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The S-Net [1] based on the R-Net [2] tries to tackle the open-domain QA problem just like the Reading Wikipedia to Answer Open-Domain Questions does. the S-Net decomposes the problem into two problems, Evidence Extraction and Answer Synthesis namely.

## Motivation

In the SQuAD dataset there is a constraint that words to answer questions are all provided in passages. Specifically, the answer is a span of words in the corresponding passage. So most works leverage the Pointer Network to predict the start point and end point of the answer in a passage. This is not the case of reality. Datasets like MS-MARCO and DuReader removes this constraint. They provide several passages for each question and the answer is generated by the human. So the answer may not be a span in a passage or not necessarily use any word in provided passages. They proposed an extraction-then-synthesis framework.

## Model

### Evidence Snippet Prediction

This evidence extraction model is treated as a multi-task learning model. It not only learns text spans of evidence but also learns to rank passage candidates. This model is alike their previous work R-NET. For a question $Q=|w^{Q}_{t}|^{m}_{t=1}$ and a passage $P=|w^{P}_{t}|^{n}_{t=1}$, all words are encoded through word embedding and character embedding. The final hidden state of a bi-directional GRU applied to charaters is used as the character embedding of corresponding word. By concatenating the two forms of embeddings ( $e$ and $char$ ) and puting them into another bi-directional GRU, a new representation of words $u^Q_i$ (for question) and $u^P_j$ (for passage) is obtained.
$$u^Q_t = \mbox{BiGRU}_{Q}(u^Q_{t-1},[e^Q_t,char^{Q}_{t}]) \\ u^P_t = \mbox{BiGRU}_{P}(u^P_{t-1},[e^P_t,char^{P}_{t}])$$

Then a SeqMatchSeq like model is used to mix up question and passage. At step $t$, first the context vector $c^{Q}_{t}$ is calculated as below
$$s^{t}_{j} = v^{T}\mbox{tanh}(W^{Q}_{u}u^{Q}_{j}+W^{P}_{u}u^{P}_{t}) \\ a^{t}_{i} = \exp (s^{t}_{i}) / \sum^{m}_{j=1}\exp(s^{t}_{j}) \\ c^{Q}_{t} = \sum^{m}_{i=1} a^{t}_{i} u^{Q}_{t}$$
Next, concatenate context vector $c^{Q}_{t}$ and current passage word $u^{P}_{t}$ and compute a gate $g_t$ to determine how much information will be used.
$$g_t = \sigma (W_g[u^{P}_{t},c^{Q}_{t}]) \\ [u^{P}_{t},c^{Q}_{t}]^{*} = g_t \odot [u^{P}_{t},c^{Q}_{t}] \\ v^{P}_{t} = \mbox{GRU}(v^{P}_{t-1},[u^{P}_{t},c^{Q}_{t}]^{*})$$

Concatenate the new representation of all passage, and compute the pointers for start position $p^1$ and end position $p^2$
$$s^{t}_{j} = v^{T}\mbox{tanh}(W^{P}_{h}u^{P}_{j}+W^{a}_{h}h^{a}_{t-1}) \\ a^{t}_{i} = \exp (s^{t}_{i}) / \sum^{N}_{j=1}\exp(s^{t}_{j}) \\ p^t = \mbox{argmax}(a^{t}_{1},…,a^{t}_{N})$$
where $h^{a}_{t-1}$ represents the last hidden state of the pointer network. For the initial hidden state, it is computed as
$$s_{j} = v^{T}\mbox{tanh}(W^{Q}_{u}u^{Q}_{j}+W^{Q}_{v}v^{Q}_{r}) \\ a_{i} = \exp (s_{i}) / \sum^{N}_{j=1}\exp(s_{j}) \\ r^{Q} = \sum^{m}_{i=1} a_{i}u^{Q}_{i}$$
where $v^{Q}_{r}$ is a parameter to learn. For other hidden states, they are obtained as
$$c_{t} = \sum^{N}_{i=1}a^{t}_{i}v^{P}_{i} \\ h^{a}_{t} = \mbox{GRU}(h^{a}_{t-1},c_{t})$$

Another task in this model is passage ranking. To get the passage representation $r^{P}$, attention mechanism is adopted once more:
$$s_{j} = v^{T}\mbox{tanh}(W^{P}_{v}v^{P}_{j}+W^{Q}_{v}r^{Q}) \\ a_{i} = \exp (s_{i}) / \sum^{N}_{j=1}\exp(s_{j}) \\ r^{P} = \sum^{n}_{i=1} a_{i}v^{P}_{i}$$
Using $r^{P}$ and $r^{Q}$, the matching score is calculated as
$$g = v^{T}_{g}(\mbox{tanh}(W_g[r^{Q},r^{P}]))$$

This answer synthesis model is a typical Seq2Seq model with attention. The encoder leverages the information extracted from the evidence extraction model by combining basic word embedding $e^{p}_{t}$ with $f^{s}_{t}$ and $f^{e}_{t}$ which indicates the start and end positions of evidence snippet respectively. The encoded questions and answers are
$$h^{P}_{t} = \mbox{BiGRU}(h^{P}_{t-1},[e^{p}_{t},f^{s}_{t},f^{e}_{t}]) \\ h^{Q}_{t} = \mbox{BiGRU}(h^{Q}_{t-1},e^{Q}_{t})$$

The decoder generates the decoding hidden state $d_{t}$ at $t$ by
$$d_{t} = \mbox{GRU}(w_{t-1},c_{t-1},d_{t-1}) \\ d_{0} = \mbox{tanh}(W_{d} [h^{P}_{1},h^{Q}_{1}] + b)$$
where $h^{P}_{1}$ and $h^{Q}_{1}$ are the last backward encoder hidden states with repsect to passage and question. $w_{t}$ is the previous decoding word embedding. The context vector $c_{t}$ is obtained by matching current decoder state $d_{t}$ with each encoder hidden state $h_{t}$ which consists of the $h^{P}_{t}$ and $h^{Q}_{t}$ (not sure how they mix these two)
$$s^{t}_{j} = v^{T}_{a} \mbox{tanh} (W_{a}d_{t-1} + U_{a}h_{j}) \\ a^{t}_{i} = \exp (s^{t}_{i}) / \sum^{n}_{j=1}\exp(s^{t}_{j}) \\ c_{t} = \sum^{n}_{i=1}a^{t}_{i}h_{i}$$
Now, construct readout state $r_t$, put it through a maxout hidden layer and predict decoding word $y_{t}$
$$r_t = W_rw_{t-1} + U_{r}c_{t} + V_{r}d_{t} \\ m_t = [\mbox{max}(r_{t,2j-1},r_{t,2j})]^T \\ p(y_{t}|y_{1},…y_{t-1}) = \mbox{softmax}(W_{o}m_t)$$

## Optimization

### Evidence Snippet Prediction

For the evidence snippet predction task, the target is to get the start and end indexes right. This objective can be expressed by cross entropy
$$L_{AP} = - \sum^{2}_{t=1} \sum^{N}_{i=1} [ y^{t}_{i} \log a^{t}_{i} + (1 - y^{t}_{i}) \log (1-a^{t}_{i}) ]$$
where $y^{t}_{i}$ is a index function.

And for the page ranking task, the goal is to make the correct passages have higher scores. Similiarly, the objective function is
$$\hat{g}_{i} = \exp (g_{i})/\sum^{k}_{j=1}(g_{i}) \\ L_{PR} = - \sum^{k}_{i=1} [ y_{i} \log \hat{g}_{i} + (1 - y_{i}) \log (1-\hat{g}_{i}) ]$$
Combine two objective functions, We have
$$L_{E} = r L_{AP} + (1-r) L_{PR}$$

The goal in answer synthesis model is to maximize the likelike of answer $Y$ given the question and related passages $Y$
$$L_{S} = -\frac{1}{|D|} \sum_{(X,Y) \in D} \log(Y|X)$$

## Experimental Setting

### Evidence Snippet Prediction

• Choose the span with the highest ROUGE-L score with the reference answer as the gold span in the training. Only use the data whose ROUGE-L score of chosen text span is higher than 0.7 (86.74% training data).
• 300-dimensional uncased pre-trained GloVe embeddings without update during training.
• Use zero vectors to represent all out-of-vocabulary words.
• Hidden vector length is set to 150.
• Dropout rate is 0.1.
• Beam search with beam size of 12.
• Thee weight r is set to 0.8.

• Choose the best span with highest ROUGE-L score as the evidence, and use the corresponding reference answer as the output. Only use the data whose ROUGE-L score of chosen evidence snippet is higher than 0.5.
• Apply the evidence extraction model to all training data to obtain the extracted span. (almost double the dataset).
• Use an identical vocabulary set for the input and output whose size is 30,000.
• Word embedding size is 300 and updated during training.
• Feature embedding size of start and end positions of the extracted snippet is 50.
• GRU hidden state sizes is 150.
• Beam search with beam size of 12.

## Analysis

The figure below shows that the multi-task framework (jointly learning the answer prediction and the passage ranking) works better than learning the evidence snippet alone. it is way better than passage ranking first and extraction.

An experiment is conducted to compare the extraction and synthesis model. For ROUGE-L score equals 1 (the answer can be found in the passage), the extraction model slightly outperforms the extraction-synthesis model. Both of them are behind the potiential ROUGE-L score.


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Drilling

# Good News for Drilling Contractors, Offshore Rigs Are Getting Harder To Find

## A new report shows that the tight offshore rig market is even tighter than most estimates reflect. This offers relief to battered drilling contractors which are now demanding higher dayrates.

Not only are offshore rigs becoming more scarce, the industry is underestimating its dwindling supply of available units.

This is according to a new report from Westwood Global Energy Group which said, “It appears the true extent of this tightening market is still not always being accurately represented across the industry.”

The energy consultancy notes that all three segments of the business—jackups, semisubmersibles, and drillships—show a contracted utilization rate below 85%. The figure represents an important industry threshold since it traditionally marks the minimum level required for dayrates to head north and for cold-stacked units to be called back into service.

However, this shift appears to be well underway per data from Westwood.

In its estimates, Westwood is figuring in the number of formerly idle rigs that are now in the process of being reactivated along with those undergoing work quayside to be prepared for future drilling projects.

In this context, the utilization rate for jackup rigs rises from 82% to 92%. Similarly, drillship utilization levels are up from 79% to 92%.

When looking beyond ongoing contracted work, Westwood said these figures reflect “a nearly sold-out market” for jackups and drillships.

Semisubmersibles are trailing with a utilization rate of 64% based on current contracts. Taking upcoming projects into account, though, Westwood said the real utilization rate is closer to 82%.

“Break this down further by region or specific rig type—including design, age, and capabilities—then these figures become even higher in some cases; starkly demonstrated by the US Gulf of Mexico and Brazilian drillship markets, which are both currently fully utilized with all rigs either currently on hire or preparing for future campaigns,” the London-based company said in a statement this week.

The increasing scarcity of rigs means drilling contractors that have suffered through two major downturns in 7 years and a wave of bankruptcies and consolidation are now firmly positioned to demand higher dayrates.

Westwood highlighted that the surging demand is being heightened by the supply concerns stemming from Russia’s ongoing invasion of Ukraine which may see dayrates “reach highs not witnessed in almost a decade." The consultancy noted that some of the industry’s most sophisticated assets, 7th-generation drillships, are enjoying average dayrates of around $400,000. In April, global drilling contractor Transocean reported that its 7th-generation drillship Deepwater Asgard is working for Shell in the Gulf of Mexico for a dayrate of$475,000. A little more than 2 years prior, that same rig was under contract for only $220,000 per day. However, in a sign of how much ground has been lost in the offshore drilling sector, the Deepwater Asgard is still earning less than what it was as a newbuild in 2014 when the drillship commanded a$600,000 dayrate.

In regard to cold-stacked and stranded newbuild rigs, Westwood also noted in its update that the reactivation process can take between 6 to 12 months. It added that drilling contractors are only interested in bringing these assets back online under multiyear contracts due to the extreme cost of such work, which for a floater ranges from around $50 million to$100 million.

“As more of these inactive units enter the fleet—and this will be the case—the committed utilization rate will continue to increase and dayrates will inevitably follow,” Westwood concluded.

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# exponents problems

+1 vote

If I try to use collect_factors for an expression that contains exponents, it does not collect them into one exponent, for example:

collect_factors($\exp(-2\sigma-2\kappa) \exp(2\sigma) \exp(2\kappa)$);

these exponents can only be removed by map_sympy(ex, "simplify"), but it can't be used with zoom function, for example. same situation with product_rule.

substitute(ex, $\exp(A??) \exp(B??)-> exp(A??+B??)$)


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# 22nd International Workshop on Weak Interactions and Neutrinos

Europe/Rome
Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
Description
The WIN 09 Workshop will be organized by the Laboratori Nazionali del Gran Sasso of INFN and will take place from September 14th to 19th, 2009. At beginning the Workshop was to be held in the city of L'Aquila, but after the earthquake that affected this region on April 6th, the Organizing Commettee decided to move the conference venue to the "Relais San Clemente" near Perugia (Italy). The philosophy of this small but time-honored series of workshops (it started in the seventies) is to provide the opportunity for the participants to discuss and interact among each other in a very informal and constructive way. The WIN 09 workshop will deal with contemporary issues on the following four major topics, discussed in dedicated groups: DG1 - Electro Weak Symmetry Breaking DG2 - Weak Decays, CP violation and CKM DG3 - Neutrino Physics DG4 - Dark Matter
Participants
• Akitaka Ariga
• Alain Blondel
• Alberto Graziano
• Aldo Ianni
• Alejandro Ibarra
• Alessandro Baldini
• Alessandro Strumia
• Alexandre Sousa
• Andre Rubbia
• Andrei Nikiforov
• Andrzej M. Szelc
• Angelo Nucciotti
• Annarita Margiotta
• Antonio Limosani
• Anze Zupanc
• Art McDonald
• Augusto Ceccucci
• Bjoern Duling
• Blair Edwards
• Bonnie Fleming
• Bruno Serfass
• Chih-hsiang Cheng
• Christina Hagemann
• Claudia Christina Strabel
• Claudia Lazzaro
• Cristiano Galbiati
• Damir Becirevic
• Dan McKinsey
• Daniele Montanino
• Dario Grasso
• Darren Grant
• David Schmitz
• Elena Guardincerri
• Eli Rosenberg
• emilio migneco
• Ennio Salvioni
• Ettore Segreto
• Fabrizio Cei
• Fanny Dufour
• Fausto Chiarizia
• Flavio Cavanna
• Francesco Di Pompeo
• Francesco Villante
• Franco Giuliani
• George Tzanakos
• Georges Vasseur
• Georgia KARAGIORGI
• Gerhard Buchalla
• Gerhard Ecker
• Giancarlo D'Ambrosio
• Gino Isidori
• Giorgio Giacomelli
• Giovanni Fiorentini
• Giulia Ricciardi
• Herbert Pietschmann
• Hidekazu Kakuno
• Hiroshi Nunokawa
• Itaru Shimizu
• Jaap Panman
• Jan Schumacher
• Jure Zupan
• Kamales KAR
• Kang Soon Park
• Karol Lang
• Luca Grandi
• Lucia Votano
• Luciano Pandola
• Majid Hashemi
• Marcella Bona
• Marcello Giorgi
• Marco Sozzi
• Marcos Dracos
• Martin Hirsch
• Masa Yamauchi
• Massimiliano Grazzini
• Massimo Casarsa
• Matthew Moulson
• Mauro PIccini
• Mayly Sanchez
• Mitchell Soderberg
• Monica Sisti
• Mu-Chun Chen
• MyeongJae Lee
• Neal Weiner
• Nicola Canci
• Olga Lalakulich
• Oliviero Cremonesi
• Ornella Palamara
• Paolo Papini
• Peter Fierlinger
• Phillip Urquijo
• Pierluigi Belli
• Piermaria Oddone
• Probir Roy
• Rainer Wanke
• Raoul Viollier
• Riccardo Barbieri
• Riccardo Cenci
• Robert Ciesielski
• Robert McElrath
• Roberto Acciarri
• Roberto Tenchini
• Sandhya Choubey
• Sandra Leone
• Sascha Mehlhase
• Sebastian Arrenberg
• Shoichi Uchinami
• Silke Nelson
• Soumitra Nandi
• Steve kettell
• Steven Barry Boyd
• Steven Biller
• Tagir Aushev
• TAKU YAMANAKA
• Takuya Hasegawa
• Teppei Baba
• Terrence Hart
• Till Kirsten
• Tim Gershon
• Vincenzo Caracciolo
• Yasuhiro Nakajima
• Yasuo Takeuchi
• Yoshinari Hayato
• Young-Kee KIM
• yves sirois
• Zurab Berezhiani
Support
• Monday, September 14
• Welcome - O. Palamara, L. Votano, P. Oddone, H. Pietschmann
• 1
Welcome
Speaker: Ornella Palamara (WIN09 Chair)
• Plenary Session: Status Reports. Chair: H. Pietschmann Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it

Plenary session

• 2
Electro Weak Symmetry Breaking: Theoretical Status
Speaker: Riccardo Barbieri
• 3
Electro Weak Symmetry Breaking: Experimental Status
Speaker: Roberto Tenchini
• 10:30 AM
coffee break
• Plenary Session: Status Reports. Chair: Y.K. Kim Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it

Plenary session

• 4
Weak Decays, CP violation and CKM: Theoretical Status
Speaker: Gino Isidori
• 5
Weak Decays, CP violation and CKM: Experimental Status
Speaker: Tim Gershon
• 12:30 PM
Lunch + Break
• Plenary Session: Status Reports. Chair: T. Kirsten Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it

Plenary session

• 6
Neutrino Physics: Theoretical Status
Speaker: Sandhya Choubey
• 7
Neutrino Physics: Experimental Status
Speaker: Art McDonald
• 5:00 PM
coffee break
• Plenary Session: Status Reports. Chair: L. Votano Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it

Plenary session

• 8
Dark Matter: Theoretical Status
Speaker: Neal Weiner
• 9
Dark Matter: Experimental Status
Speaker: Bruno Serfass
• 7:00 PM
Welcome Drink
• Tuesday, September 15
• Plenary Session: Plenary Session. Chair: A. McDonald Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it

Plenary session

• 10
Muon Collider
Speaker: Pier Oddone
• 11
Review of Current and Future Neutrino Cross-Section Experiments
Speaker: David Schmitz
• 12
Future Direct Dark Matter Searches: An Outlook
Speaker: Cristiano GALBIATI
• 10:30 AM
coffee break
• DG1 - Electro Weak Symmetry Breaking: EWK Precision Measurements and Beyond Sala Orsini

### Sala Orsini

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 13
Electroweak physics and proton structure measurements at HERA
Speaker: Andrei Nikiforov
• 14
Prospects for Z/W Physics at the LHC
Speaker: Marcella Bona
• 15
Minimal Z’ models : present bounds and early LHC Reach
Speaker: Ennio Salvioni
• 12:15 PM
Discussions
• DG2 - Weak Decays, CP violation and CKM: Present Status of the CKM Matrix. Chair: R. Wanke Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 16
CKM from semi-leptonic B decays
Speaker: Giulia Ricciardi
• 17
Measurements of alpha
Speaker: Georges Vasseur
• 18
Measurements of beta
Speaker: Chih-Hsiang Cheng
• DG3 - Neutrino Physics: Neutrino Mass and Mixing: Theory Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 19
Theory Predictions for Neutrino Masses and Mixing Angles
I will review recent developments in theoretical models for neutrino masses and mixing. Emphases are given to models based on finite group family symmetries in which Tri-bimaximal neutrino mixing pattern is generated. In particular, I will describe one recent model based on grand unification, in which both the Tri-bimaximal neutrino mixing and realistic CKM matrix are generated. The prediction for theta13 is given in terms of the Cabibbo angle and it is within the reach of Daya Bay experiment. CP violation in this model is purely geometrical in origin. Since the only non-vanishing leptonic CP phase is the Dirac phase, the model predicts a connection between the leptogenesis and CP violation in neutrino oscillation.
Speaker: Mu-Chun Chen
• 20
Testing neutrino mass models at LHC
Majorana neutrino masses can be generated in a variety of ways. They could be due to some variant of the seesaw mechanism or be generated radiatively. Also within supersymmetric models neutrino masses can be non-zero if R-parity is broken. This talk discusses different signals of various neutrino mass models one can expect for the LHC experiments.
Speaker: Martin HIRSCH
• 21
The magic of four zero neutrino Yukawa textures
Within the type-I seesaw framework with three heavy right chiral neutrinos and in the basis where the latter and the charged leptons are mass diagonal, we investigate the effects of mu-tau symmetry as well as trbimaximal mixing on the maximally allowed four zero neutrino Yukawa textures which had already led to a predicitve and highly constrained theoretical scheme. A drastic reduction of the seventy two allowed textures is found, leaving only two allowed forms of the light neutrino mass matrix, one of which is on the margin of being ruled out by the present neutrino oscillation data. The other remains a viable candidate for the actual light neutrino mass matrix chosen by nature and will be crucially tested by future measurements of. theta_{13}. Implications for leptogenesis and radiative lepton flavor violating decays are also discussed. The stabilty of these conclusions under running from a high scale, where those symmetries are imposed, to the weak scale is also demonstrated.
Speaker: PROBIR ROY
• 12:20 PM
Discussions
• 12:30 PM
Lunch + Break
• DG1 - Electro Weak Symmetry Breaking: Top Physics Sala Orsini

### Sala Orsini

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 22
Top Experimental Prospects at start-up LHC
Speaker: Sascha Mehlhase
• 23
Boosted Top at the LHC
Speaker: Seung Lee
• 24
Top Quark Physics at the LHC (Theory)
• 4:55 PM
Discussions
• DG2 - Weak Decays, CP violation and CKM: Present Status of the CKM Matrix, cont. Chair: G. Buchalla Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 25
Measurements of gamma
Speaker: Tagir Aushev
• 26
Cabibbo angle review
Speaker: Matthew Moulson
• 4:30 PM
Discussions
• DG3 - Neutrino Physics: Neutrino Mass and Mixing: Results from Oscillation Experiments Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 27
KamLAND Results
The KamLAND experiment searched for the oscillation of anti-neutrinos emitted from distant nuclear reactors. We observed the anti-neutrino spectral distortion, which was a piece of evidence of neutrino oscillations. Due to their clear spectral distortion, the neutrino mass difference was determined with 2.8% precision. In the near future, we will have a sensitivity also to the low energy solar neutrinos utilizing the characteristics of high light intensity. The precise measurement of low energy solar neutrinos helps us obtain a better understanding of fusion reactions. I will show the status of the anti-neutrino analysis and the purification of the liquid scintillator.
Speaker: Itaru Shimizu
• 28
Latest Results from the MINOS Experiment
The Main Injector Neutrino Oscillation Search (MINOS) long baseline experiment measures a muon neutrino beam in two locations: a Near detector at Fermilab, close to beam production, and a Far detector, 735 km downstream, in Northern Minnesota. Comparisons of the observed energy spectrum at the Far detector with the expectation derived from the Near detector measurement allow MINOS to study neutrino oscillation processes with high precision. In this talk, the methodology employed in the analysis of charged-current muon neutrino and anti-neutrino interactions is outlined and the most recently obtained results are described. Results from a search for oscillations into sterile neutrinos, derived from the analysis of neutral current interactions in the MINOS detectors, are also presented. A beam exposure of 3.2x10^20 protons on target is used in the analyses.
Speaker: Alexandre Sousa
• 29
First Results for Electron-Neutrino Appearance in MINOS
MINOS is a long baseline neutrino oscillation experiment designed to make precision measurements of the neutrino mixing parameters associated with the atmospheric neutrino mass splitting. Using a neutrino beam from the Main Injector (NuMI) facility at Fermilab, it compares the neutrino energy spectrum for neutrino interactions observed in two large detectors located at Fermilab and in the Soudan mine in northern Minnesota at a distance of 735km. We describe the first results for electron neutrino appearance in MINOS after two years of data-taking.
Speaker: Mayly Sanchez
• 30
Results from MiniBooNE
MiniBooNE is a short-baseline experiment located at Fermilab, sensitive to muon (anti)neutrino to electron (anti)neutrino appearance and muon (anti)neutrino disappearance oscillations at high $\Delta m^2~$ 1 eV$^2$. These oscillation searches have been motivated by the 3.8$\sigma$ excess of electron antineutrino events in a muon antineutrino beam observed by the LSND experiment in 1995. In this talk, recent antineutrino and updated neutrino oscillation results from MiniBooNE will be presented, and implications for the LSND excess will be discussed within the context of sterile neutrino oscillation models.
Speaker: Georgia Karagiorgi
• 4:50 PM
Discussions
• 5:00 PM
coffee break
• DG2 - Weak Decays, CP violation and CKM: Charm, Tau and Top Decays. Chair: G. Buchalla Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 31
Speaker: Sandra Leone
• 32
D0 mixing measurements
Speaker: Anze Zupanc
• 33
Non-LFV tau decays
Speaker: MyeongJae Lee
• DG3 - Neutrino Physics: Neutrino Mass and Mixing: Results from Oscillation, cont. Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 34
Search for Muon Neutrino Disappearance in a Short-Baseline Accelerator Neutrino Beam
Neutrino oscillation is a probe for new physics not included in the Standard Model. We search for muon neutrino disappearance at \Delta m^2 ~ 1 eV^2 using the Fermilab Booster Neutrino beamline and two experiments, SciBooNE and MiniBooNE. The neutrino fluxes are measured at SciBooNE and MiniBooNE detectors, located at 100 m and 540 m downstream from the neutrino production target, respectively. We took beam data from June 2007 through August 2008 at SciBooNE and MiniBooNE. A preliminary result of the flux measurement at SciBooNE and SciBooNE-MiniBooNE joint oscillation analysis will be presented.
Speaker: Yasuhiro Nakajima
• 35
Status of the OPERA experiment
The OPERA experiment is a long baseline neutrino experiment aiming at the first observation of the appearance of a new flavor signal predicted by the neutrino flavor-mixing oscillations hypothesis. The OPERA is designed to detect a tau neutrino appearance in the pure muon neutrino beam from CERN to Gran Sasso (CNGS beam). The target has a modular structure made of units based on Emulsion Cloud Chamber (ECC) technique. ECC is sequence of the nuclear emulsion films interleaved with 1mm-thinkness lead plates. Nuclear emulsion films are used as a tracking device with micrometric accuracy for the detection of tau decays. The ECC allows also to perform the kinematical measurements such as the momentum measurement by multiple scattering detection and the electron shower detection. Total target mass of 1.25kton was assembled by more than 150,000 units. Since 2008 the OPERA experiment has started full data taking in the CNGS beam. Around 1,700 interactions in the detector have been collected in 2008 and the 2009 run is ongoing. The status of the analysis and some fruitful outcome will be presented. The prospect of the future runs is also to be presented.
Speaker: Akitaka Ariga
• Poster Session
• 36
Strong Scaling Ansatz of flavor neutrino mass matrix and normal mass hierarchy
To find hidden structure of flavor neutrino mass matrix, we study properties of flavor neutrino mass satisfying the strong scaling Ansatz (SSA) that predicts non maximal $\nu_2 - \nu_3$ mixing, vanishing $U_{e3}$ and inverted mass hierarchy. However, we find another possibility of SSA that tiny deviation from this Ansatz permits us to realize normal mass hierarchy and tiny value of $\theta_{13}$ which allows to arise Dirac CP violation. We can clarify correlations of mass parameter and CP violating phases and compare these dependences of CP violating phases on mass parameters in the case of the normal mass hierarchy with those of the inverted mass hierarchy.
Speaker: Mr Teppei Baba (Tokai university)
• 7:00 PM
"The path to Beethovens Eroica-finale" by H. Pietschmann
• Wednesday, September 16
• DG1 - Electro Weak Symmetry Breaking: Higgs Physics Sala Orsini

### Sala Orsini

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 37
Direct Standard Model Higgs Searches at the Tevatron
Since the beginning of Run II the Fermilab Tevatron has delivered 6.9 fb-1 of proton-antiproton collisions at a center of mass energy of 1.96 TeV. Using the collected dataset the CDF and D0 experiments are pursuing extensive Physics programs which have as a main goal the search for a Standard Model Higgs boson. The contribution will review the CDF and D0 results on the direct Standard Model Higgs searches over a wide mass range, from 100 to 200 GeV/c2, including the golden channels, HW->lvbb in the lower mass region and H->WW->lvlv in the higher mass region, and a series of additional channels, ZH->llbb, ZH->vvbb, HW->WWW, and modes with tau leptons in the final state, which improve sensitively the experiments' reach. Particular prominence will be given to the recently updated measurements that analyze up to 5 fb-1 of data. The combined upper limit of CDF and D0 measurements and the projections on the production cross section limits for the final dataset of Tevatron Run II will also be shown.
Speaker: Massimo Casarsa
• 38
Prospects for SM Higgs at the LHC
Speaker: Alberto Graziano
• 10:00 AM
Discussions
• DG2 - Weak Decays, CP violation and CKM: Theory. Chair: G. Isidori Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 39
Lattice results - overview and perspectives
Speaker: Damir Becirevic
• 40
Chiral perturbation theory
Speaker: Gerhard Ecker
• 41
Rare K decays - theory
Speaker: Giancarlo D'Ambrosio
• DG3 - Neutrino Physics: Neutrino Mass and Mixing: Reactor and New Acelerator Experiments Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 42
Current Status of RENO
RENO(Reactor Experiment for Neutrino Oscillation), is under construction to measure the smallest neutrino mixing angle theta_13 using antineutrinos emitted from the Yonggwang power plant in Korea with world-second largest thermal power output of 16.4 GW. A high precision measurement of reactor neutrino oscillation can be achieved by two identical detectors. Each detector consists 16-ton Gadolinium loaded liquid scintillator as a neutrino target. The near and far detectors are placed roughly 290 m and 1.4 m from the center of the reactor array. The near detector is constructed at underground of a 70 m high hill and the far detector at underground of a 260 m high mountain. The identical detector setup will reduce systematic uncertainties to less than 1%. The experiment is planned to start data-taking in mid 2010. In this talk the current status of detector construction will be presented.
Speaker: Kang Soon Park
• 43
Measurement of theta13 at Daya Bay
The Daya Bay experiment aims to measure the last unobserved neutrino mixing angle theta_13 with a sensitivity of sin^2 (2 theta_13) < 0.01 at 90% C.L. The experiment will measure the flux and energy spectrum of reactor antineutrinos through the inverse beta-decay reaction on protons in eight detectors in three underground sites at different distances from the reactor cores. An overview and the current status of the experiment will be discussed.
Speaker: Steve kettell
• 44
The Double Chooz Experiment
The Double Chooz neutrino oscillation experiment will perform a highly sensitive measurement of the neutrino mixing angle θ13. The Double Chooz project will utilize two detectors, a near one providing the unoscillated neutrino flux coming from the two reactor cores of the Chooz nuclear plant (France), and a far detector measuring the neutrino flux after oscillation. The signal comparison between the two detectors will allow to push down the θ13 limit at very low values. Double CHOOZ far detector under construction will start taking data beginning of next year and in about 1.5 years will be able to set the limit sin2(2θ13)<0.06, the present limit being sin2(2θ13)<0.2. At that moment, the near detector will also be ready and help to set this limit at sin2(2θ13)<0.03 in less than 3 years. The mixing angle θ13 is the last non-measured oscillation parameter of the neutrino mixing matrix. The possibility of measuring the sin2(2θ13) or lowering the existing limit to a few percent would also allow selection of the best means to search for the CP-phase in the lepton sector. In this presentation a detailed description of the Double Chooz experiment will be done.
Speaker: Marcos Dracos
• 45
T2K General talk(Status and T2K general description)
T2K (Tokai-to-Kamioka) is a second generation long baseline neutrino experiment utilizing a newly built neutrino source with a MW class high energy proton accelerator complex (J-PARC neutrino facility), a near neutrino detector (ND280) to characterize the neutrino beam 280 meters from the source, and Super-Kamiokande as the far detector at 295 km. The primary motivation for T2K is the discovery of the nu_mu to nu_e conversion phenomena and, as a consequence, the finite value of the theta_13 mixing angle. It will also conduct a precise measurement of theta_23 and the mass difference of neutrino mass eigenstate. The ultimate goal for T2K is to establish the lepton flavor mixing structure. Construction of the J-PARC neutrino facility was completed in March 2009 and engineering operation of the T2K started as scheduled the following month. This talk will provide a general introduction to T2K, and present the current beam commissioning status and the status of preparations towards the start of the experiment.
Speaker: Hidekazu Kakuno
• 10:20 AM
Discussions
• 10:30 AM
coffee break
• DG2 - Weak Decays, CP violation and CKM: Theory, cont. Chair: G. Isidori Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 46
New physics models
Speaker: Bjoren Duling
• DG3 - Neutrino Physics: New and Future Experiments Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 47
The NOvA Experiment: Status and Prospects
NOvA is a next generation long baseline neutrino experiment. It has been designed to study electron neutrino appearance in a muon neutrino beam using a totally active, segmented, liquid scintillator detector located off the Fermilab NuMI beam axis. Construction of a prototype Near detector will commence this year and will be placed in the existing NuMI beam during 2010. The Far Detector will begin construction shortly thereafter and will be completed in 2014. The project will upgrade the NuMI facility from 400 kW to 700 kW. NOvA will push the search for electron neutrino appearance beyond the current limits by more than an order of mangnitude. It will also have sensitivity to the neutrino mass hierarchy and by running both neutrinos and anti-neutrinos NOvA will begin the search for CP violation in the lepton sector.
Speaker: Mayly Sanchez (Argonne National Laboratory)
• 48
Future long baseline experiments: options for Japan
As the first phase experiment utilizes J-PARC(Japan Proton Accelerator Research Complex) neutrino facility, T2K(Tokai to Kamioka Long Baseline Neutrino Experiment) starts operation. T2K is supposed to give critical information, which guides the future direction of the neutrino physics. I will discuss possible future discovery experiment next to T2K. Talk will focus especially on describing J-PARC neutrino beam upgrade plan and discussion on far detector options to maximize potential of the research.
Speaker: Takuya Hasegawa
• 49
Future long baseline experiments: options for Europe
The LAGUNA design study is presently assessing the feasibility of a new large underground infrastructure in Europe for next generation neutrino and proton decay search experiment I will describe the status of LAGUNA and discuss possible options for future long baseline experiments coupled to a potential future program at CERN.
Speaker: Andre Rubbia
• 50
Future long baseline experiments: options for US
Speaker: Mitchell Soderberg
• 12:20 PM
Discussions
• DG4 - Dark Matter: Dark Matter Direct Detection I. Chair: D. Grant Sala Orsini

### Sala Orsini

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 51
Signals from the Universe: the DAMA/LIBRA results
The highly radiopure about 250 kg NaI(Tl) DAMA/LIBRA set-up is running at the Gran Sasso National Laboratory of the I.N.F.N.. Results exploiting the model independent annual modulation signature for Dark Matter particles in the galactic halo are presented (exposure of 0.53 ton x yr). The DAMA/LIBRA data confirm the evidence for the presence of Dark Matter particles in the galactic halo as observed by the former DAMA/NaI experiment. The combined analysis of the data of the two experiments (total exposure 0.82 ton x yr) gives a C.L. at 8.2 sigma.
Speaker: Pierluigi Belli
• 52
Search for Dark Matter with the CDMS experiment
The Cryogenic Dark Matter Search Experiment (CDMS) employs a total of 30 Germanium and Silicon Detectors at the Soudan Underground Laboratory to detect weakly interacting massive particles (WIMPs) via their elastic scattering from the target nuclei. Previous CDMS results, released in February 2008, yielded a world-leading limit for spin-independent WIMP-nucleon cross sections for WIMP masses above 44 GeV/c^2, restricting significant parts of the parameter space favored by supersymmetric models. Data taken between July 2007 and October 2008 increase the exposure by a factor of 2.5. Latest results of the experiment will be presented.
Speaker: Sebastian Arrenberg
• 11:50 AM
Discussions
• 12:30 PM
Lunch + Break
• DG1 + DG4 - Combined Session: GeV scale hidden sectors/Dark Matter Sala Orsini

### Sala Orsini

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 53
The High Energy Cosmic Ray Electron Spectrum measured with the Fermi Space Telescope: some possible interpretations
The Fermi Large Area Telescope (LAT) has recently provided the measurement of the high energy (20 GeV to 1 TeV) cosmic ray electrons plus positrons (CRE) spectrum with unprecedented accuracy. The spectrum shows no prominent features and it is significantly harder than that inferred from several previous experiments. While the reported Fermi-LAT data alone can be interpreted in terms of a single (electron dominated) Galactic component, when combined with other complementary experimental results, specifically the CRE spectrum measured by H.E.S.S., and especially the positron fraction measured by PAMELA, an additional electron and positron component has to be invoked. We will show that electron-positron pairs acceleration in Galactic pulsars offer a natural interpretation of all these results and briefly mention other viable scenarios including secondary electron and positron production in supernova remnants and dark matter annihilation/decay. We also briefly discuss the possibility of discriminating among those interpretations by means of other independent measurements.
Speaker: Dario Grasso
• 54
Dark Matter and Antimatter Searches in the Cosmic Radiation and the PAMELA experiment
Speaker: Paolo Papini
• 55
GeV scale hidden sector at colliders
Speaker: Matthew Reece
• 4:45 PM
Discussions
• DG2 - Weak Decays, CP violation and CKM: Rare decays. Chair: G. Ecker Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 56
Speaker: Antonio Limosani
• 57
Leptonic B decays
Speaker: Silke Nelson
• 4:30 PM
Discussions
• DG3 - Neutrino Physics: Future Experiments and Cross sections Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 58
Novel Neutrino Beams: R&D and Design Studies in the US
A Neutrino Factory producing intense neutrino beams from muon decays is an intriguing possibility for the next generation of neutrino experiments. Determination of the neutrino mass hierarchy, precision measurements of the neutrino mixing parameters, and observation of CP violation in the neutrino sector are among the many studies possible with a Neutrino Factory. A Neutrino Factory can also be an important step toward the development of a Muon Collider. This talk discusses the current status and plans for Neutrino Factory R&D in the U.S. focusing particularly on accelerator development for generating the high luminosity parent muon beam and development of new neutrino detectors necessary for the higher neutrino intensities.
Speaker: Terrence Hart
• 59
R&D and Design work for novel neutrino beams in Europe
"R&D and Design Studies towards novel neutrino beams are advancing well in Europe. International R&D experiments (HARP, MERIT, MICE, EMMA) have been and are succesfully being performed. Design studies of high power upgrades of existing (ISIS) or planned (SPL) proton machines, of detector concepts and sites (Laguna) and of high intensity beams (EUROnu) are in progress. EUROnu aims, in particular, at the design of - a Neutrino Factory, within its International Design Study - a Betabeam neutrino facility - a low energy 4 MW neutrino Superbeam A dedicated workshop is being prepared for early October at CERN, likely to be followed by statements from the CERN Council on the role that Europe should play in a global accelerator neutrino program."
Speaker: Alain Blondel
• 60
Status of the MICE International Muon Ionisation Cooling Experiment
Muon ionization cooling provides the only practical solution to prepare high brilliance beams necessary for a neutrino factory or muon colliders. The muon ionization cooling experiment (MICE)* is under development at the Rutherford Appleton Laboratory (UK). It comprises a dedicated beam line to generate a range of input emittance and momentum, with time-of-flight and Cherenkov detectors to ensure a pure muon beam. A first measurement of emittance is performed in the upstream magnetic spectrometer with a scintillating fiber tracker. A cooling cell will then follow, alternating energy loss in liquid hydrogen and RF acceleration. A second spectrometer identical to the first one and a particle identification system provide a measurement of the outgoing emittance. In September 2009 it is expected that the beam and some detectors will be in the final commissioning phase and the time of the first measurement of input beam emittance only months away. The plan of steps of measurements of emittance and cooling, that will follow in the rest of 2009 and later, will be reported.
• 61
Resolving CP Violation by Standard and Nonstandard Interactions in Neutrino Oscillations
In neutrino oscillation with non-standard interactions (NSI) the system is enriched with CP violation caused by phases due to NSI in addition to the standard lepton Kobayashi-Maskawa phase $¥delta$. In this paper we show that it is possible to disentangle the two CP violating effects by measurement of muon neutrino appearance by a near-far two detector setting in neutrino factory experiments. Prior to the quantitative analysis we investigate in detail the various features of the neutrino oscillations with NSI, but under the assumption that only one of the NSI elements, $¥varepsilon_{e ¥mu}$ or $¥varepsilon_{e ¥tau}$, is present. They include synergy between the near and the far detectors, the characteristic differences between the $¥varepsilon_{e ¥mu}$ and $¥varepsilon_{e ¥tau}$ systems, and in particular, the parameter degeneracy. Finally, we use a concrete setting of muon energy 50 GeV and two magnetized iron detectors at the two baselines, one at $L=3000$ km and the other at $L=7000$ km, each having a fiducial mass of 50 kton to study the discovery potential of NSI and its CP violation effects. We demonstrate by assuming $4 ¥times 10^{21}$ useful muon decay for both polarities that one can identify non-standard CP violation down to $¥vert ¥varepsilon_{e ¥mu} ¥vert ¥simeq ¥text{a few} ¥times 10^{-3}$, % $¥vert ¥varepsilon_{e ¥mu} ¥vert ¥simeq ¥text{ several} ¥times 10^{-4}$, and $¥vert ¥varepsilon_{e ¥tau} ¥vert ¥simeq 10^{-2}$ at 3$¥sigma$ CL for $¥theta_{13}$ down to $¥sin^2 2¥theta_{13} = 10^{-4}$ in most of the region of $¥delta$. The impact of existence of NSI on measurement of $¥delta$ and the mass hierarchy is also worked out.
Speaker: Shoichi Uchinami
• 4:50 PM
Discussions
• 5:00 PM
coffee break
• DG2 - Weak Decays, CP violation and CKM: Rare decays, cont. Chair: G. Ecker Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 62
Rare K decays
Speaker: Rainer Wanke
• 63
QCD from K decays
Speaker: Mauro Piccini
• 6:30 PM
Discussions
• DG3 - Neutrino Physics: Cross Sections Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 64
Neutrino cross sections in GeV region
Modern long-baseline neutrino experiments, aiming at precise measurements of the neutrino oscillation parameters require a thorough understanding of neutrino cross sections in medium-energy region. An overview of neutrino-nucleon and neutrino-nucleus cross sections in a few GeV region is presented. Cross section on a nucleon target is expressed as a sum of quasi-elastic, resonance and DIS contributions. For nuclear targets, nuclear corrections are shown to be not negligible, several approaches to estimate them are surveyed.
Speaker: Olga Lalakulich
• 65
Recent results from the neutrino scattering experiments
Recent experimental studies of neutrino oscillation require more precise knowledge of the neutrino-nucleus scatterings. Therefore, several experiments to study neutrino nucleus-scattering experiments have been performed. In this presentation, I will report the recent results of neutrino scatterings from SciBooNE and MiniBooNE.
Speaker: Yoshinari Hayato
• 66
Hadro-production measurements for the T2K experiment with the NA61/SHINE detector at the CERN SPS
In this talk the NA61/SHINE detector will be presented, which is a large acceptance hadron spectrometer at the CERN SPS. It allows for a precise study of the particle production from the interactions of a 31 GeV/c proton beam on a Carbon target in order to predict the neutrino flux of the T2K experiment at J-PARC, Japan. Requirements for the T2K experiment will be discussed together with the ongoing NA61 measurements. In particular preliminary NA61 cross section measurements from the 2007 pilot run will be shown.
Speaker: Claudia Christina Strabel
• 67
T2K Cross Section Measurements
Speaker: Steve Boyd
• DG4 - Dark Matter: Dark Matter and Neutrinos. Chair: B. Serfass Sala Orsini

### Sala Orsini

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 68
Dark matter and solar neutrinos with DEAP/CLEAN
The large difference between the time constants of the two scintillation pulse com- ponents of a noble liquid like liquid Argon or Neon provide a very reliable correlation between pulse shape and type of event. This pulse shape discrimination already pro- vides the power of rejecting a background $10^{8}-10^{9}$ times larger than the signal. A 400 kg LAr detector, MiniCLEAN, is currently under construction, and a 3.6 ton detector, DEAP-3600, under development. The pro jected sensitivities of these two Dark Matter detectors are illustrated. As the DEAP/CLEAN program is also for p-p solar neu- trino detection, the sensitivity to these neutrinos expected from future larger upgrades loaded with liquid Neon is also discussed.
Speaker: Franco Giuliani
• 69
Dark Matter Searches and Fundamental Neutrino Measurements with IceCube-DeepCore
IceCube is a cubic kilometer neutrino telescope under construction at the South Pole, a successor to the first-generation AMANDA telescope designed to the search for astrophysical neutrino sources. IceCube is now three quarters complete with the full detector expected to be operating in early 2011. Data taken with the partially built detector already provides world-leading sensitivity on spin-dependent dark matter scattering cross-sections. The base design of IceCube now includes an infill array known as DeepCore, improving sensitivity to neutrinos at energies below 100 GeV. Plans for DeepCore, as well as estimates for the IceCube-DeepCore sensitivity to dark matter will be presented. Further, the possibility of studying atmospheric neutrino oscillations using will be discussed.
Speaker: Darren Grant
• 70
Mirror Dark Matter
Speaker: Zurab Berezhiani
• 6:45 PM
Discussions
• Thursday, September 17
• DG2 - Weak Decays, CP violation and CKM: LFV, EDM. Chair: T. Gershon Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 71
LFV & EDM
Speaker: Alejandro Ibarra
• 72
MEG experiment
Speaker: Alessandro Baldini
• 10:10 AM
Discussions
• DG3 - Neutrino Physics: Neutrino Properties Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 73
CUORE: the near future of neutrinoless double beta decay searches
Neutrinoless double beta decay is a powerful tool to investigate the fundamental nature of neutrinos and to determine the absolute neutrino mass scale. To date, CUORE is the only fully approved next generation 1-ton size experiment with the goal of exploring the inverted hierarchy region for the neutrino masses. CUORE is an array of 988 TeO2 cryogenic detectors containing 200 kg of Te-130 - the neutrinoless double beta decay candidate - and it is presently being built in Gran Sasso Underground Laboratory. Data taking is due to start in 2012. The feasibility of the project has been proved by CUORICINO, the pilot experiment that took data until 2008, for about five years, with 62 TeO2 cryogenic detectors in Gran Sasso Laboratory. CUORICINO will be replaced in 2010 by CUORE-0, the first CUORE tower to be installed in the CUORICINO cryogenic facility. CUORE-0 will take data until CUORE start. In this talk I will report on the final results of CUORICINO, and discuss CUORE-0 and CUORE potential and state of the art.
Speaker: Angelo Nucciotti (Dip. Fisica, Univ. di Milano-Bicocca and INFN Sez. di Milano-Bicocca)
• 74
The GERDA experiment
Neutrinoless double-beta-decay could be the key to understanding the nature of the neutrino: if observed it would prove its Majorana-nature and the half-life of the decay would be a direct measure of the yet unknown absolute scale of the neutrino mass. The GErmanium Detector Array (GERDA) experiment at the INFN, Gran Sasso Laboratory, Italy, is designed to investigate the double-beta-decay of the isotope Ge-76. Germanium crystals enriched in Ge-76, acting as source and detector simultaneously, will be submerged directly into their ultra pure cooling medium (liquid argon) that also serves as a radiation shield. A further reduction of the external background is achieved by means of a 2m-thick water buffer, which is operated as a Cherenkov detector to veto cosmic-ray muons. This concept will allow for a background reduction of up to two orders of magnitudes with respect to earlier experiments - a key requirement for further improvement of sensitivity. The design of the GERDA experiment will be introduced and the current status of the experiment will be discussed.
Speaker: Luciano Pandola
• 75
Search for neutrinoless double beta decay: latest results from NEMO-3 and plans for SuperNEMO
The NEMO-3 experiment, located in the Modane Underground Laboratory, is searching for neutrinoless double beta decay since 2003. We will present the latest results for seven isotopes used on NEMO-3: 48Ca, 96Zr, 82Se, 100Mo, 116Cd, 130Te and 150Nd. We observe no evidence for neutrinoless double beta decay and based on 100Mo data set an upper limit on the effective Majorana neutrino mass to be in the range 0.45 - 0.90 meV, depending on the nuclear matrix element. NEMO-3 data can also be interpreted in terms of alternative transition models, such as weak right-handed currents or Majoron emission. SuperNEMO is a next-generation experiment exploiting the successful tracking plus calorimetry technology of the NEMO-3 experiment. SuperNEMO will use about 100 kg of 82Se and is designed to reach sensitivity to a half-life greater 10**26 years. We will briefly describe main challenges before SuperNEMO and the current status of the project.
Speaker: Karol Lang
• 10:20 AM
Discussions
• DG4 - Dark Matter: Dark Matter Direct Detection II. Chair: C. Galbiati Sala Orsini

### Sala Orsini

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 76
DAMA/LIBRA and leptonically interacting Dark Matter
We consider the hypothesis that Dark Matter (DM) has tree-level interactions only with leptons. Such a framework, where DM recoils against electrons bound in atoms, has been proposed as an explanation for the annually modulated scintillation signal in DAMA/LIBRA versus the absence of a signal for nuclear recoils in experiments like CDMS or XENON10. However, even in such a leptophilic DM scenario there are loop induced DM-hadron interactions, where the photons emitted from virtual leptons couple to the charge of a nucleus. Using a general effective field theory approach we show that, if such an interaction is induced at one or two-loop level, then DM-nucleus scattering dominates over DM-electron scattering. This is because the latter is suppressed by the bound state wave function. One obtains a situation similar to standard DM-nucleus scattering analyses with considerable tension between the results of DAMA and CDMS/XENON10. This conclusion does not apply in the case of pseudoscalar or axial vector coupling between DM and leptons, where the loop diagrams vanish. In this case the explanation of the DAMA signal in terms of DM-electron scattering is strongly disfavored by the spectral shape of the signal. Furthermore, if DM can annihilate into neutrinos or tau leptons, the required cross sections are excluded by many orders of magnitude using the Super-Kamiokande bound on neutrinos from DM annihilations in the Sun.
Speaker: Jure Zupan
• 77
The WArP Experiment
Speaker: Ettore Segreto
• 10:15 AM
Discussions
• 10:30 AM
coffee break
• DG2 - Weak Decays, CP violation and CKM: LFV, EDM cont. Chair: T. Gershon Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 78
LFV in tau decays
Speaker: Riccardo Cenci
• 79
EDM review
Speaker: Peter Fierlinger
• 12:00 PM
Discussions
• DG3 - Neutrino Physics: Neutrino Properties, cont. Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 80
Direct neutrino Mass Measurementes
Speaker: Monica Sisti
• 81
SNO & SNO+
Speaker: Steven Biller
• 82
Kamland: double beta decay perspectives
Speaker: Itaru Shimizu
• 83
Characterization of a Nd-loaded organic liquid scintillator for neutrinoless double beta decay search of 150Nd with a 10-ton scale detector
Speaker: Aldo Ianni
• DG4 - Dark Matter: Dark Matter Direct Detection III. Chair: F. Giuliani Sala Orsini

### Sala Orsini

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 84
The ArDM experiment, a double phase liquid argon TPC for direct detection of Dark Matter
The goal of the ArDM experiment is a direct detection of weakly interacting massive particles (WIMP) in the universe using a double phase argon TPC. This detector has a fiducial volume of about 1 ton of liquid argon and the detection is done independently by collecting the scintillation light as also by the ionization charge. The Argon scintillation light has to be converted with wavelength shifters and collected by PMT, instead the electrons produced by ionization are extracted from the liquid into the gas phase of the detector, amplified and read out with Large Electron Multipliers detectors. The talk will give an overview of the design of the detector, and of the R&D of the different components, and it will describe the first light readout test with liquid argon with different gamma and neutron sources.
Speaker: Claudia Lazzaro
• 85
The LUX Dark Matter Experiment
The LUX dark matter experiment is a 350 kg two-phase xenon time projection chamber. There are several advantages of liquid xenon for WIMP detection: high scintillation yield, high ionization yield, low intrinsic radioactivity, easy purification, effective gamma-ray self-shielding, and significant A^2 enhancement of the spin-independent WIMP cross-section. LUX is currently under construction and will be installed in Fall 2009 at the SUSEL lab in South Dakota. I will review the design features of LUX, give an update on its current status, and describe our conception of the future LUX-ZEPLIN dark matter program.
Speaker: Daniel McKinsey
• 86
Depleted Argon for Future Direct Dark Matter Searches with Argon
I will discuss progress on procurement of large quantities of depleted argon from underground sources in the US, and the how this could improve future dark matter searches with argon detector. I will also discuss results from R&D efforts on two-phase argon detectors in the US and future plans.
Speaker: Cristiano Galbiati
• 87
Annual Modulation Effects Observable in a Noble Liquid Detector.
The use of liquid noble gas detectors in the search for Dark Matter is quickly becoming more widespread and they are already playing one of the major parts in the field. One of the possible signatures of the presence of Dark Matter is the so called annual modulation effect, possibly already observed by the DAMA collaboration. The liquid noble gas detectors are generally tailored to observe single WIMP interactions. Despite this, it might be interesting to investigate whether the annual modulation effect will be observable in these detectors. Another effect that is interesting is the possible influence of the Sagittarius Stream on these observations. The prospects for observing the annual modulation effect and those of the Sagittarius Stream in liquid noble gas detectors will be discussed, as well as some speculations as to what kind information can be gained by these observations.
Speaker: Andrzej Szelc
• 12:30 PM
Lunch + Break
• 3:30 PM
Free time

Free time

• 8:00 PM
Social Dinner

Social Dinner

• Friday, September 18
• DG1 - Electro Weak Symmetry Breaking: Higgs Physics and SUSY Sala Orsini

### Sala Orsini

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 88
Prospects for BSM Higgs at the LHC
Speaker: Jan Schumacher
• 89
SUSY Searches at the LHC
Speaker: Phillip URQUIJO
• 90
Non-Standard Higgs Decays
Theories of natural Electroweak Symmetry Breaking can involve light higgses. These imply significant changes in search strategies at the Tevatron, LHC, and B-factories. I will review some non-standard Higgs scenarios which involve higgs-to-higgs decays, and the experimental signatures thereof.
Speaker: Robert McElrath
• 10:25 AM
Discussions
• DG2 - Weak Decays, CP violation and CKM: Future initiatives. Chair: M. Sozzi Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 91
Flavour physics at LHC
Speaker: Jaap Panman
• 92
Super-B in Italy
Speaker: Marcello GIORGI
• 93
Super-B in Japan
Speaker: Masanori Yamauchi
• DG3 - Neutrino Physics: Cosmic/solar/Geo Neutrinos Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 94
Collective Flavor Oscillations For Supernova Neutrinos and r-Process Nucleosynthesis
The effect of collective flavor oscillations of neutrinos driven by neutrino-neutrino interaction at the very high density region of core collapse supernovae controls the emitted flux of neutrinos of different flavors. In the process one or more swaps of the flavors for both neutrinos and antineutrinos take place depending on the initial neutrino flux and distributions, particularly for the inverted mass hierarchy. We study the effect of this on the possibility of having a neutron-rich region compatible with r-process nucleosynthesis. The minimal requirement for r-process is the electron-to-nucleon ratio Ye < 0.5, but a more favorable condition may be Ye < 0.45 . In this work we consider a two flavor model, with e-type and x-type neutrinos along with their antiparticles and with the oscillation parameters mass squared difference = 0.003 eV^2 in agreement with realistic 1−3 mixing and a small effective mixing angle of 0.00001 . As in supernovae the four species, mu and tau type neutrinos and antineutrinos have identical spectra, this study itself may give indications of the real situation. Different models of neutrino energy distributions are used. For each of the distributions initial fluxes of different flavors are varied and exclusion plots for these initial neutrino fluxes show the allowed regions for r-process nucleosynthesis .The electron fraction(Ye) as a function of the radius of the core is calculated and it shows an oscillatory behavior in the bipolar region before saturating to a constant value. This shows that for the allowed fluxes one gets neutron-rich regions for r-process in the neutrino driven wind.But other considerations of baryon density and entropy in these regions need to be studied.
Speaker: Kamales Kar
• 95
Geo-Neutrinos
Speaker: Giovanni Fiorentini
• 96
SK status and perspectives (low-energy neutrinos)
Speaker: Yasuo Takeuchi
• 97
SK status and perspectives (atmospheric neutrinos)
Speaker: Fanny DUFOUR
• 10:30 AM
coffee break
• DG1 - Electro Weak Symmetry Breaking: Supersymmetry and Exotica Sala Orsini

### Sala Orsini

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 98
Higgsless models (Theory)
Speaker: Gino Isidori
• 99
Searches for New Physics at HERA
Speaker: Robert CIESIELSKI
• 12:15 PM
Discussions
• DG2 - Weak Decays, CP violation and CKM: Future initiatives, cont. Chair: M. Sozzi Sala Niccolo' V

### Sala Niccolo' V

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 100
Final Results on K+ -> pi+ nu nu_bar from BNL E949
The extremely rare decay K+ -> pi+ neutrino anti-neutrino has been studied for the last 20 years with the E787 and E949 experiments and the high-intensity proton beam of the AGS accelerator at Brookhaven National Laboratory. The E949 experiment previously reported results above the K+ -> pi+pi0 peak, in the pion momentum region [211,229] MeV/c {Phys.Rev.Lett.93:031801(2004); Phys.Rev.D77:052003(2008)}. E949 has recently reported results below the K+ -> pi+pi0 peak, in the pion momentum region [140,199] MeV/c, {Phys.Rev.Lett.101:191802(2008);Phys.Rev.D79:092004(2009)}. A total of seven candidates of this very rare process have been identified. This talk will describe the experiment and these results.
Speaker: Steve kettell
• 101
NA62 at CERN
Speaker: Augusto Ceccucci
• 102
Status of the J-PARC E14 KOTO Experiment for the KL --> pi0 nu nu-bar
The purpose of the J-PARC E14 KOTO experiment is to search for new physics beyond the standard model that enhances the branching ratio of the $$K_L \to \pi^0 \nu \bar{\nu}$$ decay. The experiment looks for two and only two photons from the decay that have a finite transverse momentum. The detector consists of a pure CsI calorimeter and hermetic photon veto counters placed in a vacuum tank. The beamline has been constructed, and the beam will be tested starting this fall. New CsI calorimeter will be installed next year, and rest of the detector will be installed in the following year. The features and status of the experiment will be presented.
Speaker: Taku Yamanaka
• DG3 - Neutrino Physics: Cosmic/solar/Geo Neutrinos, cont. Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it
• 103
Borexino is a liquid scintillator, solar neutrino detector running underground at the Laboratori Nazionali del Gran Sasso, Italy. Thanks to the extreme radio-purity achieved throughout its whole set-up Borexino is capable of detecting solar neutrinos below 2 MeV. Among the main results obtained so far are the first measurement of the 7Be solar neutrinos flux, whose value is in agreement with the LMA-MSW scenario for neutrino oscillations, and the measurement of the 8B solar neutrino flux with an energy threshold of 2.8 MeV. These results are reported here and the future prospects discussed.
Speaker: Elena Guardincerri
• 104
High energy neutrino astronomy: towards a km3 neutrino telescope in the Mediterranean Sea
Speaker: Emilio Migneco
• 105
The ANTARES detector: status and first results
ANTARES is an underwater neutrino telescope, located about 2500 m under the sea, 40 km off the coast of Toulon/France. Consisting of about 900 optical modules it is currently the largest neutrino detector in the northern hemisphere and has been taking data in its final configuration since May 2008. In this talk the physics potential and the detection technique are described and first results are presented. The future of neutrino astronomy in the Mediterranean Sea is discussed.
Speaker: Annarita Margiotta
• 12:10 PM
Discussions
• 12:30 PM
Lunch + Break
• Plenary Session: Discussion Group Summaries. Chair: T. Yamanaka Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it

Plenary session

• 106
Electro Weak Symmetry Breaking: Theoretical Summary
DG1 - Summary and Perspectives
• 107
Electro Weak Symmetry Breaking: Experimental Summary
DG1 - Summary and perspectives
Speaker: Yves Sirois
• Plenary Session: Discussion Group Summaries. Chair: T. Yamanaka Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it

Plenary session

• 108
Weak Decays, CP violation and CKM: Theoretical Summary
Speaker: Gerhard Buchalla
• 109
Weak Decays, CP violation and CKM: Experimental Summary
Speaker: Marco Sozzi
• 5:00 PM
coffee break
• Plenary Session: Discussion Group Summries. Chair: E. Migneco Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it

Plenary session

• 110
Neutrino Physics: Theoretical Summary
DG3 - Summary and perspectives
Speaker: Hiroshi Nunokawa
• 111
Neutrino Physics: Experimental Summary I
Speaker: Bonnie Fleming
• 112
Neutrino Physics: Experimental Summary II
DG3 - Summary and perspectives
Speaker: Oliviero Cremonesi
• Saturday, September 19
• Plenary Session: Discussion Group Summaries. Chair: G. Fiorentini Sala Gregorio VI

### Sala Gregorio VI

#### Strada Passo dell'Acqua, 34 - 06134 Bosco PERUGIA, Italy

http://win09.lngs.infn.it

Plenary session

• 113
Dark Matter: Theoretical Summary


text:
×

# Live Challenge

I know that pressure is continuously increasing on Brilliant Staff. But I just wanted to why no Live Challenge was conducted for the last 5-6 months( as on Brilliant Blog ) . I think there wasn't much pressure on the Staff until January. It seems fun for level 5 users . Also, the staff will be a little relieved once they release the problem banks tomorrow so they may take this (or the Problem of the Week which I posted earlier) into consideration.

Note by Garvil Singhal
4 years, 8 months ago

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Yes please, I too wish to participate in these Live Challenges, the questions are interesting!

- 4 years, 8 months ago

Yup, i agree. Even i was going to start one discussion . Though pressure is high, but their development and contribution is highly appreciable.

- 4 years, 8 months ago

have faith they will start soon

- 4 years, 8 months ago


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# Union of topologies as subbase of a topology is the weakest topology that is also stronger than each of summed topologies

Let $$\mathcal{T}_\alpha$$ be topologies on space X for some $$\alpha$$ acting as arbitrary index. Define $$V_\alpha \mathcal{T}_\alpha$$ as topology with $$\bigcup_\alpha\mathcal{T}_\alpha$$ as its subbase. Prove that $$V_\alpha \mathcal{T}_\alpha$$ is the weakest topology that is also stronger than every $$\mathcal{T}_\alpha$$.

Proof:

1. $$V_\alpha \mathcal{T}_\alpha$$ is stronger than every $$\mathcal{T}_\alpha$$

To prove that it is sufficient to show that for a fixed $$\alpha$$ we have $$T \in \mathcal{T}_\alpha \implies T \in V_\alpha \mathcal{T}_\alpha$$. Knowing that subbase of $$V_\alpha \mathcal{T}_\alpha$$ is $$\bigcup_\alpha\mathcal{T}_\alpha$$ and both $$X \in \bigcup_\alpha\mathcal{T}_\alpha$$ and $$T \in \bigcup_\alpha\mathcal{T}_\alpha$$ for every $$T \in \mathcal{T}_\alpha$$, then we may say $$T = X \cap T$$. Now, $$X \cap T$$ is a finite intersection so it is in $$V_\alpha \mathcal{T}_\alpha$$ by the definition of subbase. We thus conclude $$V_\alpha \mathcal{T}_\alpha$$ is stronger than $$\mathcal{T}_\alpha$$ for every $$\alpha$$.

1. $$V_\alpha \mathcal{T}_\alpha$$ is the weakest topology with that property

Here I got stuck and appreciate any help.

Also, is the proof number 1 correct?

I'll call your $$V_\alpha \mathcal{T}_\alpha$$ topology just $$\mathcal{T}$$, with subbase $$\mathcal{S} := \bigcup_\alpha \mathcal{T}_\alpha$$
For each $$\alpha$$ we have $$\mathcal{T}_\alpha \subseteq \mathcal{T}$$ because $$\mathcal{T}_\alpha \subseteq \mathcal{S} \subseteq \mathcal{T}$$. This takes care of 1.
Now suppose $$\mathcal{T'}$$ is any topology on $$X$$ that is stronger than every $$\mathcal{T}_\alpha$$. We have to show that $$\mathcal{T} \subseteq \mathcal{T}'$$ to show $$\mathcal{T}$$ is weakest. This is quite easy: let $$O \in \mathcal{T}$$ and let $$x \in O$$. Because the finite intersections of members of $$\mathcal{S}$$ is a base for $$\mathcal{T}$$, we have finitely many $$S_1, S_2, \ldots, S_n \in \mathcal{S}$$ such that $$x \in (S_1 \cap S_2 \cap \ldots \cap S_n) \subseteq O$$. By definition of $$\mathcal{T}$$ all $$S_i$$ are members of $$\bigcup_\alpha \mathcal{T}_\alpha$$ and so members of $$\mathcal{T}'$$ (because $$\mathcal{T}'$$ contains all $$\mathcal{T}_\alpha$$ by assumption!) and so, as topologies are closed under finite intersections, $$(S_1 \cap S_2 \cap \ldots \cap S_n) \in \mathcal{T}'$$, so $$x$$ is an interior point of $$O$$ w.r.t. $$\mathcal{T}'$$ and as $$x \in O$$ was arbitrary, $$O \in \mathcal{T}'$$, which is what we had to show.
Show that if $$\tau$$ is stronger than any $$\mathcal{T}_\alpha$$, then $$\tau$$ contains $$\bigcup_\alpha\mathcal{T}_\alpha$$. The basis generated by $$\bigcup_\alpha\mathcal{T}_\alpha$$ would then be contained in $$\tau$$, and then by the definition of a basis, every open set in $$V_\alpha \mathcal{T}_\alpha$$ would be in $$\tau$$.


text:
# Non-Archimedean non-standard models for R

Let $\langle \mathbb{R}, 0, 1, +, \cdot, <\rangle$ be the standard model for $R$, and let $S$ be a countable model of $R$ (satisfying all true first-order statements in $R$). Is it true that the set $1,1+1,1+1+1,\ldots$ is bounded in $S$? My intuition says "no", but I am yet to find a counter example. I read something about rational functions, but I cannot verify it is, indeed, a non-standard model of R.

-
Why does a routine application of compactness not yield the model you want? –  Juris Steprans Apr 18 '12 at 11:42
@Juris Steprans: I'm sorry, I can't see what you're implying. Could you be more precise, please? –  Dave Apr 18 '12 at 11:51
Dave, due to the high intersection of users between the MO and MSE communities it is considered impolite to post a question on both sites simultaneously. Please remember that for future reference. (Cross posted on math.SE math.stackexchange.com/q/133418/622) –  Asaf Karagila Apr 18 '12 at 15:50

If $S$ is the set of real algebraic numbers then $1, 1+1, 1+1+1, \dots$ is unbounded in $S$. On the other hand, by compactness of first order logic (as Juris points out), there are models $S$ for which $1, 1+1, 1+1+1, \dots$ is bounded.
Thanks for your answer. I'm afraid I don't understand how compactness comes in here. I'm guessing I should add the statements $\exists{x} x>1$, $\exists{x} x>1+1$ and so on to my set, and show that there are models that satisfy that set. But I think it only gives me that there are greater elements than any element of the type $1+1+\ldots+1$; I need one element that "covers them all". –  Dave Apr 18 '12 at 12:13


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# When do natural leap years occur, on years based on the northern winter solstice?

I'm designing a new calendar and I want the first day of the year to be the first day after the longest night of year in the northern hemisphere. Under this plan, "natural" leap years, where the first day of two adajacent years are 366 days apart, will occur every four years, but sometimes with a five year gap instead.

If I want this calendar to be embraced by the software engineers of the world, the position of those leap years will need to be predictable with a mathematical function that operates only on the year number, making no reference to observed astronomy.

Given a year number y, how can I predict if that year is a leap year, as defined above?

For the puroses of this question, my year numbers are the same as standard years for the bulk of the year, with the exception of the period between the winter solstice and new years day where they will differ by one.

Also, for the purposes of this question, assume that the Earth's cycles will remain as they are now indefinitely.

Every year, there is day that's the first one after the longest night in the northern hemisphere. Most years, that day will be 365 days after the previous one.

Some years, that day will be 366 days later. I'll call them "natural leap years".

These "natural leap years" will mostly be four years apart, but will sometimes be five years apart instead.

How can I calculate, assuming the Earth's current cycles, when these "natural leap years" will occur?

-
If y is evenly divisible by 4, by 100 and 400, then you have a leap year. Modulo is your friend, in this case. –  Grant Thomas Dec 22 '11 at 13:47
@Mr.Disappointment - Those are standard leap years. I'm looking for a way to calculate natural leap years as I've outlined. –  billpg Dec 22 '11 at 13:52
These "natural" leap years would follow exactly the same pattern as the Gregorian leap years, except possibly with some offset to the pattern, assuming that you are assuming perfect regularity, in logical extension of the assumption quoted at the bottom of the question and the comment about software engineers. You can calculate that offset yourself from ephemerides. –  Andrew Dec 22 '11 at 14:00
hmmm,It's not clear to me what your asking. Are you looking mathematical functions that fit the Earth's orbit or for details about Earth's orbit? –  Nic Dec 22 '11 at 14:01
PS- As long as you're working on this, will you please make each month exactly 28 days, plus a "February" or holiday feast at the end of the year to take up all the slack? Kthxbye –  Andrew Dec 22 '11 at 14:05

The Gregorian calendar that we use today is a compromise between having leap years spaced as evenly as possible and keeping the calculations relatively simple. (The older Julian calendar, which had a leap year every 4 years, was also such a compromise, one with a greater bias toward simplicity.)

In the Gregorian calendar, every 4th year is a leap year, except that every 100th year is not a leap year, except that every 400th year is a leap year. The cycle repeats every 400 years, with 97 leap years in each cycle. But they're not distributed as evenly as they could be; most successive leap years are 4 years apart, but some of them are 8 years apart.

The Gregorian calendar assumes that a year is 365.2425 days. In fact, it's slightly less than that; WolframAlpha says it's 365.242190419 days.

What you're suggesting, I think, is a calendar in which leap years are always either 4 or 5 years apart, and are as evenly distributed as possible, so that the difference between the calendar and the astronomical year are minimized.

To do this, start with a 365-day year, and keep track of the offset between the calendar the astronomical year. This offset increases by 0.242190419 days every year.

• Year 0: offset = 0
• Year 1: offset = 0.242190419 days
• Year 2: offset = 0.484380838 days
• Year 3: offset = 0.726571257 days
• Year 4: offset = 0.968761676 days
• Year 5: offset = 1.210952095 days
• Add a leap day, recompute offset = 0.210952095 days
• ...

Every time the offset exceeds 1 day, add a leap day and subtract 1 day from the offset.

This will keep the offset as small as possible over time, but at the expense of easy predictability; it becomes much more difficult for those who are not mathematically inclined to understand when the next leap year is going to be. And the calendar would presumably have to be adjusted as the number of days in a solar year is computed more precisely, or as it changes as the Earth's rotation rate changes slightly.

Assuming that the figure of 365.242190419 is correct, your calendar would repeat itself, not every 400 years, but every billion years.

An alternative would be to keep the Gregorian calendar's assumption of 365.2425 days per year, and just distribute the leap years more evenly through the 400-year cycle. For example, starting from 2000, you have leap years in 2000, 2004, 2008, 2012, ..., 2128, 2133, 2137, 2141, ..., 2265, 2270, 2274, ....

In either case, most of the time leap years would not be years that are multiples of 4, which is a nice property of the Julian and Gregorian calendars.

That's the math, but it's absolutely trivial compared to the politics. I'm afraid there's very little chance that such a system would be widely accepted. The Gregorian calendar, which was a clear improvement over the Julian calendar, was introduced in 1582, but it wasn't fully adopted worldwide until the late 1920s (and there are still pockets of resistance).

There are tremendous advantages in the fact that almost the entire world uses the Gregorian calendar, and it will remain within a day or so of the astronomical year for the next several thousand years. Your proposed calendar has the advantage of remaining very slightly closer to the astronomical year, but the costs of universal adoption would be tremendous, and the costs of partial adoption would be even worse.

-
As long as they keep adding(/subtracting) leap seconds, you don't need that caveat about "... for the next several thousand years" in the last paragraph –  ThePopMachine Jan 3 '12 at 0:10
@ThePopMachine: I don't think so. Leap days and leap seconds solve different problems. –  Keith Thompson Jan 3 '12 at 3:57


text:
# what is Z[x] in the sense of a unique factorization domain

What does the notation Z[something] mean?

• is Z[x] the same as the set of all ax+b where a and b come from Z ? or is it more subtle than that? Mar 10 '20 at 10:25
• No, it is more complicated. Think of fields first, i.e., of $\Bbb Q(\zeta_n)$. Since this is a vector space of dimension $\phi(n)$ over $\Bbb Q$, we have a basis $(1,\zeta_n,\zeta_n^2,\cdots)$, which can have more than $2$ elements. Mar 10 '20 at 10:28

Given a ring extension $$S\supseteq R$$ and an element $$\alpha\in S$$ one denotes by $$R[\alpha]$$ the smallest subring of $$S$$ containg $$R$$ and $$\alpha$$.
In the above example, $$R=\Bbb Z$$ and $$\alpha=\zeta_n$$, a primitive $$n$$-th root of unity.
The ring $$\Bbb Z[\zeta_n]$$ is the ring of integers in the cyclotomic field $$\Bbb Q(\zeta_n)$$.


text:
- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
 A fundamental theorem of calculus for the $M_{\alpha}$-integral Commun. Korean Math. Soc. 2022 Vol. 37, No. 2, 415-421 https://doi.org/10.4134/CKMS.c210043Published online March 29, 2022Printed April 30, 2022 Abraham Perral Racca Adventist University of the Philippines Abstract : This paper presents a fundamental theorem of calculus, an integration by parts formula and a version of equiintegrability convergence theorem for the $M_{\alpha}$-integral using the $M_{\alpha}$-strong Lusin condition. In the convergence theorem, to be able to relax the condition of being point-wise convergent everywhere to point-wise convergent almost everywhere, the uniform $M_{\alpha}$-strong Lusin condition was imposed. Keywords : $M_{\alpha}$-integral, $M_{\alpha}$-$SL$, fundamental theorem of calculus, integration by parts Downloads: Full-text PDF   Full-text HTML

 Copyright © Korean Mathematical Society. (Rm.1109) The first building, 22, Teheran-ro 7-gil, Gangnam-gu, Seoul 06130, Korea Tel: 82-2-565-0361  | Fax: 82-2-565-0364  | E-mail: paper@kms.or.kr   | Powered by INFOrang Co., Ltd


text:
# Numerical differentiation in D using finite difference

Author: Dr Chibisi Chima-Okereke Created: October 3, 2020 00:46:07 GMT Published: October 3, 2020 00:46:07 GMT

## Introduction

The D programming language semantics lends itself to building a wide range of application types, one of these is technical and scientific computing. However, the number of packages available for numerical computing is low. This article demonstrates how to create a basic numerical differentiation scheme in D using finite differences to calculate first and second partial derivatives when given an expression. We use template functions and some basic template declarations in this article, so if you are unfamiliar with these concepts please visit a previous article that covers them.

## Preamble

In this session, we need to define a number of items before we start writing finite difference code. Our examples use the following mathematical functions:

import core.stdc.math: sin = sinf, sin, sin = sinl,
cos = cosf, cos, cos = cosl,
pow = powf, pow, pow = powl;

We use the C-library math functions in the D compiler since they often perform better than their Phobos D counterparts. Define $$\pi$$ as:

enum pi = 3.14159265358979323846;

In this scheme we will treat single and double precision (and above) separately. In fact, only the difference implementation for double precision and above will be presented. Below we have some basic traits to differentiate the various relevant floating points.

enum isFloat(T) = is(T == float);
enum isHighFloat(T) = is(T == double) || is(T == real);
enum isFloatingPoint(T) = is(T == float) || is(T == double) || is(T == real);

Finite difference uses steps $$h$$, which is different for single float compared with more precise float types. Some references use a value of the square root of machine epsilon for double float type and others use the cube root of the machine epsilon. Below is our step implementation of the step function for double and higher precision:

template step(T)
if(isHighFloat!(T))
{
enum T step = (T.epsilon)^^0.5;
}

For single precision, the step implementation below needs to be multiplied by the function parameter $$x$$ before it becomes the actual step:

template step(T)
if(is(T == float))
{
enum T step = T.epsilon^^(0.5f);
}

For more details see the wikipedia article on numerical differentiation.

## Finite difference for first order partial derivative

Many different books on Numerical Analysis discuss finite different schemes. The difference equations given here are adapted from chapter 8 of Numerical methods for engineers and scientists, 3rd Edition, by Amos Gilat, and Vish, Subramaniam.

### Two-point central finite difference for partial differentiation

The simple univariate equation to numerically evaluate the two-point central difference derivative for function $$f(x)$$at $$x = x^{(k)}$$ for the first derivative is given by:

$$\frac{\partial f(x)}{\partial x} \bigg|_{x = x^{(k)}} \approx \frac{f(x^{(k + 1)}) - f(x^{(k - 1)})}{2h}$$

where $$x^{(k + 1)} = x^{(k)} + h$$ and $$x^{(k - 1)} = x^{(k)} - h$$.

For the partial derivative of a single component of a multivariable $$\textbf{x}^{(k)} = \left(x_{1}^{(k)}, x_{2}^{(k)}, \ldots, x_{i}^{(k)}, \ldots, x_{n}^{(k)}\right)$$, where the subscript $$i$$ denotes the position in the array and the superscript $$k$$ denotes the value $$x = x^{(k)}$$, we then have:

$$\frac{\partial f(\textbf{x})}{\partial x_{i}} \bigg |_{\textbf{x} = \textbf{x}^{(k)}} \approx \frac{f(x_{1}^{(k)}, x_{2}^{(k)}, \ldots, x_{i}^{(k + 1)}, \ldots, x_{n}^{(k)}) - f(x_{1}^{(k)}, x_{2}^{(k)}, \ldots, x_{i}^{(k - 1)}, \ldots, x_{n}^{(k)})}{2h}$$

The D code for carrying out this numerical partial derivative is given below:

auto ND(alias fun, size_t order, T, U...)(T[] x, U u)
if(isHighFloat!(T) && (order == 1))
{
enum T h = step!(T); //evaluated at compile time
auto df = new T[x.length];
for(long i = 0; i < df.length; ++i)
{
auto xp = x.dup;
auto xm = x.dup;
xp[i] += h;
xm[i] -= h;
df[i] = (fun(xp, u) - fun(xm, u))/(2*h);
}
return df;
}

In the above code, we pass the function fun as a compile time item, and we constraint the order to be 1 for the first derivative, and T to be one of double (64 bit) or real (80 bit) floats.

### Four-point central finite difference for partial differentiation

Let’s look at another example, a more accurate scheme than the above two point system. For brevity lets change notation so that $$f(x_{i}^{(k)}) = f(x_{1}^{(k)}, x_{2}^{(k)}, \ldots, x_{i}^{(k)}, \ldots, x_{n}^{(k)})$$, the four-point central finite difference for partial differentiation, is given by:

$$\frac{\partial f(\textbf{x})}{\partial x_{i}} \bigg |_{\textbf{x} = \textbf{x}^{(k)}} \approx \frac{f(x_{i}^{(k - 2)}) - 8f(x_{i}^{(k - 1)}) + 8f(x_{i}^{(k + 1)}) - f(x_{i}^{(k + 2)})}{12h}$$

so an alternative implementation of the first order partial difference formula is:

auto ND(alias fun, size_t order, T, U...)(T[] x, U u)
if(isHighFloat!(T) && (order == 1))
{
enum T h = step!(T);
auto df = new T[x.length];
for(long i = 0; i < df.length; ++i)
{
// Written this way for clarity
auto x2m = x.dup;
auto x1m = x.dup;
auto x1p = x.dup;
auto x2p = x.dup;
x2p[i] += 2*h;
x2m[i] -= 2*h;
x1p[i] += h;
x1m[i] -= h;
df[i] = (fun(x2m, u) - 8*fun(x1m, u) + 8*fun(x1p, u) - fun(x2p, u))/(12*h);
}
return df;
}

## Finite difference for second order partial derivatives

We shall present one scheme for evaluating numerical second order partial derivatives. This scheme uses three points for the diagonal elements and four points for the off-diagonal elements. Also we shall use D’s nested arrays as the matrix output in this exercise (for brevity of implementation).

The difference equation for the diagonal ($$i$$) elements of the curvature (Hessian) matrix is given by:

$$\frac{\partial f^{2}(\textbf{x})}{\partial x_{i}^2} \bigg |_{\textbf{x} = \textbf{x}^{(k)}} \approx \frac{f(x_{i}^{(k - 1)}) - 2f(x_{i}^{(k)}) - f(x_{i}^{(k + 1)})}{h_i^2}$$

the off-diagonal ($$i, j$$) terms are given by:

$$\frac{\partial f^{2}(\textbf{x})}{\partial x_{i}\partial x_{j}} \bigg |_{\textbf{x} = \textbf{x}^{(k)}} \approx \frac{[f(x_{i}^{(k + 1)}, x_{j}^{(k + 1)}) - f(x_{i}^{(k - 1)}, x_{j}^{(k + 1)})] - [f(x_{i}^{(k + 1)}, x_{j}^{(k - 1)}) - f(x_{i}^{(k - 1)}, x_{j}^{(k - 1)})]}{2h_{i} 2h_{j}}$$

which is implemented below:

auto ND(alias fun, size_t order, T, U...)(T[] x, U u)
if(isHighFloat!(T) && (order == 2))
{
enum T h = step!(T)^^0.5;
enum T hsq = h*h;
auto curv = new T[][](x.length, x.length);
for(long i = 0; i < x.length; ++i)
{
for(long j = i; j < x.length; ++j)
{
if(i == j)
{
auto xp = x.dup; auto xm = x.dup;
xp[i] += h; xm[i] -= h;
auto f1m = fun(xm, u);
auto f1p = fun(xp, u);
auto f = fun(x, u);
auto tmp = (f1m - 2*f + f1p)/hsq;
curv[i][j] = tmp;
}else{
auto x1 = x.dup;
x1[i] += h;
x1[j] += h;
auto x2 = x.dup;
x2[i] -= h;
x2[j] += h;
auto x3 = x.dup;
x3[i] += h;
x3[j] -= h;
auto x4 = x.dup;
x4[i] -= h;
x4[j] -= h;
auto f1 = fun(x1, u);
auto f2 = fun(x2, u);
auto f3 = fun(x3, u);
auto f4 = fun(x4, u);
auto tmp = ((f1 - f2) - (f3 - f4))/(4*hsq);
curv[i][j] = tmp;
curv[j][i] = tmp;
}
}
}
return curv;
}

## Demonstrations

### Example 1

In this first example, we look at the first and second partial differentials for the equation:

$$f(\textbf{x}) = 50 - x_{0}^{2} - 2x_{1}^{2}$$

the first order partial derivative is given by the equation:

$$\frac{\partial f(x)}{\partial x_0} = -2x_{0}; \qquad \frac{\partial f(x)}{\partial x_1} = -4x_{1}$$

the second order partial derivative is given by the equation:

$$\frac{\partial^{2} f(x)}{\partial x_0^2} = -2; \qquad \frac{\partial^{2} f(x)}{\partial x_{0} \partial x_{1}} = 0$$

$$\frac{\partial^{2} f(x)}{\partial x_{1}\partial x_{0}} = 0; \qquad \frac{\partial^{2} f(x)}{\partial x^{2}_{1}} = -4$$

### Code for example 1

The code for the above example is given below:

double[] point = [10.0, 10];
auto f = (double[] x) => 50 - x[0]*x[0] - 2*x[1]*x[1]; //this is lambda in D
auto df = ND!(f, 1)(point);
writeln("First (partial) derivative: ", df);
auto cf = ND!(f, 2)(point);
writeln("Second (partial) derivative: ", cf);

output:

First (partial) derivative: [-20, -40]
Second (partial) derivative: [[-2, 0], [0, -4]]

From the above equations, clearly, the first order derivatives at [10.0, 10.0] is given by:

$$\frac{\partial f(\textbf{x})}{\partial x_0} = -20; \qquad \frac{\partial f(\textbf{x})}{\partial x_1} = -40$$

clearly the output of second derivative compares well with the analytical values.

### Example 2

The equation to be differentiated in this second example is given below:

$$f(\textbf{x}) = x_{0}\sin{x_{1}} + 1$$

meaning that the first order partial differential is given by:

$$\frac{\partial f(\textbf{x})}{\partial x_{0}} = \sin{x_{1}}; \qquad \frac{\partial f(\textbf{x})}{\partial x_{1}} = x_{0}\cos{x_{1}}$$

and the curvature matrix is given by:

$$\frac{\partial^{2} f(\textbf{x})}{\partial x_0^2} = 0; \qquad \frac{\partial^{2} f(\textbf{x})}{\partial x_{0} \partial x_{1}} = \cos{x_{1}}$$

$$\frac{\partial^{2} f(\textbf{x})}{\partial x_{1}\partial x_{0}} = \cos{x_{1}}; \qquad \frac{\partial^{2} f(\textbf{x})}{\partial x^{2}_{1}} = -x_{0}\sin{x_{1}}$$

### Code for example 2

The code for example 2 is given below:

point = [pi/2, pi/2];
f = (double[] x) => x[0]*sin(x[1]) + 1;
df = ND!(f, 1)(point);
cf = ND!(f, 2)(point);
writeln("First (partial) derivative: ", df);
writeln("Second (partial) derivative: ", cf);

which gives the output:

First (partial) derivative: [1, -4.96705e-09]
Second (partial) derivative: [[0, 0], [0, -1.5708]]

From the above equations, clearly, the first order derivatives at $$[\frac{\pi}{2}, \frac{\pi}{2}]$$ is given by:

$$\frac{\partial f(\textbf{x})}{\partial x_0} = 1; \qquad \frac{\partial f(\textbf{x})}{\partial x_1} = 0$$

and the curvature matrix is given by:

$$\frac{\partial^{2} f(\textbf{x})}{\partial x_0^2} = 0; \qquad \frac{\partial^{2} f(\textbf{x})}{\partial x_{0} \partial x_{1}} = 0$$

$$\frac{\partial^{2} f(\textbf{x})}{\partial x_{1}\partial x_{0}} = 0; \qquad \frac{\partial^{2} f(\textbf{x})}{\partial x^{2}_{1}} = -\frac{\pi}{2}$$

which is (approximately) what we have in the numerical case.

### Example 3

The equation to be differentiated in this second example is given below:

$$f(x) = cos(2\pi x_{0})\cos(2\pi x_{1})\cos(2 \pi x_{2})\cos(10(x_{0} + x_{1} + x_{2}))$$

The analytical equations of the gradient and curvature are too long to recite here (this is not a thesis, just a tutorial), but their code and outputs will be presented along with those of finite difference.

### Code for example 3

The code to calculate the finite difference for example 3 is given below:

point = [0.2, 0.5, 0.1];
f = (double[] x) => cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2]));
df = ND!(f, 1)(point);
writeln("First (partial) derivative: ", df);
cf = ND!(f, 2)(point);
writeln("Second (partial) derivative: ", cf);

which gives the output:

First (partial) derivative: [1.76999, 2.4734, 2.30734]
Second (partial) derivative: [[-100.733, -51.4672, -59.5472],
[-51.4672, -5.07351, -14.9286],
[-59.5472, -14.9286, -27.6556]]

The code to calculate the analytical first order partial derivative:

auto df1 = (double[] x) => -10*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2])) - 2*pi*sin(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2]));
auto df2 = (double[] x) => -10*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2])) - 2*pi*sin(2*pi*x[1])*cos(2*pi*x[0])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2]));
auto df3 = (double[] x) => -10*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2])) - 2*pi*sin(2*pi*x[2])*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(10*(x[0] + x[1] + x[2]));
writeln("First partial derivative: ", [df1(point), df2(point), df3(point)]);

which gives the output:

First partial derivative: [1.76999, 2.4734, 2.30734]

The code for the second order partial derivative is given below:

auto cf11 = (double[] x) => -4*(pi^^2)*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) - 100*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 40*pi*sin(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2]));
auto cf21 = (double[] x) => -100*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 4*(pi^^2)*sin(2*pi*x[0])*sin(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 20*pi*sin(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2])) + 20*pi*cos(2*pi*x[0])*sin(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2]));
auto cf31 = (double[] x) => -100*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 4*(pi^^2)*sin(2*pi*x[0])*cos(2*pi*x[1])*sin(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 20*pi*sin(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2])) + 20*pi*cos(2*pi*x[0])*cos(2*pi*x[1])*sin(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2]));
auto cf12 = (double[] x) => -100*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 4*(pi^^2)*sin(2*pi*x[0])*sin(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 20*pi*sin(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2])) + 20*pi*cos(2*pi*x[0])*sin(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2]));
auto cf22 = (double[] x) => -4*(pi^^2)*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) - 100*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 40*pi*cos(2*pi*x[0])*sin(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2]));
auto cf32 = (double[] x) => -100*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 4*(pi^^2)*cos(2*pi*x[0])*sin(2*pi*x[1])*sin(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 20*pi*cos(2*pi*x[0])*sin(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2])) + 20*pi*cos(2*pi*x[0])*cos(2*pi*x[1])*sin(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2]));
auto cf13 = (double[] x) => -100*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 4*(pi^^2)*sin(2*pi*x[0])*cos(2*pi*x[1])*sin(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 20*pi*sin(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2])) + 20*pi*cos(2*pi*x[0])*cos(2*pi*x[1])*sin(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2]));
auto cf23 = (double[] x) => -100*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 4*(pi^^2)*cos(2*pi*x[0])*sin(2*pi*x[1])*sin(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 20*pi*cos(2*pi*x[0])*sin(2*pi*x[1])*cos(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2])) + 20*pi*cos(2*pi*x[0])*cos(2*pi*x[1])*sin(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2]));
auto cf33 = (double[] x) => -4*(pi^^2)*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) - 100*cos(2*pi*x[0])*cos(2*pi*x[1])*cos(2*pi*x[2])*cos(10*(x[0] + x[1] + x[2])) + 40*pi*cos(2*pi*x[0])*cos(2*pi*x[1])*sin(2*pi*x[2])*sin(10*(x[0] + x[1] + x[2]));
writeln("Hessian: ", [[cf11(point), cf21(point), cf31(point)],
[cf12(point), cf22(point), cf32(point)],
[cf13(point), cf23(point), cf33(point)]]);

which gives the output:

Hessian: [[-100.733, -51.4672, -59.5472],
[-51.4672, -5.07353, -14.9286],
[-59.5472, -14.9286, -27.6556]]

so the numerical and analytical gradient vectors and curvature matrices agree with each other.

## Conclusion

I hope that the contents of this article has persuaded the reader that writing numerical code in D is a reasonably easy and straightforward process. D’s template and meta-programming system lends itself easily to building such systems. Performance optimization considerations have been left out, and they will perhaps be visited some other time. I have had fun writing the code for this article and I hope to revisit this and other topics in numerical analysis in the future.