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zbMATH — the first resource for mathematics

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On some von Neumann topological algebras. (English) Zbl 1200.46038

The authors consider unital algebras $A$ with the following property: for each $x$, there exists $y$ with $x=xyx$ $\left(x,y\in A\right)$. Their main result states that such a ${B}_{0}$-algebra (completely metrizable locally convex algebra) with an open group of invertible elements is finite-dimensional. Using this result, the authors show that a locally ${C}^{*}$-algebra with the above property is an inverse limit of finite-dimensional algebras. Another result states that such an $F$-algebra (completely metrizable algebra) is a finite product of division algebras of type $F$.

Reviewer’s remark. It remains open whether such a division algebra must be finite-dimensional, i.e., equal to $ℝ,ℂ$ or $ℍ$.

MSC:
 46H20 Structure and classification of topological algebras 46L05 General theory of ${C}^{*}$-algebras


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# Examples on Conditional Trigonometric Identities Set 2

Go back to  'Trigonometry'

Example - 29

Prove that  $$\Delta ABC$$  is equilateral if and only if

$\tan A + \tan B + \tan C = 3\sqrt 3$

Solution: This is a two-way implication of which one is very easy to prove. Namely, if $$\Delta ABC$$ is equilateral, then

$\tan A + \tan B + \tan C = \sqrt 3 + \sqrt 3 + \sqrt 3 = 3\sqrt 3$

The main part of this problem is proving the converse.

From the previous example, we have

$\tan A + \tan B + \tan C = \tan A\tan B\tan C$

On the other hand, by the  $${\text{A}}{\text{.M}}\;{\text{–}}\;{\text{G}}{\text{.M}}$$  inequality,

$\tan A + \tan B + \tan C \geqslant {(\tan A\tan B\tan C)^{1/3}}$

The A.M. of $$\tan A,\;\tan B,\;\tan C\;{\text{is}}\;\sqrt 3$$, whereas the G.M $${(3\sqrt 3 )^{1/3}} = \sqrt 3$$. is, i.e., the same as the A.M. This can happen only if

$\tan A = \tan B = \tan C\;\;\;\;\;\;\; \Rightarrow A = B = C$

Example - 30

In a  $$\Delta ABC$$  if  $$\sin 3A + \sin 3B + \sin 3C = 0$$, then show that at least one of the angles is  \begin{align}\frac{\pi }{3}\end{align}.

Solution: Following the approach in previous examples, we can prove that

$\sin 3A + \sin 3B + \sin 3C = - 4\cos \frac{{3A}}{2}\cos \frac{{3B}}{2}\cos \frac{{3C}}{2}$

For this to be zero, one of the cos terms on the RHS must be zero, which means that one of the angles must be  \begin{align}\frac{\pi }{3}\end{align}.

Example - 31

In a   $$\Delta ABC$$, if

$\cos A\cos B\cos C = \frac{{\sqrt 3 - 1}}{8},\;\sin A\sin B\sin C = \frac{{3 + \sqrt 3 }}{8}$

Find the angles of the triangle

Solution: From the given relations,

$\tan A\tan B\tan C = \frac{{3 + \sqrt 3 }}{{\sqrt 3 - 1}}$

Using the result of Example - 28, we have

$\tan A + \tan B + \tan C = \frac{{3 + \sqrt 3 }}{{\sqrt 3 - 1}}$

Now, since \begin{align}A + B + C = \pi \end{align}

\begin{align}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cos (A + B + C) = - 1 \\ \Rightarrow\quad &\qquad\;\;\cos A\cos B\cos C\left\{ {1 - \sum (\tan A\tan B)} \right\} = - 1 \\ \Rightarrow\quad &\qquad\;\; \sum \left( {\tan A\tan B} \right) = 1 + \frac{8}{{\sqrt 3 - 1}} = 5 + 4\sqrt 3 \\ \end{align}

Finally, for  $$\tan A,\tan B,\tan C$$, we have their sum, their sum taken pairwise and their product, and thus we can say that these three are the roots of the equation

\begin{align}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x^3} - \left( {\frac{{3 + \sqrt 3 }}{{\sqrt 3 - 1}}} \right){x^2} + (5 + 4\sqrt 3 )x - \left( {\frac{{3 + \sqrt 3 }}{{\sqrt 3 - 1}}} \right) = 0\\ \Rightarrow\quad &\qquad\;\;(x - 1)\left( {x - \sqrt 3 } \right)\left( {x - (2 + \sqrt 3 )} \right) = 0 \\ \Rightarrow \quad &\qquad x = 1,\;\sqrt 3 ,\;2 + \sqrt 3 \\ \end{align}

The angles are \begin{align}\frac{\pi }{4},\;\frac{\pi }{3},\;\frac{\pi }{{12}}\end{align}

Example - 32

In a  $$\Delta ABC$$, suppose that  \begin{align}\tan \frac{A}{2},\;\tan \frac{B}{2},\;\tan \frac{C}{2}\end{align}  are in H.P.. What is the minimum possible value of \begin{align}\cot \frac{B}{2}\end{align}?

Solution: Refer to the result of Example - 28 Part (b). If we divide throughout by \begin{align}\tan \frac{A}{2}\tan \frac{B}{2}\tan \frac{C}{2}\end{align},

we have,

$\cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2} = \cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \ldots \left( 1 \right)$

Now, since \begin{align}\tan \frac{A}{2},\tan \frac{B}{2},\tan \frac{C}{2}\end{align}  are in H.P., their reciprocals, i.e. \begin{align}\cot \frac{A}{2},\;\cot \frac{B}{2},\;\cot \frac{C}{2}\end{align}, are in A.P. So,

$\qquad\cot \frac{A}{2} + \;\cot \frac{C}{2} = 2\cot \frac{B}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \ldots \left( 2 \right)$

Using (2) in (1), we obtain

$\cot \frac{A}{2}\;\cot \frac{C}{2} = 3$

Finally, we use the A.M-G.M. inequality on \begin{align}\cot \frac{A}{2},\;\cot \frac{C}{2}\end{align}:

\begin{align}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{\cot \frac{A}{2} + \cot \frac{C}{2}}}{2} = \cot \frac{B}{2} \geqslant \sqrt {\cot \frac{A}{2}\cot \frac{C}{2}} = \sqrt 3 \\ \Rightarrow\quad &\qquad\;\;\;\cot \frac{B}{2} \geqslant \sqrt 3 \\ \end{align}

Therefore, the minimum value of  \begin{align}\cot \frac{B}{2}\end{align}  is  $$\sqrt 3$$. From this, we can further infer that the maximum value of  \begin{align}\tan \frac{B}{2}\end{align} is \begin{align}\frac{1}{{\sqrt 3 }}\end{align}  which implies that  \begin{align}\frac{B}{2} \leqslant \frac{\pi }{6} \Rightarrow B \leqslant \frac{\pi }{3}\end{align}

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text:
This document describes the goals and the approaches for automating the management of the infrastructure used by Apertis. It will focus in particular on release branching since the new release flow implies that Apertis will need to go through that process two or three times more than in the past on each quarter.

# Goals

## Data-driven

Separating the description of the desired infrastructure state from the tools to apply it nicely separates the two concerns: in most cases the tools won’t need to be updated during branching, only the desired infrastructure state changes.

## Git-controlled

Basing everything on configuration stored in a Git repository has several advantages:

• all the changes are tracked over time
• the standard Apertis workflows based on GitLab merge requests can be used to review changes
• fine access controls can be configured via GitLab

## Idempotent

Every tool should compare the current state with the desired one and not produce errors when they already match. Administrators should be able to run the tools at any time, multiple times, without any ill effect.

## Scalable

The Apertis infrastructure is composed of enough services that a centralized list of things to update when branching is doomed to be outdated every quarter.

## Single source of truth

The duplication of the same information between modules should be minimized, such that updating the single source of truth automatically produces effects on the depending modules.

## Reproducible

Running the tools in a standardized, easily reproducible environment enables all the administrators to easily deploy changes without any special setup.

## Explicit

All the needed information should be explicitly encoded in metadata repository. The tools using it should strive to not make any assumption on the data and derive more information out of it. This is another facet of ensuring that the metadata repository remains the single source of truth.

# Basic approach

The basic approach aims at improving the current branching scripts to make them easier to test by developers, enabling more people to work on them, and to extend them to fully handle the complete branching process.

## Add test mode for current branching scripts

At the moment the quarterly release branching is done through a set of scripts that get invoked manually by one the Apertis infrastructure team member from their machine.

They act directly on the live services using the caller’s accounts.

The first step for improving the branching automation is to add a “dry-run” mode to all the current release scripts to let developers and admin run them

## Improve coverage of current branching scripts

The scripts currently in charge of reducing the manual intervention during the branching process do not cover all services and repositories which are part of Apertis.

Once the “dry-run” mode is in place, new steps need to be added to the branching scripts to cover the missing services and repositories.

# Longer term approach

Larger refactorings are needed to align the current infrastructure to the goals previously described.

The following sections describe the steps needed to further improve the infrastructure maintenance to make it more robust and require less effort to manage.

A new git repository contains the principal metadata about the whole Apertis infrastructure describing:

• the currently active release branches
• e.g. v2020pre, v2019, etc.
• their phase in the release lifecyle
• e.g. development, preview, stable
• their release status
• e.g. frozen, release-candidate, released
• the release from which they get branched from:
• e.g. 2019pre for both v2019 and v2020dev0
• the matching git branch name
• e.g. apertis/v2019
• the APT components they ship
• e.g. target, development, sdk, hmi
• etc.

This provides a git-controlled single source of truth: tools are updated to fetch the information they need from this repository.

For instance, the creation of OBS projects should be handled by a tool that:

• fetches the above YAML
• checks the current OBS configuration
• computes the changes needed compared to the desired state, if any
• applies the changes, if any, to align the actual state to the desired state, providing an idempotent solution
• runs from a GitLab pipeline, providing a reproducible environment that can be either triggered by changes in the main data repository or manually

The current infrastructure encodes a lot of information about the releases in several places: tools should be changed to fetch such information on the fly from the main data repository or GitLab pipelines should be configured to monitor the main data repository and automatically apply changes accordingly.

For instance, the LAVA job templates encode the branch name of the release they track in multiple places. Either the templates can be enhanced to fetch the information on the fly from the main data repository, or a pipeline should be configured in a dedicated branch in the repository to monitor the main data repository and branch/update the repository accordingly.

The change compared to the current approach is to minimize the amount of information that needs to be branched and distribute the branching logic closer to the entity to be branched. This is meant to avoid the recurring issues where the current centralized branching scripts failed to branch things properly or did not include new components to be branched at all.

## Per-repository branching operations

For most repositories it is sufficient to add a new git ref when branching for a new release. In particular, nearly all the the packaging ones do not need any change to the repository contents and creating a new ref is enough.

Other repositories need instead some changes to be applied to the contents once a new release branch is created. A common reason is that the release name is encoded in some file, which means that the file needs to be updated and the change needs to be committed and pushed.

By making branching self-contained in the repositories, moving and renaming them no longer cause breakage. It also gives full control over the branching of a repository to the people managing that repository, rather than those who manage the centralized repository. This can be especially useful for components not managed by the core Apertis team, owned instead by product-specific teams.

In general, keeping the branching operation in the same place as the rest of the contents helps in keeping them coeherent and makes testing easier.

# Implementation

## Add test mode for current branching scripts

Setting the NOACT=y environment variable causes the branching scripts to run in test mode, without actually launching the branching commands.

## Improve coverage of current branching scripts

New actions need to be taken when branching a new release.

This is a non exhaustive list:

• branch LAVA job templates;
• update the configuration on GitLab repositories to create the new release branch, make it the default, etc.;
• create the relevant :snapshots repositories on OBS;
• add support for the security, updates and backports repositories when branching stable releases.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48   .common_components: &common_components - target - development - sdk - hmi projects: apertis: releases: v2019: lifecycle: stable status: released branched-from: v2019pre branch-name: apertis/v2019 upstream: debian/buster obs-build-suffix: v2019.0 suites: v2019: obs-pattern: '$project:$release:$component' components: *common_components v2019-updates: obs-pattern: '$project:$release:updates:$component' components: *common_components v2019-security: obs-pattern: '$project:$release:security:$component' components: *common_components infrastructure-packages: obs: apertis:infrastructure:v2019 suite: infrastructure-v2019 components: - buster v2020dev0: lifecycle: development status: frozen branched-from: v2019pre branch-name: apertis/v2020dev0 upstream: debian/buster obs-build-suffix: v2020dev0 suites: v2020dev0: obs-pattern: '$project:$release:$component' components: *common_components infrastructure-packages: obs: apertis:infrastructure:v2019 suite: infrastructure-v2019 components: - buster
A release-branching step should be added to the GitLab CI pipeline YAML in the repository with the purpose of ensuring that the release-specific contents match the branch name.
GitLab does not provide any way to execute an action only when a new ref is created so the best strategy is to ensure that the release-branching script is idempotent and gets run when changes land to any apertis/* refs: if no changes are detected the step succeeds with no further operations, otherwise it commit and push the changes automatically, or it submits a MR to be reviewed before landing the changes.


text:
## Elementary Algebra

RECALL: $|a| = k \longrightarrow a = k \text{ or } a = -k$ In the given equation, $k=0$. Thus, using the rule above gives: $|x+5| = 0 \longrightarrow x+5 = 0$ Solve the equation to obtain: $x+5=0 \\x=-5$ Thus, the solution set of the given equation is $\left\{-5\right\}$. Therefore, the given statement is false.


text:
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IEEE Transactions on Electron Devices

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• IEEE Transactions on Electron Devices publication information

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• Miracle Workers—A Round of Much Deserved Applause for the T-ED and EDL Staff

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• Design, Simulation, and Fabrication of Metal–Oxide–Semiconductor Field-Effect Transistor (MOSFET) With New Termination Structure

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In this paper, we introduce a new type of termination structure utilizing semi-insulating polycrystalline silicon (SIPOS) structures in conjunction with P- junction extension in order to reduce the area of termination device structure and increase the breakdown voltage. In SIPOS structures, one high-resistance layer is deposited between electrodes on two terminals such that the v... View full abstract»

• A Closed-Form Analytical Transient Response Model for On-Chip Distortionless Interconnect

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This paper presents a closed-form analytical transient response model for an on-chip distortionless interconnect considering resistance/capacitance loads via solving a semi-infinite transmission line equation. As verified by the simulation results, this transient response model has high accuracy, which could be used to derive the characteristics of the transmitted signal for facilitating the desig... View full abstract»

• Thermoelectric Performance of $hbox{Si}_{0.8} hbox{Ge}_{0.2}$ Nanowire Arrays

Publication Year: 2012, Page(s):3193 - 3198
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The output power of a thin-film thermoelectric generator consisting of a Cu-20- μm nanowire (NW) array (NWA)-Si bulk-Cu sandwich with Si or Si0.8Ge0.2 NWs is measured and compared to Cu-Si bulk-Cu for small temperature differences around room temperature. The array of NWs is made by metal-assisted chemical etching that retains the Ge concentration in the wires. The con... View full abstract»

• Impact of Quantum Confinement on Stress-Induced nMOSFET Threshold Voltage Shift

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In this paper, we propose a comprehensive model to express nMOSFET threshold voltage shift induced by stress, ranging from a high tensile one to a high compressive one. Using this model, the quantum confinement effect, combined with large out-of-plane stress, is shown to play an important role to cause the threshold voltage shift as large as about 80 mV induced by high-film-stress contact etch-sto... View full abstract»

• Impact of Quantum Confinement on Gate Threshold Voltage and Subthreshold Swings in Double-Gate Tunnel FETs

Publication Year: 2012, Page(s):3205 - 3211
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We investigate how the inclusion of quantum confinement in double-gate tunneling field-effect transistors (DG-TFETs) modifies the conventional behavior of electrical parameters of utmost importance in these devices, such as subthreshold swings (point and average) and the gate threshold voltage. We make use of a simple approach that allows us to incorporate a quantum-mechanical description in which... View full abstract»

• High-Throughput Screening of Amorphous $hbox{Y}_{2} hbox{O}_{3}$–$hbox{TiO}_{2}hbox{/}hbox{SiO}_{2}$ Higher $kappa$ Gate Dielectric Layers

Publication Year: 2012, Page(s):3212 - 3216
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In this paper, an approach using native SiO2 to make amorphous higher dielectric constant films based on the Y2O3-TiO2/SiO2/Si compositional spread libraries by combinatorial pulsed laser deposition is reported. The key feature of the experiment is that combinatorial methodology is used to quickly screen the potential high-dielectric-constant ... View full abstract»

• An Analytical Charge Model for Double-Gate Tunnel FETs

Publication Year: 2012, Page(s):3217 - 3223
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An analytical charge model for double gate (DG) tunnel FETs (TFETs) is proposed. By splitting the TFET into a series combination of a gated tunnel diode and a DG MOSFET, we solved the Poisson equation with matching boundary conditions to obtain a surface potential model for the DG TFET. Based on that, the source depletion charge and the mobile channel charge are derived. Comparisons between the pr... View full abstract»

• Single Germanium Quantum-dot Placement Along With Self-Aligned Electrodes for Effective Management of Single Charge Tunneling

Publication Year: 2012, Page(s):3224 - 3230
Cited by:  Papers (3)
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We demonstrated the controlled placement of a Ge quantum dot (QD) along with tunnel-junction engineering in a self-organized approach for the effective management of single charge tunneling. In this approach, a single-Ge-QD ( ~ 11 nm) self-aligning with nickel-polycide electrodes is realized by thermally oxidizing a SiGe nanorod that bridges a 15-nm-wide nanotrench in close proximity to electrodes... View full abstract»

• Influence of Edge Defects, Vacancies, and Potential Fluctuations on Transport Properties of Extremely Scaled Graphene Nanoribbons

Publication Year: 2012, Page(s):3231 - 3238
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Atomistic quantum transport simulations of a large ensemble of devices are employed to investigate the impact of different sources of disorder on the transport properties of extremely scaled (length of 10 nm and width of 1-4 nm) graphene nanoribbons. We report the dependence of the transport gap, on- and off-state conductances, and on-off ratio on edge-defect density, vacancy density, and potentia... View full abstract»

• Experimental Study of Self-Heating Effects in Trigate Nanowire MOSFETs Considering Device Geometry

Publication Year: 2012, Page(s):3239 - 3242
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Temperature rise by self-heating effects in nanowire (NW) transistors (NW Trs.) is systematically studied with respect to their dependence on the structural parameters. Temperature rise in NW Tr. is found to be independent of the NW size in sub-100-nm regions when compared at the same total power consumption. This is because the heat generated by the drain current is spread to the area larger than... View full abstract»

• Balancing SET/RESET Pulse for $>hbox{10}^{10}$ Endurance in $hbox{HfO}_{2}hbox{/Hf}$ 1T1R Bipolar RRAM

Publication Year: 2012, Page(s):3243 - 3249
Cited by:  Papers (44)
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By tuning the SET/RESET pulse amplitude conditions, the pulse endurance of our 40-nm HfO2/Hf 1T1R resistive-random-access-memory devices demonstrates varying failure behaviors after 106 cycles. For unbalanced SET/RESET pulse amplitude conditions, both low-resistance state (LRS) and high-resistance state (HRS) failures may occur, while varying the pulsewidths influences the LR... View full abstract»

• Monolayer $hbox{MoS}_{2}$ Transistors Beyond the Technology Road Map

Publication Year: 2012, Page(s):3250 - 3254
Cited by:  Papers (77)
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The performance of a 5-nm gate length monolayer MoS2 transistor is benchmarked against an ultrathin body Si transistor of similar dimensions and the ITRS requirements for 2026 low operating power (LOP) technology. The MoS2 transistor has a subthreshold slope of 70 mV/dec, an on -/off-current ratio of 4.8 × 104, a drive current of 238 μA/μm, a p... View full abstract»

• Numerical Simulations of Automatic Change of Threshold Voltage Shift in SRAM With Double-Floating-Gate Structures

Publication Year: 2012, Page(s):3255 - 3262
| | PDF (1101 KB) | HTML

We theoretically investigate the self-adjustment mechanism of the threshold voltage shift of the static random access memory (SRAM) based on the double-floating-gate (DFG) structure by considering the capacitive coupling between the neighboring DFGs. We numerically show that the threshold voltage shift is enhanced by the interference between DFGs through the capacitive coupling. The static noise m... View full abstract»

• An Investigation of Linearity Performance and Intermodulation Distortion of GME CGT MOSFET for RFIC Design

Publication Year: 2012, Page(s):3263 - 3268
Cited by:  Papers (13)
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In this paper, an extensive study on the intermodulation distortion and the linearity of gate-material-engineered cylindrical-gate MOSFET (GME CGT MOSFET) has been done, and the influence of technology variations such as channel length and gate material workfunction variations is explored using an ATLAS 3-D device simulator. The simulation results reveal that the GME CGT MOSFET design displays a s... View full abstract»

• Microstructures and Properties of SnZn Lead-Free Solder Joints Bearing La for Electronic Packaging

Publication Year: 2012, Page(s):3269 - 3272
Cited by:  Papers (12)
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In this paper, the effect of rare earth (RE) La on the properties of Sn9Zn solder was carried out. The results indicate that RE La plays an important role not only in the wettability and the structure of the solder but also in the mechanical property of the solder joint. The results show that adding trace amount of RE La can remarkably improve the wettability and oxidation resistance of SnZn solde... View full abstract»

• N-Channel Dual-Workfunction-Gate MOSFET for Analog Circuit Applications

Publication Year: 2012, Page(s):3273 - 3279
Cited by:  Papers (9)
| | PDF (1069 KB) | HTML

Analog behaviors of n-channel metal-oxide-semiconductor field-effect transistors (MOSFETs) with dual-workfunction-gate (DWFG) structure are presented. The gate of the n-channel DWFG MOSFET is composed of p+ and n+ poly-Si along the channel carrier flowing direction. To investigate the impact of the proportional length of p- and n-type-doped poly-Si on analog behaviors, they a... View full abstract»

• Spatial Composition Grading of Binary Metal Alloy Gate Electrode for Short-Channel SOI/SON MOSFET Application

Publication Year: 2012, Page(s):3280 - 3287
Cited by:  Papers (21)
| | PDF (672 KB) | HTML

An overall performance comparison analysis based on 2-D Poisson's equation solution has been presented here both for silicon-on-insulator (SOI) and silicon-on-nothing (SON) MOSFET structures. In this paper, for the first time, an idea of work function engineering with continuous horizontal mole fraction variation in a binary alloy gate has been proposed and implemented analytically to reduce rollo... View full abstract»

• Flexible Logic Gates Composed of Si-Nanowire-Based Memristive Switches

Publication Year: 2012, Page(s):3288 - 3291
Cited by:  Papers (8)
| | PDF (554 KB) | HTML

The flexible logic circuit configured using Si-based memristive switches is demonstrated. The memristive switches consisting of Ag/a-Si/heavily doped p-type Si are constructed on plastic using top-down-fabricated Si nanowires. The logic gate analyses provide insight toward logic circuits having multifunctionality and high connectivity through a crossbar-array architecture. With the strong s... View full abstract»

• Surface-Potential-Based Drain Current Model for Long-Channel Junctionless Double-Gate MOSFETs

Publication Year: 2012, Page(s):3292 - 3298
Cited by:  Papers (33)
| | PDF (1292 KB) | HTML

A surface-potential-based model is developed for the symmetric long-channel junctionless double-gate MOSFET. The relationships between surface potential and gate voltage are derived from some effective approximations to Poisson's equation for deep depletion, partial depletion, and accumulation conditions. Then, the Pao-Sah integral is carried out to obtain the drain current. It is shown that the m... View full abstract»

Aims & Scope

IEEE Transactions on Electron Devices publishes original and significant contributions relating to the theory, modeling, design, performance and reliability of electron and ion integrated circuit devices and interconnects.

Full Aims & Scope

Meet Our Editors

Editor-in-Chief

Giovanni Ghione
Politecnico di Torino,
10129 Torino, Italy


text:
How to confront phone scammers

Today I received another one of those SSN admin fake phone calls. I did not do a good job scaring them away because as usual my ad-hoc performance sucks. So I decided to write a blurb for future reference:

1. Are you aware that this is illegal?
2. which country are you calling from?
3. Do you have family? Do you want to be recognized by your family as a scam artist? A criminal? A parasite of society?
4. How many people have you scammed in the past month? Do you understand that your victims are the most vulnerable sector of society, and that they can be your parents, your grandparents, and your children as well?
5. I understand that your line of business is very lucrative. Even if you only snatch up 1% of victims, you can walk away with a lot of money at the end of the day. Use that money wisely.

Flu Tips

It was never fun to have a flu, especially after duly getting the flu shot at the earliest opportunity. a week after my China trip, however, I finally fell for it on the 24th of January this year, and it’s not too surprising: I was sitting next to a coughly colleague for 12 hours on the plane. He even reported having fevers for several days prior to that, and had sickness lasting after we returned to the US together.

Anyway a few tips I learned along the way of recovery that should help with future speedy recovery:
1. do not worry about electricity bills from heating, it’s well worth it especially at night in the bay area. Several times I woke up with a sore throat and frozen limbs, and wished I had turned the heater on all night. A cheaper alternative is to wear down jacket and ski suites to bed, but it’s simply easier to crank up the heat.

2. Stay hydrated, especially before bed.

3. Don’t stay up late and leave plenty of time before bed for entertainment, like 2 hours for a regular movie.

4. Try ibuprofen to keep temperature in check, and not wait til the fever has taken a toll on the judgment.

Fortunately I was paying attention to number 1-3 in the second half of my recovery phase, so I didn’t have to suffer too much from the lingering cough when the fever subsided. But I wish I had remember the importance of staying warm from the very beginning to avoid the fever altogether. Better revisit the notes next time.

When faced with dire uncertainty, Lord is the best/only friend

I am deciding whether to leave a comfortable 6 & 1/2 figure salary to pursue a startup. Some of my most capable friends seem unsure. I don’t have a stellar track record in terms of judgment: think gold hype, bitcoin bubble, housing peak. But invariably every time I let greed take over reason, and most importantly perhaps, I never consulted with our Lord in those situations. On a micro scale, I suspect if I prayed to the Lord every time I consume a meal, my GI woes would also be mitigated at least. There is no denying that the biological me is an extremely irresponsible, reptilian creature with no sense of long term purpose. The pursuit of math, for example, is a mere satisfaction of spontaneous curiosity and intellectual hubris. The vexation over some tricky coding bug,  similarly results from the bedeviling by hubris and control-freakdom. If my brain is truly wired like a neural net, the lack of divine guidance can be viewed as a supervised learning exercise with the objective given by whatever the environment happens to offer: think food, hubris arousal, money temptation, anger, etc. While the true purpose of life maybe cleverly hidden from us mortals, its vicissitudes of embodiment are readily accessible at strategic checkpoints of life.  It is our mortals to pay attention and discover such divine offerings and realize our fuller potentials.

How to deal with GI woes

It has become a habit of late that I would suffer from GI issues (either diarrhea or indigestion) on a weekly basis. This is clearly no fun as the stomach can take up resource from the brain. Further, it limits my stamina up side because fasting can solve the problem temporarily but with the dire consequence of lost productivity.

My wife as usual complains about my lethargy in looking after the obnoxious kids. My excuse of stomach ill is habitually greeted with contempt, callousness and antipathy. I perfectly understand how passive I appear on family matters, but I simply have a far more cruel inner master to appease.

After no seriously detectable symptoms from decade long repeated hospital visits and struggling of recurring episodic suicidal thoughts, I finally feel determined to trust my own guts on this matter. It’s not that what I eat is terribly western or contaminated by despicable Chinese food vendors, since people around me are doing fine most of the time (granted my company did get involved relatively recently in a food poisoning incident that went viral on wechat). The crux of my curse is that it takes physical energy and leisure (time) to properly digest food. So it is not surprising that I get diarrhea from time to time after eating in front of my computer, especially if I also happen to overeat, and if the food is somewhat irritating (e.g. mildly spicy).

While food poisoning in the bay area is unfortunately still an existential threat, the fact that people eating the same food around me are not complaining as much indicates scientifically that my own predisposition is to blame. Just because colleagues eat at their desk doesn’t mean I need to follow the suit. My current job is comfortable enough to avoid such peer pressure, and I should exploit other occasions for longer work hours instead of the precious time after meal reserved for my organs.

Interesting findings of the week

A. This morning reading through colleague’s weekly report, I see this figure 5kw and got initially puzzled. Then it became clear that k stands for 1,000 and w for 10,000 (the Chinese convention of powers of 10 of course). Sometimes people don’t realize how creative they get with everyday mundane work.

B. After talking to a former classmate and near-miss colleague of mine, I realize that the reason of my current dissatisfaction in life could well be one of the following:

1. I don’t know what I truly want in terms of priority, and therefore don’t know which ones to sacrifice.

2. I am not learning enough from people around me.

3. I fantasize too much about a life burdened with research OKRs, and have not had enough opportunity simply building things from scratch.

I currently don’t have a method to address any of these. But perhaps will in a few months. In addition, I have some major complaint leveled against my current company food. At least the overtime meal feels like a bare minimum of starch and protein, with negligible amount of flora. It can take a toll on health in the long term. Buffet style food, no matter how poorly cooked, is still a luxury in comparison.

C. One mysterious debugging trick: often I run into a bug that has no reason to exist, and spend half an hour or so deliberating where to insert the print statement, only to find out that the bug is not reproducible, that is, it disappears on second attempt. This does not mean I should be less careful with the debugging process, but only that better editor sync is needed.

Amazing breakthrough in theory of computation

I learned from a colleague at JD.com that a professor Huang Hao from Emory math department made breakthrough just weeks ago in the following conjecture of Gotsman and Linial:

Given any induced subgraph $G$ of the hypercube graph $Q_n$, with $V(G) \geq 2^{n-1} + 1$, its maximum degree is at least $\sqrt{n}$. This by itself is already a big deal since the previous best result by famous people like Fan Chung was on the order of $\log(n)$, and presumably not obvious either.

Since the prof is Chinese, it was well advertised in the Chinese blogging community, though not to the extent of Yitang Zhang’s prime gap result.

The connection to other areas of complexity theory is even more remarkable, and well-exposed in the author’s original paper. In fact, I found that the subsequent advertising blogs by other mathematicians and journalists to be largely not adding much expository value.

A notion of complexity is given by the so-called sensitivity $s$. For a boolean function $f: F_2^n \to F_2^1$, where $F_2 := \{0, 1\}$ viewed as a set, the sensitivity function is defined by $s(f, x) := |\{i \in [n]: f(x^{\{i\}}) \neq f(x)\}|$, where $x^{\{i\}}$ stands for the binary vector $x$ with the $i$-th coordinate flipped. In other words, point-wise sensitivity is the number of ways the function $f$ can be perturbed by a single coordinate, or more metaphorically, the co-dimension of its null tangent space at $x$. Global sensitivity of $f$ is simply $s(f) := \max \{s(f, x): x \in F_2^n\}$. This is a natural measure of complexity of $f$ since if $f$ is constant, sensitivity is clearly $0$. But maybe it’s not fundamental enough to let us tackle P versus NP because for a simple function like the parity function $f(x) := |\{i \in [n]: x_i = 0\}| % 2$, $s(f) = n$ already achieves maximum sensitivity.

Now the connection between sensitivity and maximum degree is proved by Gotsman and Linial in a quarter-page proof, so I encourage everyone to read it. But here is my digested version:

The following two statements are equivalent:

1. The max degree of $G$ and the induced subgraph on $Q_n \setminus G$ is bounded below by $h(n)$, provided $|V(G)| \neq 2^{n-1}$.
2. The sensitivity of a boolean function $f: F_2^n \to F_2$ with $d(f) = n$ satisfies $s(f) >= h(d(f))$ where $d(f)$ is the degree of $f$, viewed as a polynomial. More precisely, $\max \{|\{i \in [n]: x_i = 1\}|: f(x) = 1\}$.

To see the equivalence, simply consider the indicator function $g$ of $V(G)$ with domain $F_2^n$, multiplied by the parity function $p$. The degree of $gp$ at $x \in Q_n$ is clearly related to its sensitivity at $x \in F_2^n$. Indeed the parity makes so that neighbors in the original graph $G$ change the function value.

The requirement of  $|V(G)| \neq 2^{n-1}$ corresponds to $\mathbb{E} g \neq 0$, which in turn corresponds to $d(f) = n$. The rest should be pretty straightforward.

Statement 2 can be equivalently relaxed to the case where $d(f) < n$. The reason, as pointed out to me by Gil Kalai, is that given a function of degree $d < n$, there is a monomial $x_1 \ldots x_d$ of exactly $d$ variables. Let $g(x_1, \ldots, x_d) = f(x_1, \ldots, x_d, 0, \ldots 0)$. Then $g$ has degree $d$ and sensitivity less than or equal to that of $f$. But $g$ has full degree (equal to the dimension of its domain), so 2 applies. And we get

$s(f) \geq s(g) \geq h(d(g)) = h(d) = h(d(f)).$

I will devote another post for Huang’s actual proof.

Click a form button to open a new tab in flask

Unfortunately online support for flask is somewhat limited. It is not treated as a mature web programming language. But I use it mainly for internal demo purpose, and it simplifies things tremendously. The trick is often that a technique is not specific to flask, but html in general.

It is relatively easy to redirect to a new tab with a hyperlink in html:

<a href=”google.com” target=”_blank”> my awesome link </a>

will do that.

With forms, target is a top level attribute that controls rendering of the entire form

<form target=”_blank”> … <input type=”submit” value=”submit”> … </form>

To redirect to new tab for an individual button within the form, try

<form> … <input type=”submit” value=”submit” formtarget=”_blank” > … </form>

I also found that POST actions do not support Ctrl + Click, since the target url cannot infer the POST parameters directly.


text:
## Developers' Forum

### Default java encoding

by Anton Dil -
Number of replies: 4

I believe it depends on the operating system defaults, but we have found that our compiler is expecting US-ASCII encoding for java source code.

Would it be worth making -encoding UTF-8 the default setting in the question prototypes? (I know I can set this using Advanced Customisation settings.)

Test code:

System.out.println("Default Charset=" + java.nio.charset.Charset.defaultCharset());

Thanks,

Anton

In reply to Anton Dil

### Re: Default java encoding

by Tim Hunt -

Just to give the origin of this, it came from a question where the answer needed to be something like

System.out.println("The prize was £10.");

which is currently giving an error like

Moodle tends to force everything to UTF-8 encoding, so it there are simple compiler and/or run-time flags to get Java to expect that, then then that is probably the best thing to do.

In reply to Tim Hunt

### Re: Default java encoding

by Richard Lobb -

I'm not sure I understand the full context here. But it looks like the Jobe server on which this job was run hasn't had the locale set properly as per the install instructions here. Certainly the example code you've given works fine when I run it on my development system (and I expect on all other systems we run, too).

In reply to Richard Lobb

### Re: Default java encoding

by Tim Hunt -

Strangely, when we run

echo \$LANG

from the shell on the Jobe server, it says en_GB.UTF-.

However, if I go to a Java CodeRunner question, and run the code

int sumSquares(int[] data) {    System.out.println(String.format("LANG environment variable: %s", System.getenv("LANG")));    System.out.println(String.format("file.encoding: %s", System.getProperty("file.encoding")));    System.out.println(String.format("defaultCharset: %s", java.nio.charset.Charset.defaultCharset().name()));    return 0;}

That gives

LANG environment variable: Cfile.encoding: ANSI_X3.4-1968defaultCharset: US-ASCII

We are continuing to investigate, but in the mean time, if you have any more clues, that would be appreciated.

In reply to Tim Hunt

### Re: Default java encoding

by Richard Lobb -

I think the problem is that the job execution is initiated by Apache. The Jobe install instructions show how to set the Apache envvars (environment variables) to change from the default C/ASCII environment to something that's UTF-8 aware. It's those environment variable settings that are controlling the jobe task execution, not the OS/Shell environment.


text:
改进的半经典动力学模拟二苯乙烯光致顺反异构化反应

1. a 合成与天然功能分子化学教育部重点实验室 西北大学化学与材料科学学院 西安 710069;
b 西北大学现代物理研究所 西安 710069
• 投稿日期:2012-04-21 发布日期:2012-08-06
• 通讯作者: 朱超原
• 基金资助:

项目受国家自然科学基金(Nos. 21003100, 21033001, 21103136, 21173166)资助.

New Implementation of Semi-classical Dynamic Simulation on the Photoisomerization of cis- and trans-Isomers of Free Stilbene

Lei Yiboa,b, Zhu Chaoyuanb, Wen Zhenyia,b, Lin Sheng-Hsienb

1. a Key Laboratory of Synthetic and Natural Functional Molecule Chemistry of Ministry of Education, The College of Chemistry & Materials Science, Shaanxi Key Laboratory of Physico-Inorganic Chemistry, Northwest University, Xi'an 710069;
b Institute of Modern Physics, Northwest University, Xi'an 710069
• Received:2012-04-21 Published:2012-08-06
• Supported by:

Project supported by the National Natural Science Foundation of China (Nos. 21003100, 21033001, 21103136, 21173166).

New implementation of semi-classical trajectory surface hopping dynamic simulation has been developed and applied to the photoisomerization of cis- and trans- isomers on the gas phase. This method not only uses the exponential model to the modification of the originally analytical non-adiabatic transition probability formula, but also involves the constrained Hamiltonian system into the constrained molecular dynamic simulation. Two-dimensional potential energy surfaces of ground S0 and excited S1 states are constructed analytically fitting to ab initio calculations in terms of torsion angle and one dihedral angle around the central ethylenic C＝C bond as variables, and the other internal coordinates are all fixed at configuration of one-bond flip conical intersection. The analytical PESs are quite accurate and the mean absolute error is less than 2.4 kcal·mol-1, and much less than 1.0 kcal·mol-1 around conical intersection region. A straight seam line is found on potential energy surfaces that simply separates the cis-area with the trans-area. The constrained Hamiltonian system is employed to run trajectories in the Cartesian coordinate system and surface hopping in terms of the two internal dihedral angles. Typical trajectories are found in which the torsion angle changes monotonically for both cis- to trans- and trans- to cisisomerizations. This is an exact picture of one-bond flip mechanism of photoisomerization around the conical intersection. Quantum yield for trans- to cis- isomerization is simulated as 60.45% in very good agreement with experimental value 55.0%, while quantum yield for cis- to trans- isomerization is simulated as 42.3% in comparison with experimental value 35.0%. As the S1 energy in local minimum of cis-area is higher than that in trans-area, and thus cis- to trans- isomerization is quite possible to access to another Hula-Twist conical intersection. These simulation results demonstrate that the computed cumulative quantum yield and reaction mechanism are consistent with the previously experimental and theoretical results. This means that the present trajectory surface hopping method would be good at the dynamic simulation on the large system with or without constraint Hamiltonian in comparison with the quantum molecular dynamics.


text:
# Payday financing legislation is found on the ballot in Nebraska amid sky-high finance interest rates

Payday financing legislation is found on the ballot in Nebraska amid sky-high finance interest rates

A sudden dollars crisis pushed Richard Blocker to get a payday loan in 2014.

To get his or her epilepsy in order, the man needed treatment — yet the costs have been improved. And even though he was working in the deposit market and had great cover, the fee had been troublesome.

“I had been having difficulty keeping up with the meds and having to pay your other expenditures,” this individual retrieve in interviews with Yahoo funds. “So we went along to a payday bank consideration, perfectly, it’s only going to be one rapid funding and I’ll buy it reduced, and I’ll be good. That’s not what taken place.”

Blocker acknowledged the risks of payday credit, however the financial institution isn’t travelling to give him limited funding to tide your on. So he visited the EZ cash part with his region near Omaha, Nebraska, and took out a $500 debt with a$15 price every couple of weeks.

“There’s about $30 million in charges on your own that payday financial institutions rate and more than the program of one year,” Ken Gret, economic justice application movie director during the Nebraska Appleseed Center for rule when you look at the community Interest, told Yahoo funds. “The nightmare has-been really, really clear.” Smith’s class — along with several others — are actually driving for a project that dreams to introduce a cap on rates at 36per cent, which happens to be a “level that many other shows in the nation have actually.” The man put there got a “long string” of destruction that party had to endure from payday creditors who planned to stop the issue from being to the ballot. One suit explained these people decided not to need to be recognized as payday financial institutions, as it may have a damaging meaning. (containing since become resolved.) The offer to cover APR at 36% will show up on the vote on Election night. “This is similar common-sense assess that voters not too long ago authorized in Southern Dakota and Colorado,” believed past Consumer Investment policies agency (CFPB) manager Richard Cordray. “We wish corporations with a purpose to produce small-dollar finance, nevertheless must be lending that help customers, perhaps not hurt all of them.” Jennifer Bellamy, legislative advise by using the United states Civil Liberties Union (ACLU), assured Yahoo finances that “if Nebraskans happened to be to choose and support this action, this will be an enormous help suitable course.” The payday credit business, at the same time, takes into account the step to be geared towards “eliminating managed small-dollar credit during the county while creating nothing to meet Nebraskans’ very real economic demands,” stated Ed D’Alessio, executive director of INFiN, a nationwide business organization which shows the industry. ## ‘An immoral interest rate has been in existence for millennia’ Gradually, scores of North americans have left to payday creditors and were in times when her first two-week funding drift over into months and in many cases many years. In certain extreme situations, as Yahoo fund before reported, professionals revealed a “debt-to-jail” pipeline concerning lenders suing debtors in small claims courtroom and receiving all of them arrested whenever they don’t show up. ProPublica in addition exposed specific problems. “The doubt precisely what happens to be a bad rate of interest has been in existence for centuries, and is particularly actually attended to during the Bible,” Aaron Klein, a financial learning fellow during the nonprofit think-tank Brookings, instructed Yahoo fund. In reaction to predatory creditors, “many say governments has died monthly interest caps, or usury limits,” this individual put, no matter his or her governmental orientation, particularly in South Dakota, or even in Ca. (U.S. lawmakers forbidden debtors prisons in 1833.) The CFPB, which happens to be tasked with monitoring predatory lenders, enjoys mostly already been defanged from Trump administration. The organization had truly moved to secure owners in 2017, by providing a rule to condemn the way in which these loan providers made personal loans regardless of individuals’ power to payback. But in July in 2010, the CFPB rescinded that regulation. That struggle is definitely ongoing: On monday, community person and Center for important loaning prosecuted the CFPB to create right back measures the organisation embraced in 2017 to guard American owners from both payday loans or auto-title financial products. With all the elections emerging, a Biden government could “restore the CFPB, place right back commonsense guidelines, like simply bring to the people you think that have actually to be able to pay out back,” added Klein. A 2019 statement proposed by Rhode isle Senator Sheldon Whitehouse and Senate advocate Whip penis Durbin referred to as the preserving buyers from unrealistic loan prices work of 2019 would reduce rate and also to 36%. Other individuals have got introduced the experts and users reasonable credit score rating operate, which will expand the 36per cent rate hat security ready for experts and their family members to all or any people. ## ‘The rich obtain deeper, together with the bad create poorer’ For 43-year-old Nebraskan Phil Davis, a quick payday loan appeared like the best option any time his own auto broke down so he am short of wealth to celebrate Christmas time along with his family members about ten years ago. “They’re merely in a remove mall, but you walk in top house, you visit the table, in addition they simply take a check, note down the ideas, supply cash… and also you accomplish your enterprise,” remembered Davis, exactly who is effective inside the dining establishment discipline. Exactly what Davis couldn’t count on were to staying stuck with those money for more than just a two-week years: His original$500 loan, having its $75 costs due all 2 weeks, ballooned to about$5,800 by the time they at some point spent it all 3 years afterwards with a tax return.

“It’s the installation with the rates that in some way extends to consumers,” Davis believed.

Davis has actually since get an advocate for any 36per cent hat, thinking that payday credit practices are perpetuating profit difference.

“personally i think for example the abundant see deeper, along with very poor bring poorer,” Davis claimed. “It upsets me additional right now, as well as contemplate what amount of cash we provided to the consumers… to consider what other people who are getting those funds and don’t posses \$150 monthly.”

Along with an affordable cover on percentage of interest, the guy put in, the choice remained offered to those in demand for short-term wealth while making they “a little more relaxing for men and women to crawl away from the hole these are typically in.”


text:
Varieties of small Kodaira dimension whose cotangent bundles are semiample
Compositio Mathematica, Tome 84 (1992) no. 1, p. 43-52
@article{CM_1992__84_1_43_0,
author = {Fujiwara, Tsuyoshi},
title = {Varieties of small Kodaira dimension whose cotangent bundles are semiample},
journal = {Compositio Mathematica},
volume = {84},
number = {1},
year = {1992},
pages = {43-52},
zbl = {0763.14015},
mrnumber = {1183561},
language = {en},
url = {http://www.numdam.org/item/CM_1992__84_1_43_0}
}

Fujiwara, Tsuyoshi. Varieties of small Kodaira dimension whose cotangent bundles are semiample. Compositio Mathematica, Tome 84 (1992) no. 1, pp. 43-52. http://www.numdam.org/item/CM_1992__84_1_43_0/

[1] J. Bertin and P. Teller, Invariants symetriques et fibrés vectoriels sur les courbes. Math. Ann. 264 (1983) 423-436. | MR 716258 | Zbl 0504.14011

[2] T. Fujita, Semipositive line bundles. J. Fac. Sci. Univ. Tokyo 30 (1983) 353-378. | MR 722501 | Zbl 0561.32012

[3] D. Gieseker, p-ample bundles and their Chern classes. Nagoya Math. J. 43 (1971) 91-116. | MR 296078 | Zbl 0221.14010

[4] R. Hartshorne, Algebraic Geometry. Graduate Texts in Math. 52, Springer, 1977. | MR 463157 | Zbl 0367.14001

[5] S. Iitaka, On D-dimensions of algebraic varieties. J. Math. Soc. Japan 23 (1971) 356-373. | MR 285531 | Zbl 0212.53802

[6] Y. Kawamata, Characterization of abelian varieties. Compositio Math. 43 (1981) 253-276. | Numdam | MR 622451 | Zbl 0471.14022

[7] K. Kodaira, On deformations of complex analytic structures I, II. Ann. Math. 67 (1958) 328-466. | MR 112154 | Zbl 0128.16901

[8] K. Kodaira, On compact analytic surfaces II. Ann. Math. 77 (1963) 563-626. | Zbl 0118.15802

[9] I. Nakamura and K. Ueno, An addition formula for Kodaira dimensions of analytic fibre bundles whose fibres are Moišezon manifolds. J. Math. Soc. Japan 25 (1973) 363-371. | MR 322213 | Zbl 0254.32027

[10] M. Reid, Bogomolov's theorem c21 ≤ 4c2. Intl. Symp. on Algebraic Geometry, Kyoto (1977) 623-642. | Zbl 0478.14003


text:
A fast and precise DFT wavelet code

# SCF loop input parameters

(Redirected from Input.mix)

The "mix:" block in input.yaml contains the parameters required to use a density (or potential) mixing scheme for the self consistent field loop. This loop may imply a smearing of the occupancies for finite temperature DFT calculations. The old input scheme used a separate file named input.mix.

## Example

mix:
iscf: 7          # Mixing parameter.
itrpmax: 50      # Maximum number of diagonalisation iterations.
rpnrm_cv: 1.e-11 # Stop criterion on the residue of potential or density.
norbsempty: 100  # number of extra orbitals ( unoccupied ).
tel: 1.e-3       # Electronic temperature.
alphamix: 0.95   # Proportion of old density kept. Must be larger than 1/2.
alphadiis: 2.    # Multiplying factors for the electronic DIIS.

## Example of an input.mix file (old input scheme)

7             # Mixing scheme type
50            # Maximum number of diagonalisation iterations
1.e-11        # Stop criterion on the residue of potential or density
100 1e-3 1    # No. of additional bands, elec. temperature, smearing method
0.95 2.0      # Multiplying factors for the mixing and the electronic DIIS

## Entry descriptions

Descriptions of the keys:

• iscf: An integer giving the mixing scheme and the mixing target. It follows the ABINIT convention. For values lower than 10, the potential is mixed, while for values greater than 10, the density is mixed. A recommended option is Pulay mixing (7 or 17 for potential or charge mixing, respectively)
• itrpmax: Maximum number of repeats for the mixing scheme. Each iteration implies an optimization of the wavefunctions at constant potential according to the settings in input.dft.
• rpnrm_cv: The stop criterion on the residue of the potential or the density. Typical values are in the order of 1e-10 to 1e-12.
• norbsempty: The maximum number of additional (initially unoccupied) bands considered. If a larger value is given, The maximum of orbitals available via the input guess procedure will be used automatically
• Tel: The electronic temperature in energy units (Ha).
• occopt: The smearing function to determine the occupation. The conventions differ from those used by ABINIT. 1 = Gaussian, 2 = Fermi, 3 = Marzari (bumb minimization), 4 = Marzari (monotonic tail), 5 = Methfessel and Paxton.
• alphamix : The closer this value gets to one, the more cautious will be the mixing of the charge or potential. In the case of straight mixing, it corresponds to Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \alpha
according to Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)):  \rho_{new} ~\leftarrow ~\alpha~\rho_{old} + (1-\alpha)~\rho_{new}

. alphamix must be larger than one half. A typical choice for metals is 0.95.

• alphadiis : The DIIS multiplier used for preconditioning the wavefunctions at constant potential.

## Related DFT variables

When performing a calculation with a mixing scheme as described above, it is important to adjust some DFT settings accordingly in the "dft:" block (or in the file input.dft for the old input scheme).

• itermax: Few iterations are used in the inner loop to optimize the wavefunctions at constant potential, typically 2 to 4.
• nrepmax: An additional repeat loop with diagonalisation does not make sense, thus nrepmax should be kept at 1.
• idsx: The DIIS history length should be set to zero.
• gnrm_cv: This value should be kept small, e.g. 1e-8, such that the inner loop will do itermax iterations and not less.


text:
Numerical ordinary differential equations

Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). This field is also known under the name numerical integration, but some people reserve this term for the computation of integrals.

Many differential equations cannot be solved analytically, in which case we have to satisfy ourselves with an approximation to the solution. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution.

Ordinary differential equations occur in many scientific disciplines, for instance in mechanics, chemistry, ecology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.

 Contents

The problem

We want to approximate the solution of the differential equation

where f is a function that maps [t0,∞) × Rd to Rd, and the initial condition y0 ∈ Rd is a given vector.

The above formulation is called an initial value problem (IVP). The Picard-Lindelöf theorem states that there is a unique solution, if f is Lipschitz continuous. In contrast, boundary value problems (BVPs) specify (components of) the solution y at more than one points. Different methods need to be used to solve BVPs, for example the shooting method, multiple shooting or global methods like finite differences or collocation.

Note that we restrict ourselves to first-order differential equations (meaning that only the first derivative of y appears in the equation, and no higher derivatives). However, a higher-order equation can easily be converted to a first-order equation by introducing extra variables. For example, the second-order equation y'' = −y can be rewritten as two first-order equations: y' = z and z' = −y.

Methods

Two elementary methods are discussed to give the reader a feeling for the subject. After that, pointers are provided to other methods (which are generally more accurate and efficient). The methods mentioned here are analysed in the next section.

The Euler method

Starting with the differential equation (1), we replace the derivative y' by the finite difference approximation

which yields the following formula

This formula is usually applied in the following way. We choose a step size h, and we construct the sequence t0, t1 = t0 + h, t2 = t0 + 2h, ... We denote by yn a numerical estimate of the exact solution y(tn). Motivated by (3), we compute these estimates by the following recursive scheme

[itex] y_{n+1} = y_n + hf(t_n,y_n). [itex]

This is the Euler method, named after Leonhard Euler who described this method in 1768.

The backward Euler method

If, instead of (2), we use the approximation

[itex] y'(t) \approx \frac{y(t) - y(t-h)}{h}, [itex]

we get the backward Euler method:

[itex] y_{n+1} = y_n + hf(t_{n+1},y_{n+1}). [itex]

The backward Euler method is an implicit method, meaning than we have to solve an equation to find yn+1. One often uses functional iteration or (some modification of) the Newton-Raphson method to achieve this. Of course, it costs time to solve this equation; this cost must be taken into consideration when one selects the method to use.

Generalisations

The Euler method is often not accurate enough. In more precise terms, it only has order one (the concept of order is explained below). This caused mathematicians to look for higher-order methods.

One possibility is to use not only the previously computed value yn to determine yn+1, but to make the solution depend on more past values. This yields a so-called multistep method. Almost all practical multistep methods fall within the family of linear multistep methods, which have the form

[itex] \alpha_k y_{n+k} + \alpha_{k-1} y_{n+k-1} + \cdots

+ \alpha_0 y_n[itex]

[itex] = h \left( \beta_k f(t_{n+k},y_{n+k}) + \beta_{k-1}

f(t_{n+k-1},y_{n+k-1}) + \cdots + \beta_0 f(t_n,y_n) \right). [itex]

Another possibility is to use more points in the interval [tn,tn+1]. This leads to the family of Runge-Kutta methods, named after Carle Runge and Martin Kutta. One of their fourth-order methods is especially popular.

Both ideas can also be combined. The resulting methods are called general linear methods.

A good implementation of one of these methods for solving an ODE entails more than the time-stepping formula.

It is often inefficient to use the same step size all the time, so variable step-size methods have been developed. Usually, the step size is chosen such that the (local) error per step is below some tolerance level. This means that the methods must also compute an error indicator, an estimate of the local error.

An extension of this idea is to choose dynamically between different methods of different orders (this is called a variable order method). Extrapolation methods are often used to construct various methods of different orders.

Other desirable features include:

• dense output: cheap numerical approximations for the whole integration interval, and not only at the points t0, t1, t2, ...
• event location: finding the times where, say, a particular function vanishes.

Alternative methods

Many methods do not fall within the framework discussed here. Some classes of alternative methods are:

• multiderivative methods, which use not only the function f but also its derivatives. This class includes Hermite-Obreschkoff methods and Fehlberg methods.
• methods for second order ODEs. We said that all higher-order ODEs can be transformed to first-order ODEs of the form (1). While this is certainly true, it may not be the best way to proceed. In particular, Nyström methods work directly with second-order equations.
• geometric integration methods are especially designed for special classes of ODEs (e.g., symplectic integrators for the solution of Hamiltonian equations). They take care that the numerical solution respects the underlying structure or geometry of these classes.

Analysis

Numerical analysis is not only the design of numerical methods, but also their analysis. Three central concepts in this analysis are convergence (whether the method approximates the solution), order (how well it approximates the solution), and stability (whether errors are damped out).

Convergence

A numerical method is said to be convergent if the numerical solution approaches the exact solution as the step size h goes to 0. More precisely, we require that for every ODE (1) with a Lipschitz function f and every t* > 0,

[itex] \lim_{h\to0+} \max_{n=0,1,\dots,\lfloor t^*/h\rfloor}

\| y_{n,h} - y(t_n) \| = 0. [itex]

All the methods mentioned above are convergent. In fact, convergence is a condition sine qua non for any numerical scheme.

Order

Suppose the numerical method is

[itex] y_{n+k} = \Psi(t_{n+k}; y_n, y_{n+1}, \dots, y_{n+k-1};

h). [itex]

The method is said to have order p if

[itex] \Psi \left( t_{n+k}; y(t_n), y(t_{n+1}), \dots, y(t_{n+k-1}); h \right)

The quantity on the left-hand side is called the local error of the method. The (forward) Euler method and the backward Euler method introduced above both have order 1. Most methods being used in practise attain higher order.

The local error is the error committed in a single step. A related concept is the global error, the error sustained in all the steps one needs to reach a fixed time t. Explicitly, the global error at time t is yN - y(t) where N = (t-t0)/h. The global error of a pth order one-step method (that is, a method of the form (4) with k = 1) is O(hp); in particular, such a method is convergent. This statement is not necessarily true for multi-step methods.

Stability and stiffness

Main article: Stiff equation

For some differential equations, application of standard methods such as the Euler method, explicit Runge-Kutta methods, or multistep methods such as Adams-Bashforth methods exhibit instability in the solutions obtained by these methods; however other methods do behave correctly with these "difficult-behaving" equations (however, in fact these equations may be quite simple). This "difficult-behaviour" in the equation is described as stiffness, and is often caused by the presence of different time scales in the underlying problem. Stiff problems are ubiquitous in chemical kinetics, control theory, solid mechanics, weather prediction, biology, and electronics.

History

Below is a concise timeline of some important developments in this field.

References

• Ernst Hairer, Syvert Paul Nørsett and Gerhard Wanner, Solving ordinary differential equations I: Nonstiff problems, second edition, Springer Verlag, Berlin, 1993. ISBN 3-540-56670-8.
• Ernst Hairer and Gerhard Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, second edition, Springer Verlag, Berlin, 1996. ISBN 3-540-60452-9.
(This two-volume monograph systematically covers all aspects of the field.)
• Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996. ISBN 0-521-55376-8 (hardback), ISBN 0-521-55655-4 (paperback).
• John Denholm Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, 1991. ISBN 0-471-92990-5.
(Textbook, slightly more demanding than the book by Iserles.)

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy


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# Wydawnictwa / Czasopisma IMPAN / Dissertationes Mathematicae / Wszystkie zeszyty

## Dissertationes Mathematicae

Artykuły w formacie PDF dostępne są dla subskrybentów, którzy zapłacili za dostęp online, po podpisaniu licencji Licencja użytkownika instytucjonalnego. Czasopisma do 2009 są ogólnodostępne (bezpłatnie).

## On convergence in distribution of the Markov chain generated by the filter kernel induced by a fully dominated Hidden Markov Model

### Tom 514 / 2016

Dissertationes Mathematicae 514 (2016), 1-67 MSC: Primary 60J05; Secondary 60F05. DOI: 10.4064/dm739-9-2015 Opublikowany online: 21 March 2016

#### Streszczenie

Consider a Hidden Markov Model (HMM) such that both the state space and the observation space are complete, separable, metric spaces and for which both the transition probability function (tr.pr.f.) determining the hidden Markov chain of the HMM and the tr.pr.f. determining the observation sequence of the HMM have densities. Such HMMs are called fully dominated. In this paper we consider a subclass of fully dominated HMMs which we call regular.

A fully dominated, regular HMM induces a tr.pr.f. on the set of probability density functions on the state space which we call the filter kernel induced by the HMM and which can be interpreted as the Markov kernel associated to the sequence of conditional state distributions.

We show that if the underlying hidden Markov chain of the fully dominated, regular HMM is strongly ergodic and a certain coupling condition is fulfilled, then, in the limit, the distribution of the conditional distribution becomes independent of the initial distribution of the hidden Markov chain and, if also the hidden Markov chain is uniformly ergodic, then the distributions tend towards a limit distribution.

In the last part of the paper, we present some more explicit conditions, implying that the coupling condition mentioned above is satisfied.

#### Autorzy

• Thomas KaijserDepartment of Mathematics


text:
# By Using the Properties of Definite Integrals, Evaluate the Integrals Int_0^(2x) Cos^5 Xdx - Mathematics

By using the properties of definite integrals, evaluate the integrals

int_0^(2x) cos^5 xdx

#### Solution

Is there an error in this question or solution?

#### APPEARS IN

NCERT Class 12 Maths
Chapter 7 Integrals
Q 14 | Page 347


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Isogenies between Tate curves

Let $q$ and $q'$ be complex numbers with $0<|q|,|q'|<1$, and let $m$ and $n$ be positive integers.
Suppose that $q^m={q'}^n$. Then the map $$f:\mathbb{C}^\times/q^{\mathbb{Z}} \to \mathbb{C}^\times/{q'}^{\mathbb{Z}}\qquad \text{defined by}\qquad u\mapsto u^m$$ gives an isogeny of (analytic) elliptic curves over $\mathbb{C}$.

The Tate curve $\mathrm{Tate}(q)$ is an (algebraic) elliptic curve over the Laurent series ring $\mathbb{Z}((q))$ which can be used to give a uniformization of the curve $\mathbb{C}^\times/q^\mathbb{Z}$ by means of certain well known explicit formulae.

My question is:

Does there exist an isogeny $\mathrm{Tate}(q)\to \mathrm{Tate}(q')$ of elliptic curves defined over $\mathbb{Z}((q,q'))/(q^m-{q'}^n)$ which "lifts" the map $f$ above, and if so, how do you prove it exists?

It should suffice to construct such an isogeny for $(m,n)=(m,1)$, and use dual isogenies and composition to get the general case.

(I'm being vague about "lifts", because one has to worry about convergence somewhere. Probably you want to say that the isogeny is defined over some subring of $\mathbb{Z}((q,q'))/(q^m-{q'}^n)$ of power series which are analytically convergent near $q=0$, or something like that.)

I presume (though I probably can't prove) that the existence of the analytic isogenies means that such a map of schemes is defined over $\mathbb{C}((q,q'))/(q^m-{q'}^n)$, so that this is just a question about integrality.

This is very closely related to exercise 5.10 in Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. There, he apparently asks us to show that for a $p$-adic field $K$, if $q,q'\in K$, $0<|q|,|q'|<1$, and $q^m={q'}^n$, then the function $\overline{K}^\times/q^\mathbb{Z}\to \overline{K}^\times/{q'}^{\mathbb{Z}}$ defined by $u\mapsto u^m$ lifts to an isogeny $E_q\to E_{q'}$ of elliptic curves over $K$, where $E_q$ and $E_{q'}$ are defined by the Tate curve equations. (An answer to my question solves his exercise, right?)

Unfortunately, I have no idea how to do Silverman's exercise either (he marks it as difficult). Any hints?

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The answer is "yes", but only way I understand such things is via formal schemes & generalized ell. curves over rings (such as $\mathbf{Z}[[q,q']]/(q^n - {q'}^m)$, & then invert $q$ and $q'$ to get to ell. curves). Methods in AECII & rigid-analytic geometry are useless when base ring is not field. I asked Tate if he knew how to rigorously understand structure of $N$-torsion on Tate curve over $\mathbf{Z}((q))$ without hard alg. geom; he said "no". Saying it's "just" integral refinement on analytic theory misses too many subtleties. "Black box" approach in Katz-Mazur hides difficulties too. –  BCnrd Aug 20 '10 at 0:19
To do the exercise, a natural method is to use rigid-analytic geometry and GAGA, much as one would do over the complex numbers. In effect, one is trying to show that an "analytic" viewpoint allows one to construct "algebraic" maps. Think how one would do the complex-analytic version, and the ideal would be to carry out the same reasoning in the non-archimedean case. However, if you don't know rigid-analytic geometry, it might inspire you to try to learn about that topic (that sure worked for Tate!). There might be a brute force argument with power series, but is that illuminating? –  BCnrd Aug 20 '10 at 2:59
Thanks, BCnrd. Can you say a little more about the approach via "formal schemes & generalized elliptic curves"? (Is something like this in Deligne-Rappaport?) That sounds closer to things I know than rigid-analytic things (of which my ignorance is total). (I mentioned AEC2 because it's the only place I could find a statement along the lines I want, not because I care about its methods so much.) –  Charles Rezk Aug 20 '10 at 4:32
Yes, in D-R. For some highlights from D-R with extras tossed in, see sec. 2.1, 2.2 (especially 2.2.4), and 2.5 in my paper "Arith. moduli of gen. ell. curves" (on my webpage), which includes reference to section of D-R which discusses Raynaud construction of Tate curve via formal schemes. (Raynaud procedure is in more concrete terms in sec. 2.5.) Must be careful about rigor when relating things over $\mathbf{Z}((q))$ and over $\mathbf{C}$ (since can't evaluate formal series). Simpler for non-arch. fields, but algebraization of formal schemes and then generic fiber will give what you want. –  BCnrd Aug 20 '10 at 6:27

No matter how you define Tate(q), it should have the following properties:

(a) for any $n$ it contains a subgroup $M_n$ canonically isomorphic to $\mu_n$ (which corresponds tho $\mu_n\subset\mathbb{C}^\times$ in the complex model),

(b) the (co)tangent space along the unit section is canonically trivialized (by $d\log u$ in the complex model).

Let me first treat the case $n=1$, as Charles suggests. The sought-for isogeny Tate($q$)$\to$Tate($q^m$) is characterized by two conditions:

(a') its kernel is $M_m$ (i.e. it induces an isomorphism Tate($q$)$/M_m\to$Tate($q^m$)),

(b') it induces multiplication by $m$ on the tangent space, modulo the identification (b).

Consider the scheme $X\to S:=\mathrm{Spec}\:\mathbb{Z}((q))$ parametrizing isomorphisms Tate($q$)$/M_m\to$Tate($q^m$). This is an unramified $S$-scheme, and in fact it is finite because Tate($q$) has no complex multiplication in any fiber over $S$ (I guess this has to be checked). Since it has a section over $\mathrm{Spec}\:\mathbb{C}((q))$ it is dominant, hence surjective over $S$. Since $S$ is normal it suffices to find a section at the generic point. But by flat descent, condition (b') guarantees that the above section over $\mathrm{Spec}\:\mathbb{C}((q))$ descends to the fraction field of $\mathbb{Z}((q))$. QED.

Remark: I am a bit uncomfortable about the "no CM" stuff, but we can probably avoid it by noting that $X\to S$ satisfies the valuative criterion of properness, even when it is not of finite type. This (together with unramifiedness) is enough to imply that a section at the generic point extends over a normal base.

For arbitrary $n$, observe that we have just constructed $\alpha_m:\text{Tate}(q)\to \text{Tate}(q^m)$ with kernel killed by $m$, so multiplication by $m$ factors as $\beta_m\circ\alpha_m$ for some $\beta_m:\text{Tate}(q^m)\to \text{Tate}(q)$. You can now treat the general case by taking the composition $$\text{Tate}(q)\to \text{Tate}(q^m)=\text{Tate}(q'^n)\to \text{Tate}(q')$$ where the two maps are $\alpha_m$ and $\beta_n$ respectively.

-
Very nice. The verification of (a) and (b) is the sort of "black box" as in Katz-Mazur. Ironically,even though K-M basically avoid ever alluding to generalized elliptic curves or moduli interpretation along cusps, to justify stuff like (a) and (b) (and formal group analogue) over $\mathbf{Z}((q))$ which underlies their analysis of structure along the cusps it seems to has to haul out the D-R approach to things anyway. –  BCnrd Aug 20 '10 at 15:47
My feeling is that we lack a good a priori definition of Tate curves (e.g. by a universal property, or a list of axioms such as (a) and (b)). All we have, it seems, is "constructions". Remember, for instance how Grothendieck's definition of the Picard functor made it easier to use Jacobians, which previously were just constructed. The difficulty in doing this, I think, is the "scheme vs formal scheme" bit, i.e. the algebraization process. In terms of functors, it makes it hard to guess on what category the represented functor should live. –  Laurent Moret-Bailly Aug 21 '10 at 10:54
Dear Laurent: The only nice viewpt I know is as univ. deformation of standard 1-gon over a field. Doesn't get around formal vs. algebraic issue, but suggests the formal aspect shouldn't be hidden. Deformation viewpt also loses contact with $\mathbf{Z}$, & more importantly with element $q$ in base ring (which is part of what makes it useful, as in algebraic theory of $q$-expansions). So in the end I think of it as the version for standard 1-gon over $\mathbf{Z}$ of what deformation theory provides over a field, but the formal construction is needed to do anything serious with it. Sad, but true. –  BCnrd Aug 22 '10 at 3:38
Dear Brian: You are probably right, but I should add that the picture is not complete without the classical analytic side (this is the convergence question raised by Charles). Specifically, the construction should prove that the Tate curve over $\mathbb{Z}[[q]]$ descends to the subring $R$ of series converging on the unit disk. Then so do our isogenies becouse $R$ is an integrally closed (albeit non-noetherian) domain. –  Laurent Moret-Bailly Aug 22 '10 at 7:45
Thanks for your help, both of you. I had not realized how opaque Katz-Mazur's "black box" (as you call it, BCnrd) is. –  Charles Rezk Aug 22 '10 at 16:20