The overlap probabilities can be written as where \(\tau\) is the dead-time, and T is the duration of the detection gate.  The reference point can be the fission fragments in the case of a fissioning nucleus, or the prompt \(\gamma\)-rays coming from a nuclear reaction21. where k is the probability that the neutron will eventually be captured in the gadolinium, \(\beta\) is a “slowing down” parameter which depends mainly on the actual amount of the loading material present in the liquid scintillator and g(t) is a probability density representing the neutron thermalization time and subsequent capture time of the form where the reciprocal of \(\lambda\) is a “slowing down” parameter, with 2/\(\lambda\) being the mean time for neutrons to become moderated to G, and g(t)dt is the probability that a neutron is moderated to energy G in a time interval (t, \(t+dt\)). The dead-time corrected multiplicity probability distribution, \(D_j\), is obtained by solving the following equations, where \(N_i\) is the background corrected neutron multiplicity probability distribution, and \(S_{i,j}\) represents the probability that only i neutrons will be counted when j neutrons are actually present within the detection window, where f(t) is the neutron capture time distribution of Fig. 8, and the function \(G_{i,j}=1\) if \(i=j\), \(G_{i,j}=0\) if \(i>j\), and for \(i<j\), where The multidimensional integral in \(S_{i,j}\) was solved numerically using the MC integration algorithm VEGAS25 implemented in the ROOT class ROOT::Math::GSLMCIntegrator. You can also search for this author in PubMed Google Scholar Contributions J.G.D. conceived the NN method, created the simulations, analyzed the results and prepared the original manuscript; J.T.H. formulated general research objectives, contributed to project supervision, the reviewing and editing of the manuscript; Y.H.M., contributed to the analysis of the results, and to the reviewing and editing of the manuscript. Using the \(^{252}\)Cf standard neutron multiplicity distribution from the work of Santi17 and the characteristics of the NeutronSTARS array16, our investigations revealed that at a dead-time of \(\tau = 1 \, \upmu s\) nearly 29% of triggers resulted in the loss of one neutron, while 5% lost two, underscoring the critical importance of using an accurate dead-time correction technique (Fig. 2). 

4.
URL: https://www.nature.com/articles/s41598-024-61165-5
Title: Simulating worm feeding patterns with computational models | Scientific Reports - Nature.com
Content: This is shown in Fig. 4 by population size up to 2 M. Figure 4 shows that as the population limit increases, the proportion of worms in category R decreases and the proportion of worms in category T increases, but the proportion of worms in category L stays roughly consistent. Advertisement Explore content About the journal Publish with us Search Quick links Scientific Reports (Sci Rep) ISSN 2045-2322 (online) nature.com sitemap About Nature Portfolio Discover content Publishing policies Author & Researcher services Libraries & institutions Advertising & partnerships Professional development Regional websites  On this new proposed square grid, there are a huge number of worms of unknown outcome in category R. Of all the possible 1033 worms, it was found that approximately 2% continue running chaotically in category R beyond a million population size with unknown outcome, which currently leaves about 1031 worms with unknown outcome, for future research to explore.  Diagram showing the categories of all possible worms up to a rule length of 9. Using recursive search, all worms to a rule depth of 11 were simulated to population limit of 250,000 to identify their categories. For each of the 128 binary 7 digit path distributions, the number of paths available is the Hamming weight of the binary string (inverted so that a 1 represents an available path and a 0 represents a traversed path).

5.
URL: https://www.nature.com/articles/s41598-024-61959-7
Title: Mapping livestock density distribution in the Selenge River Basin of Mongolia using random forest | Scientific Reports - Nature.com
Content: The larger the \(q\)- value, the greater the influence of the factor on livestock density and the stronger the explanatory power; \(N\) is the number of samples in the study area, and \({N}_{h}\) is the number of samples in strata \(h\); \({{\sigma }_{h}}^{2}\) and \({\sigma }^{2}\) are the variance of livestock density in strata \(h\) and the study region, respectively; and \(L\) is the number of strata. The main objectives of the study were to: (1) decompose the soum-level livestock density to the grid scale using the RF model, obtain high-precision gridded distribution data for total livestock, sheep & goats, cattle, and horses, and fill in the gaps of the livestock densities gridded data in 2020; (2) identify the drivers of livestock distribution and quantifying their contributions; (3) provide a reference method for mapping the spatial distribution of livestock density not only in Mongolian Plateau but also in other steppe areas.  The factor detector in the geodetector is mainly used to analyze the determining force of the independent variable on the dependent variable, and we used the factor detector to quantitatively detect the influence mechanism of each environmental factor on the spatial stratified heterogeneity of sheep & goats, horse and cattle density, which can be measured using the q-statistic50: where \(q\) is the explanatory or determining power of each environmental factor on livestock density, and the range of values is 0–1. Author information Authors and Affiliations College of Geoscience and Surveying Engineering, China University of Mining & Technology (Beijing), Beijing, 100083, China Yaping Liu & Keming Yang State Key Laboratory of Resources and Environmental Information System, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, 100101, China Yaping Liu & Juanle Wang College of Resources and Environment, University of Chinese Academy of Sciences, Beijing, 100049, China Juanle Wang Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing, 210023, China Juanle Wang Environmental Engineering Laboratory, Department of Environment and Forest Engineering, School of Engineering and Applied Sciences and Institute for Sustainable Development, National University of Mongolia, Ulaanbaatar, 14201, Mongolia Altansukh Ochir  In the specific implementation idea based on the OOB method, a trained decision tree \({n}_{1}\) is constructed using OOB samples, and the difference between the original value of the sample and the predicted value is used to obtain the out-of-pocket data error \({e}_{1}\), after which the other columns are kept unchanged, permutating or adding noise to the characteristic value of the column \(i\) feature to obtain the out-of-pocket data error \({e}_{i2}\).


Assistant: