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Mon, Nov 11, 2019 page16
Entrepreneurs gather for business forum in Taipei
Staff writer, with CNA
About 20 business leaders from African countries, including Eswatini — Taiwan's sole diplomatic ally in Africa — South Africa and Nigeria, are to attend the Taiwan-Africa Business Forum in Taipei this week, the Ministry of Foreign Affairs said.
The Eswatini delegation, which is on a 10-day visit to Taiwan until Wednesday, is headed by Sibani Mngomezulu, chief executive officer of the Eswatini Investment Promotion Authority, the ministry said in a recent press briefing.
At the forum, entrepreneurs from the African nations are to meet with their Taiwanese counterparts to explore business and investment opportunities, the ministry said, adding that there is great potential for the development of trade and economic relations between Taiwan and Africa.
With an average 5.5 percent GDP growth, huge land mass, abundant natural resources and a young labor force, Africa has emerged as one of the fast-growing economic areas in the world, the ministry said.
The African Continental Free Trade Area, which took effect in May among 54 of 55 the African Union nations, is the largest of its kind in the world in terms of participating countries, after the WTO, the ministry said.
In view of these factors, Taiwan is hoping to build closer links with African nations, it said.
The ministry said that it would host a presentation on market conditions and business opportunities in Africa tomorrow, the second day of the business forum, which begins today and runs through Friday.
The forum was organized by the Taipei-based Africa Taiwan Economic Forum, the Ministry of Economic Affairs' Taiwan-Africa Trade Promotion Office and the Taiwan External Trade Development Council.
Major crash hits memorychip firms
This time was supposed to be different. The memorychip sector, famous for its boom-and-bust cycles, had changed its ways. A combination of more disciplined management and new markets for its products — including 5G technology and cloud services — would ensure that companies delivered more predictable earnings. Yet, less than a year after memory companies made such pronouncements, the US$160 billion industry is suffering one of its worst routs ever. There is a glut of the chips sitting in warehouses, customers are cutting orders and product prices have plunged. "The chip industry thought that suppliers were going to have better control," said
FDA approves first local-made vaccine for EV-71
Enimmune Corp (安特羅生技) has obtained marketing approval from the Food and Drug Administration (FDA) for its EnVAX-A71 vaccine for enterovirus 71 (EV-71), becoming the nation's first enterovirus vaccine completely made in Taiwan, it said yesterday. After spending 13 years and NT$1.5 billion (US$49.77 million) on the research and development of the vaccine, Enimmune plans to start manufacturing and marketing it by the end of March, the company said in a statement, without disclosing customer order figures. "It is possible that the vaccine would not be included in a national vaccination program initially, and consumers would need to pay for it themselves," parent
By Kao Shih-ching
Vaccine misinformation spawns 'pure blood' movement
Vaccine skeptics blocking transfusions for life-saving surgeries, Facebook groups inciting violence against doctors and a global search for unvaccinated donors — COVID-19 misinformation has bred a so-called "pure blood" movement. The movement spins anti-vaccine narratives focused on unfounded claims that receiving blood from people inoculated against COVID-19 "contaminates" the body. Some have advocated for blood banks that draw from "pure" unvaccinated people, while medics in North America say they have fielded requests from people demanding transfusions from donors who have not received a vaccine. In closed social media groups, vaccine skeptics — who brand themselves as "pure bloods" — promote violence against doctors
By Anuj Chopra and Marisha Goldhamer
Asteroid mining start-up to launch first missions
Asteroid mining start-up AstroForge Inc is planning to launch its first two missions to space this year as it seeks to extract and refine metals from deep space. The first launch, scheduled for April, is to test AstroForge's technique for refining platinum from a sample of asteroid-like material. The second, planned for October, would scout for an asteroid near Earth to mine. The missions are part of AstroForge's goal of refining platinum-group metals from asteroids, with the aim of bringing down the cost of mining these metals. It also hopes to reduce the massive amount of carbon emissions that stem from mining
By Loren Grush


There are a surprising number of places you can check for Latvian learning aids.
I'll be updating this frequently over the next few weeks as I add in all of my links and books.
I strongly recommend Jānis Roze. They will ship anywhere for close to the actual cost charged by Latvijas Pasts and have an absolutely stellar selection. If a book is really expensive here in the US, chances are you're paying a lot to those middlemen to import it.
Also, Zvaigzne ABC, a Latvian publisher, is now offering a small collection of ebooks through their Grāmatu Klubs (book club). The books seem to be only offered in Adobe Digital Editions format, many readers support this format, but check yours before buying.
Baltic Shop offers all sorts of merchandise from the Baltic countries. They have quite a few books, but I would check Jānis Roze and calculate shipping/currency conversion charges before buying.
The American Latvian Association has a fairly extensive bookstore with reasonable prices. They also have information on classes and schools offered around the country.
I am fully English and have been married to a Latvian for 9 years. I made a huge effort to learn and I would like to share some details of this journey with the blogger by e-mail.
Sure! I was going to direct you to the contact form (but you've handily worked around that, so I shan't worry about it) which is supposed to be on the menu but I don't know why it isn't. Thanks for drawing that to my attention and I'll be in touch with you shortly!


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/* This source file is distributed under the MIT license
 * (see attached LICENSE.txt file for details)
 */

#pragma once

#include "Control.h"

class NullControl : public Control
{
public:
    NullControl(QObject *parent = 0) : Control(parent) { /* Do nothing */ }

    virtual void processRequest(const Request& ) { /* Do nothing */ }
};

redpajama_set_name: RedPajamaGithub


figures is unambiguously positive for the Canadian dollar."
The Canadian dollar jumped sharply against the U.S.
dollar, which itself gained against the Japanese yen.
C$0.9735, its strongest level since Sept. 21.
cents, to yield 1.816 percent.
interest rates, an intention it repeated Thursday.
later and maintain its hawkish tone towards raising rates,"


Montblanc, held in Geneva in the organization of SHH-Time World Hemispheres series introduced for the first time. Then in the second SHH exhibition in Hong Kong this series Sinospher newly added models presented the of watch lovers. When we look at the clock 24-hour period emerges with sharp lines. Different cities of different countries about the time zone specified by giving the name of the city that we see. But here most interesting details of the Chinese name is highlighted in red. So China's main starting point was the time we can say that model. Already a clock face casing from the elements of until all other China made according to. The time red part of the night, the gray part represents the day. There are 24-hour period in the middle of the map of China. Besides all these, the image resembles sword indicators covering the deposit box brown crocodile leather strap red gold and the model gave a different feel. Again hour automatic caliber 4810/410 was used. As it is known that caliber, has complications WorldTimer.
Jewellery and watches skillfully combining rooted past when we say quality watch with one of the names that come to mind first Vacheron Constantin Malte Tourbillon Watches, Swiss brand is one of the most exclusive and attentive.Besides quality and robustness of the model is the product is completely handmade, processed diamonds in an invisible way, with an eye-catching curved model will attract everyone's attention. Quite a difficult job signatories designers, clock inch by inch in a skillful way by manipulating individually work on all the stones and they have signed the this wonderful watch models. 2795 The two-day power reserve caliber and with this model, which consists of 196 parts, black crocodile leather strap and perfect the mechanism of a high quality are waiting for loving users.


JERUSALEM/AMMAN (Reuters) - Hundreds of Syrian "White Helmet" rescue workers and their families fled advancing government forces and slipped over the border into Jordan overnight with the help of Israeli soldiers and Western powers, officials said.
Israeli Prime Minister Benjamin Netanyahu said in a brief video statement on Sunday he had helped the evacuation at the request of U.S. President Donald Trump and other leaders - and there had been fears that the workers' lives were at risk.
The group, known officially as Syria Civil Defence, has been widely hailed in the West and credited with saving thousands of people in rebel-held areas during years of bombing attacks by Damascus and its allies.
Its members, known for their white helmets, say they are neutral. But Syrian President Bashar al-Assad and his backers, including Russia, have dismissed them as Western-sponsored propaganda tools and proxies of Islamist-led insurgents. There was no immediate response from Damascus on Sunday.
A Jordanian government source said 422 people were brought from Syria, over the Israeli-occupied Golan Heights frontier and into Jordan, down from a figure of 800 announced earlier by the foreign ministry in Amman.
The evacuees will be kept in a "closed" location in Jordan and resettled in Britain, Germany and Canada within three months, the source said.
A second non-Jordanian source familiar with the agreement said the original plan had been to evacuate 800 people, but only 422 made it out as operations were hampered by government checkpoints and the expansion of Islamic State in the area.
Jordan's Foreign Minister Ayman al Safadi was quoted in a foreign ministry statement as saying he had discussed in a phone call with Russian Foreign Minister Sergei Lavrov the details of the mission to bring the rescue workers to Jordan.
Syria and its Russian allies have launched an offensive on rebels in the sensitive southwestern border zone.
Netanyahu said Trump, Canadian Prime Minister Justin Trudeau and others had spoken to him recently asking for help in extracting the White Helmets. "The lives of these people, who have saved lives, were now in danger. I therefore authorised their transfer via Israel to other countries as an important humanitarian gesture," Netanyahu said.
Trump did not mention the operation during a series of tweets on Sunday.
"While in Jordan, UNHCR in close collaboration with the involved governments, is supporting their temporary stay in anticipation of their onward relocation to third countries," the statement said.
Britain hailed the evacuation, saying it and other allies had requested it.
"Fantastic news that we - UK and friends - have secured evacuation of White Helmets and their families - thank you Israel and Jordan for acting so quickly on our request," tweeted British Foreign Secretary Jeremy Hunt.
German weekly magazine Bild, which broke news of the evacuation and published footage of buses used to transport the Syrians across Golan, said 50 of them would be granted asylum by Berlin.
"Humanity dictates that many of these brave first-aiders should now find protection and refuge, some of them in Germany," it quoted German Foreign Minister Heiko Maas as saying.
A spokeswoman for the German interior ministry said Berlin would take in eight White Helmets plus their families. It was not immediately clear whether that amounted to the same 50 people.
Russia's embassy in the Netherlands welcomed the White Helmets' departure. "Definitely there will be less chances of new so-called #CW (chemical weapon) attacks in Syria after forced evacuation by the collective West of the notorious #WhiteHelmets," it tweeted.
Damascus and its allies have accused rebels and their supporters of staging chemical attacks in an attempt to frame the government. Western powers have accused Syria of attacking rebel-held areas with illegal chemical weapons.


See the entirety of Ibiza's stunning coastline, take a swim, see old town Ibiza and return by magical sunset on this superb all-day trip.
Take advantage of a special early booking discount of 15% off the total price when you book before 30 April 2019.
Have an amazing day on the water for much less.
Seeing Ibiza from the coastline is an experience you won't forget and there is no better way to see it than to spend a whole day travelling around it. You will marvel at stunning beaches, secret coves, dramatic cliff faces and outlying islands in this terrific all-inclusive and great value trip.
You'll start on the virgin east coast of the island, taking in the beautiful and changing scenery as you sweep past, stopping to swim in front of the pretty islet of Tagomago before cruising on to Cala Martina for a paella lunch on-board.
Next it's an anchor drop in Ibiza Town to see the old town and ancient mediaeval citadel. If you don't feel like walking then you can simply browse one of the many boutiques there. Then it's a quick sail to Playa d'en Bossa for some water sports, including a terrific jet ski trip, that's co-piloted by a professional, so beginners need not fear. After that, it's a trip to see the marine area of Formentera's Ses Salines nature reserve.
You'll then continue back down Ibiza's southern coast for a stop close to the historic site of Atlantis before heading on to the mystical and magical Es Vedrà where you could get to touch the magic rock. Here you'll have another chance to take a dip in the sea. Happy and refreshed the trip continues into sunset, when you'll be treated to a barbecue and a live DJ session to end your perfectly sublime island voyage.
The Island Tour departs from San Antonio at 09:00 every Monday to Friday.
The boats used on the excursions are traditional Ibiza boats made by local craftsmen. Excursiones Ibiza is active in rescuing and preserving these boats and every year invests in restoring and maintaining them so you can enjoy an original Ibiza boating experience.
NOTE: Departures also available from Portinatx and San Miguel.
Recommendations: sun cream, sun hat, sun glasses, swimwear, towel, camera, a jacket or cardigan (temperatures drop when sailing back in the evening) and if you suffer from sea sickness Biodramina (the drug sold in Spanish pharmacies for sea and travel sickness).
Cancellation policy: If you cancel more than 24 hours before your date of departure as shown on your voucher, you will be fully refunded.


text:
At Long Last, Welcome to Eater Philly
Share All sharing options for: At Long Last, Welcome to Eater Philly
Eater Philly has arrived.
We're thrilled to launch here in Philly, where the most earnest and talented crop of young chefs in the country make their home and open restaurants. It's a bold statement, and absolutely true. But, Philly's food scene isn't just about the cooks and industry folks. We also boast the most dedicated, borderline-obsessed food fanatics that game restaurant openings like Hollywood movie premieres, and celebrate local Beard Award wins like World Series Championships. We're excited to be a part of it.
At Eater Philly, you won't find original restaurant reviews, recipes, or too much food porn; but we have you covered with unique reporting, curated round-ups of what the food people around town—and the nation—are talking about, and commentary on the absurdity of it all. Because there's plenty of that.
Eater Philly is happy to join the Eater National family of sites, including New York, Chicago, LA, SF, Miami, Austin, DC, Portland, and our fellow new kid on the block: Las Vegas. Do give them a click when you get a free moment, but not quite yet.
A little about me, your Eater Philly editor, Collin Flatt: I used to be the editor of The Feast, the now-defunct food site that was gobbled up by the ever-present Daily Candy. I can still be found teaching wine and beer classes at The Wine and Beer School of Philadelphia, and consulting on craft beer lists for various bars and restaurants around town.
While Eater Philly is your source for anything and everything worth hearing about in the Philly foodscape, we can't do it without our readers—whether you are an avid eater-outer or a pro with commentary on the industry or your very own workplace. No opening, closing, back-of-the-house mishap, crazy signage, menu typo, health code violation, celebrity sighting, shady behavior, or weirdness is off-limits. Actually, we like weird, and we love absurd.
So, do keep in touch with us and send us anything you think we'd love to hear about via our tipline: philly@eater.com. Help us help you. This will be fun.

redpajama_set_name: RedPajamaCommonCrawl


Do you have oily skin? Do you often complain about a shiny face or greasy, oily skin? Read on for the causes of oily skin and tips from Aloette experts on how to treat oily skin. Indeed, while problems don't usually begin until around puberty, for many people, oily skin persists long after the senior prom has faded into memory. For some, oily skin can last a lifetime. The flow of sebum or oil increases during adolescence and starts decreasing with age. The most important thing you can do for oily skin is to keep your skin clean using warm water and soap, or a soapless cleanser. Use only noncomedogenic moisturizers and sunscreen products.
Clean your face with astringent pads if frequently washing your face leaves your skin irritated. The right method is to clean the skin with a mild detergent that contains no fatty substances or oils. These kinds of detergents remove the excess sebum that cause grime accumulation, at the same time.
Causes of Oily skin Common causes of Oily skin: * Heredity. * Humidity and hot weather. Essential Oils for Oily Skin Citrus essential oils, such as lemon, grapefruit, tangerine and orange, can help with oily skin problems without the harsh effects of using the actual juice from the fruits themselves. Find an all-natural herbal lotion base and add 3 to 4 drops per ounce of the citrus essential oil of choice to the base. Use it on the face in the evening to help prevent oily skin.
Use only water-based or oil-free cosmetics if you have oily skin. If you have sensitive skin, use white or rose-colored clays. Diet probably has little, if anything, to do with whether or not you have oily skin. Try using a clay or mud mask.
When cleansing, massage your face well with your fingertips, using an upward and outward motion. Be careful not to rub soap into the skin; it can clog pores. Keep your skin very clean. Limit washing your face to two or three times a day.
Too much washing will stimulate your skin to produce more oil. Use hot water when washing your face. Hot water dissolves skin oil better than lukewarm or cold water. Drink plenty of water as oily skin may also be dehydrated - the oiliness of the skin will hide the lack of moisture. Try to drink 2 litres of water a day.
Wash Face Gently Do not overwash the face. Washing the face too frequently can actually cause the body to make more oil because it tricks the body into thinking the face and oil ducts are dry. When washing the face, avoid harsh soaps or chemicals, and settle for more natural alternatives or simply use warm water and a soft washcloth. Do not over scrub, but lightly and gently rub the face with the cloth.
Prevention is better than cure. Proper cleansing is necessary if you want to keep your oily skin clear. I have oily skin and suffered from acne. That cleared when I followed my doctors advice to wash my face morning, noon and at bedtime everyday, using an antibacterial face wash she dispensed. Actually, any antibacterial face wash would do.
If you don't have pimples, then any good cleanser meant for oily skin would do the trick.


Greystone Castle, the 2015 Orvis Wingshooting Lodge of the Year, is Texas' premier hunting destination, offering world-class shooting opportunities to an international clientele. Situated on a 6000 acre property 90 miles west of the Dallas/Fort Worth Metroplex, Greystone Castle is a sportsman's oasis in the small town of Mingus, TX. Dedicated upland gunners have long considered Greystone to offer some of the South's finest opportunities for cackling roosters and coveys of bobwhite quail. Guests follow elegant pointing dogs through acres of tended sorghum, millet, and switch grass cover, making memories and filling game bags before retiring to the castle for an unrivaled lodging experience. Hunters round out the day with hors d'oeuvres and cocktails in Greystone's state-of-the-art pavilion, comparing tales of great points and heroic shots, then retire to the formal castle dining room for a gourmet meal prepared by our executive chef. After dinner, guests enjoy a cigar and sunset beside the fire, or challenge friends to a game of pool or a poker game in the gun room.
A bird hunt at Greystone Castle is far more than a bird hunt; it is the stuff that memories are made of. Join Orvis in February, 2019 to enjoy this unique, world-class destination, and make some memories of your own.
"Exceptional guides, accommodations, and food. Facilities are the best of any lodge I have stayed at..." - James K.
PACKAGE INCLUDES: all meals, open bar, private room for two (2) nights, one (1) round of Sporting Clays, golf cart for Sporting Clays group, three (3) half-day hunts with unlimited birds, warm-up clays, transportation to and from the field, experienced guides & dogs, and all cleaning & packaging of game. Also includes all tax and gratuity.
DOES NOT INCLUDED: license fees, ammunition or gun rentals.


Soledrifter turns in another stellar single for Atlantic Standard Recordings, his third in the last 6 months alone. Beautiful beds of bass bump along while shiny chords and sultry vocal stabs deliver a slow reveal of mournful synth in a warm embrace. Stumbling ascending punches of modulated bass over an expanding canvas of sustained strings and distant voices will transport you to the far corners of the dancefloor.


text:
\section{Introduction}

Logarithmic Gromov--Witten theory concerns cycles on moduli spaces of stable maps from pointed curves $(C,p_1,\ldots, p_n)$ to a pair $(Z,D_Z)$, where $Z$ is a smooth variety and $D_Z$ is a simple normal crossings divisor. The logarithmic structure prescribes the tangency orders of each point $p_i$ along each divisor $D_j$. This tangency order is equal to the scheme theoretic contact order for non-degenerate maps, but also remains locally constant in families. The resulting moduli space $\mathsf K_\Gamma(Z)$ is equipped with a virtual fundamental class and invariants are integrals of evaluation cycles from $Z$ and tautological classes against the virtual class~\cite{AC11,Che10,GS13}.

A basic property of the virtual structure of the spaces $\mathsf K_\Gamma(Z)$ that has appeared in recent work in the subject is the behaviour under products~\cite{Herr,HPS19,LQ18,NR19}. We revisit the question in this note from the viewpoint of tropical moduli theory. Let $(X,D_X)$ and $(Y,D_Y)$ be smooth projective varieties equipped with simple normal crossings divisors, and let $(Z,D_Z)$ be the product.

\subsection{Modified products} Fix the genus, curve class, marked points, and their contact orders for maps to $Z$. This determines discrete data for maps to $X$ and $Y$ as well. We will package the discrete data in the symbol $\Gamma$, and use it flexibly to name the discrete data on moduli spaces of maps to $X,Y$, and $Z$. The moduli space of maps from \textit{smooth curves} to $Z$ with given contact orders is naturally identified with the fiber product the mapping spaces for $X$ and $Y$, over ${\cal M}_{g,n}$. This product decomposition breaks down over $\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$.

For simplicity, assume that the genus and markings are in the stable range. Given spaces $\mathsf K^\dagger$ and $\mathsf K$ equipped with virtual class, if there is a morphism $\mathsf K^\dagger\to \mathsf K$ such that pushforward identifies virtual classes, we say that $\mathsf K^\dagger$ is a \textit{virtual birational model} of $\mathsf K$. Our main result explains that the product formula continues to hold provided the moduli spaces are replaced by virtual birational models.

\begin{customthm}{A}\label{thm: product-formula}
There exists an explicit logarithmic modification of the moduli space of stable curves
\[
\begin{tikzcd}
\overline{{\cal M}}\vphantom{{\cal M}}_{\Gamma}\arrow{r}{\pi} &\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}
\end{tikzcd}
\]
and virtual birational models $\mathsf K_\Gamma(W)^\dagger\to\mathsf K_\Gamma(W)$ for $W = X,Y,Z$, fitting into a diagram
\[
\begin{tikzcd}
\mathsf K_\Gamma(Z)^\dagger\arrow{r}{\vartheta} & \mathsf P_\Gamma(X\times Y) \arrow{d}\arrow{r}\arrow[dr, phantom, "\square"] & \mathsf K_\Gamma(X)^\dagger\times \mathsf K_\Gamma(Y)^\dagger \arrow{d} \\
&\overline{{\cal M}}\vphantom{{\cal M}}_{\Gamma}\arrow{r}[swap]{\Delta} &\overline{{\cal M}}\vphantom{{\cal M}}_{\Gamma}\times \overline{{\cal M}}\vphantom{{\cal M}}_{\Gamma}. &
\end{tikzcd}
\]
There is an equality of virtual classes
\[
\vartheta_\star [\mathsf K_\Gamma(Z)^\dagger]^{\mathrm{vir}} = \Delta^![\mathsf K_\Gamma(X)^\dagger\times \mathsf K_\Gamma(Y)^\dagger]^{\mathrm{vir}} \ \ \mathrm{in} \ \ A_\star(\mathsf P_\Gamma(X\times Y);\QQ).
\]
\end{customthm}

\subsection{Geometry of the modification} The necessity of this modification can be easily captured in examples, and we comment further on it below. The modification $\overline{{\cal M}}\vphantom{{\cal M}}_\Gamma\to \overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$ is easily described. The morphisms $\pi_W: \mathsf K_\Gamma(W)\to \overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$ are typically ill-behaved at the level of logarithmic structures. This manifests concretely as follows. If $S$ is a codimension $k$ stratum of $\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$, its preimage, which is a logarithmic stratum, may not have virtual codimension $k$ in $\mathsf K_\Gamma(W)$. This is equivalent to the statement that the morphism  $\pi_W$ is not \textit{integral and saturated} as a logarithmic morphism, or that it is not flat with reduced fibers at the level of Artin fans~\cite{AW}.

Combinatorially, in the morphism of tropical moduli stacks associated to $\pi_W$, there are cones which do not map surjectively onto their target cone. The tropical moduli stack ${\cal M}_{g,n}^\mathrm{trop}$ can be subdivided until the image of $\pi_W$ is a union of cones for each $W = X,Y,Z$, i.e. \textit{along the image of the cycle of tropical maps of type $\Gamma$}. Toroidal geometry furnishes a birational modification $\overline{{\cal M}}\vphantom{{\cal M}}_\Gamma$ of $\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$, and there is a compatible refinement for mapping spaces.

In the next section we describe how such modifications can be interpreted as moduli functors on logarithmic schemes, carrying natural perfect obstruction theories, whose virtual classes have the expected properties.

\subsection{Toric contact cycles} We briefly recall that given a vector of integers $A\in \mathbb Z^n$ with vanishing sum, the \textit{double ramification cycle} $\mathsf{DR}_g(A)$ is cycle on the moduli space of curves whose restriction to $\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$ is the locus of curves that admit a map to $\mathbb P^1$ with ramificaition orders over $0$ and $\infty$ given by the positive and negative entries in $A$ along the corresponding markings. Let $\mathsf K_{g,A}(\PP^1)_{\mathbb G_m}$ be the moduli space of genus $g$  logarithmic stable maps to $\mathbb P^1$ relative to $0$ and $\infty$ with contact orders $A$, up to $\mathbb G_m$ scaling on the target. The double ramification cycle is the pushforward of the virtual class of this mapping space under
\[
\mathsf K_{g,A}(\PP^1)_{\mathbb G_m}\to \overline{{\cal M}}\vphantom{{\cal M}}_{g,n}.
\]
The class is known to lie in the tautological ring and a formula for it has been calculated~\cite{FP,JPPZ}.

We consider the following generalization. Fix contact order data $A_1$ and $A_2$ in $\mathbb Z^n$. Let $\mathsf K_\Gamma(\mathbb P^1\times \PP^1)_{T}$ be the moduli space of genus $g$ logarithmic stable maps to $\PP^1\times\PP^1$ with contact orders along the toric boundary given by $(A_1,A_2)$. We consider two maps equivalent if the differ by scaling under the $2$-dimensional dense torus action. Define the \textit{toric contact cycle} $\mathsf{TC}_g(A_1,A_2)$ to be the pushforward of the virtual fundamental class under the morphism
\[
\mathsf K_\Gamma(\mathbb P^1\times \PP^1)_{T}\to \overline{{\cal M}}\vphantom{{\cal M}}_{g,n}.
\]

\begin{customthm}{B}\label{thm: HPS-theorem}
There exists an explicit logarithmic modification of the moduli space of stable curves
\[
\begin{tikzcd}
\overline{{\cal M}}\vphantom{{\cal M}}_{\Gamma}\arrow{r}{\pi} &\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}
\end{tikzcd}
\]
and explicit lifts of the cycles $\mathsf{DR}(A_i)$ and $\mathsf{TC}(A_1,A_2)$ in the the Chow groups of $A_\star(\overline{{\cal M}}\vphantom{{\cal M}}_\Gamma;\QQ)$, such that, denoting the lifts by hats, there is an equality
\[
\widehat{\mathsf{TC}}(A_1,A_2) = \widehat{\mathsf{DR}}(A_1)\cdot \widehat{\mathsf{DR}}(A_2) \ \ \mathrm{in} \ A_\star(\overline{{\cal M}}\vphantom{{\cal M}}_{\Gamma};\QQ).
\]
\end{customthm}

The analogous statements for multifold products also holds. Note that a marked point may have nonzero contact with multiple divisors, so the statement for multifold products of $\PP^1$ implies the analogous result for arbitrary toric varieties, by virtual birational invariance~\cite{AW}.

\subsection{Recent results} The double ramification cycle can also be defined using a resolution of the Abel--Jacobi section from the moduli space of curves to the universal Picard variety, and in that context, the statement of Theorem~\ref{thm: HPS-theorem} has been proved recently in an elegant paper of Holmes--Pixton--Schmitt~\cite{HPS19}. In particular Section~8 of loc. cit. provides a concrete example exhibiting the failure of multiplicativity of the toric contact cycles. Our second result is a proof of their result from the Gromov--Witten viewpoint. The state of the art on resolutions of the Abel--Jacobi section may be found in~\cite{AP19, Hol17, MW17}.

Even more recently, Leo Herr has proved the product formula in logarithmic Gromov--Witten theory as an application of an elegant and general framework of logarithmic normal cones and logarithmic perfect obstruction theories. These yield a notion of \textit{virtually logarithmically smooth morphisms}~\cite{Herr}. If the logarithmic obstruction theories are unwound into a piece coming from an ordinary perfect obstruction theory and another coming from a logarithmic blowup, one is led to a formula of the form presented here. In contrast, the results here are presented using the traditional framework of \textit{virtually smooth} morphisms. We hope that the presentation here explains the need and naturality of a logarithmic Chow theory in the work of Herr, as well as earlier work of Barrott~\cite{Bar18,Herr}.

The idea of using tropical geometry to correct the product formula appeared also in our earlier work with Nabijou, proving the local/logarithmic conjecture~\cite{NR19}. The basic strategy in that paper is to establish a version of the product formula for different logarithmic structures over the same base manifold, and therefore relate the logarithmic Gromov--Witten theory with the older relative theory for smooth pairs.

Finally, this paper serves as a companion to~\cite{R19}, which establishes the degeneration formula for logarithmic Gromov--Witten invariants using expanded degenerations. Products of smooth pairs provide basic examples of normal crossings pairs, and the central strategy required to prove the product formula appears in loc. cit. In particular, we rely on the correcting ``naive'' intersections of classes by strict transform calculations attached to subdivisions to obtain both results.

\subsection{Earlier work on products}  When the logarithmic structures on $X$ and $Y$ are both trivial, the tropical part of the geometry disappears and we recover Behrend's product formula for ordinary Gromov--Witten theory, and indeed its proof~\cite{Beh97}. The product formula also holds for orbifolds~\cite{AJT}. If $X$ alone has trivial logarithmic structure, the product formula holds without birational modifications, and was proved by Lee and Qu~\cite{LQ18}. In genus $0$, for toric targets there is a product decomposition for the moduli space of logarithmic maps, using the logarithmic torus~\cite{RW19}.

\subsection*{Acknowledgements}  My view on this subject has been shaped by conversations with Davesh Maulik in the context of earlier work on the degeneration formula, and I'm grateful to him for his generosity with time. I thank Dan Abramovich for the comment, ``it would all be fine if the stabilization morphism was flat, but it probably fails in general'', which finally clarified the geometry. I have also benefited from conversations with Tom Graber, Andreas Gross, Navid Nabijou, Rahul Pandharipande, and Jonathan Wise. I thank Jonathan Wise for keeping me informed of the parallel work of his student Leo Herr.

\section{Subdivisions of moduli problems}\label{sec: subdivisions}

Given a moduli space ${\cal M}$ representing a fibered category over schemes, a blowup $\widetilde{{\cal M}}$ does not come equipped with a natural modular interpretation. However, if ${\cal M}$ is a logarithmic scheme representing a fibered category over \textit{logarithmic schemes}, a logarithmic blowup can be described as a \textit{subcategory} of the fibered category ${\cal M}$. This fact was observed by Kato~\cite{Kato-LogMod}, and utilized in logarithmic moduli problems in~\cite{MW17,R19,RSW17A,RSW17B}. We begin by reviewing these ideas.

\subsection{Subdivisions: cone complexes and logarithmic schemes} Let $\Sigma$ be a rational polyhedral cone complex. This defines a functor from the category of rational polyhedral cones to sets:
\begin{eqnarray*}
F_\Sigma: \mathbf{Cones}&\to& \mathbf{Sets}\\
\tau&\mapsto& \mathbf{Hom}(\tau, \Sigma).
\end{eqnarray*}
A key observation is to note that if $\widetilde\Sigma\to \Sigma$ is a subdivision of cone complexes, then $F_{\widetilde \Sigma}$ is a \textit{subfunctor} of $F_\Sigma$.

The link to logarithmic geometry arises as follows. Assume that $\Sigma$ is a fan embedded in a vector space, and let $X = X(\Sigma)$ be the toric variety associated to $\Sigma$. The scheme $X$ inherits a natural logarithmic structure from the toric boundary divisor, and therefore defines a functor on logarithmic schemes
\begin{eqnarray*}
F_X: \mathbf{LogSch}&\to& \mathbf{Sets}\\
S&\mapsto& \mathbf{Hom}(S, X),
\end{eqnarray*}
where the morphisms are taken to be in the category of logarithmic schemes.

Each logarithmic scheme $S$ comes equipped with a tropicalization $S^{\mathrm{trop}}$, which we may assume is a single cone. There is a natural tropicalization map carrying a logarithmic scheme to its underlying polyhedral functor~\cite{CCUW,Kato94}. Kato observed that there is an identification
\[
F_{\widetilde X} = F_X\times_{F_{\Sigma}} F_{\widetilde\Sigma}.
\]
In other words, just as a a subdivision of a cone complex is a subfunctor of the original cone complex, a toric modification of a toric variety, interpreted as a functor on logarithmic schemes, is a subfunctor of the original functor.

This formalism does not require $X$ to be toric, only that it has a logarithmic structure. A logarithmic structure on $X$ gives rise to a functor on the category of cone complexes whose $\RR_{\geq 0}$ points is its tropicalization. Given a logarithmic scheme $X$, a \textit{logarithmic modification} is by pulling back the subfunctor defined by a refinement on its tropicalization. It is a basic observation that these subfunctors are in turn obtained from schemes with logarithmic structure. That is, they are representable by schemes with logarithmic structure~\cite{Kato-LogMod}.

\begin{remark}
The interpretation of logarithmic modifications as subfunctors, or monomorphisms of logarithmic schemes, is lost once we pass to the representing ordinary scheme. This manifests combinatorially. Passing from the logarithmic scheme to its underlying scheme is analogous to replacing a fan with its partially ordered set of faces. Any subdivision of polyhedral complexes is injective, but the map of fans induces a map on partially ordered sets that is no longer an injection.
\end{remark}

\subsection{Logarithmic and tropical moduli of curves} In our examples, the logarithmic scheme above will be replaced by a moduli space of curves or of logarithmic stable maps to a logarithmic scheme $X$. The fan side is controlled by tropical moduli theory, which we briefly recall. We follow standard conventions for tropical curves~\cite{CCUW}.

\begin{definition}\label{def: trop-curve}
An \textbf{$n$-marked tropical curve} $\scalebox{0.8}[1.3]{$\sqsubset$}$ is a finite graph $G$ with vertex set $V$ and edge set $E$, enhanced with
\begin{enumerate}
\item \textbf{markings} $m: \{1,\ldots,n\}\to V$,
\item a \textbf{vertex genus function} $g:V\to \NN$,
\item an \textbf{edge length function} $\ell: E\to \RR_{+}$.
\end{enumerate}
The \textbf{genus} of $\scalebox{0.8}[1.3]{$\sqsubset$}$ is equal to
\[
g(\scalebox{0.8}[1.3]{$\sqsubset$}) = h_1(G)+\sum_{v\in V} g(v).
\]
\end{definition}

The edge length function gives $\scalebox{0.8}[1.3]{$\sqsubset$}$ the structure of a metric space, where marked points are realized by attaching copies $\RR_{\geq 0}$ to appropriate vertices, as ``legs''. There is a notion of a family of tropical curves.

\begin{definition}
Let $\sigma$ be a cone with dual cone $S_\sigma$. A \textbf{family of tropical curves over $\sigma$} is a tropical curve in the sense above, but whose length function takes values in in $S_\sigma$.
\end{definition}

A point of $\sigma$ is a monoid homomorphism $\varphi: S_\sigma \to \RR_{\geq 0}$, and applying it to the edge length $\ell(e)\in S_\sigma$, we obtain a positive real length for each edge and thus a tropical curve.

Given a family of logarithmic curves $\mathscr C/S$, there is a tropicalization $\scalebox{0.8}[1.3]{$\sqsubset$}/S^\mathrm{trop}$, which is a family of tropical curves with the natural genus and marking data. When $S$ is a single point with logarithmic structure, the tropicalization is obtained by decorating the edges of the dual graph of $\mathscr C$ with the data of the deformation parameters of the nodes. The procedure is treated in detail by~\cite{CCUW}

We may view $\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$ as a category fibered in groupoids over logarithmic schemes, and to emphasize this, we temporarily introduce the notation ${\cal M}_{g,n}^{\mathrm{log}}$. There is a tropical moduli stack ${\cal M}_{g,n}^\mathrm{trop}$. This tropical moduli stack is a category fibered in groupoids over the category of rational polyhedral cone complexes~\cite{CCUW}. There is a surjective morphism
\[
{\cal M}_{g,n}^{\mathrm{log}}\to {\cal M}_{g,n}^\mathrm{trop}
\]
by taking a family of logarithmic curves to its tropicalization.

\subsection{Minimality and tropical deformations} The moduli space of stable curves $\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$ can be recovered as a subcategory of ${\cal M}_{g,n}^{\mathrm{log}}$. Let $S = \spec(P\to \CC)$ and let $\mathscr C/S$ be a logarithmic curve. The dual graph of $\mathscr C$ determines a cone of the tropical moduli space ${\cal M}_{g,n}^\mathrm{trop}$, which we denote $\sigma(\mathscr C)$.

The logarithmic curve over a logarithmic point is \textit{minimal} if the dual monoid $\mathrm{Hom}(P,\RR_{\geq 0})$ is isomorphic to $\sigma(\mathscr C)$. That is, in the corresponding family of tropical curves, there are no unexpected relationships between the edge lengths. A family is said to be minimal if it is minimal at every point. F. Kato characterizes the moduli space of stable curves as follows.

\begin{theorem}[{~\cite{Kat00}}]
The moduli space of stable curves $\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$ can be identified with the open substack of ${\cal M}_{g,n}^{\mathrm{log}}$ parameterizing minimal logarithmic curves.
\end{theorem}

In other words, there are no relations between the deformation parameters of the nodes of $\mathscr C$.

\subsection{Mapping spaces} We also require the parallel statements for the moduli spaces of stable maps to logarithmic schemes. Let $\Sigma$ be a fan and let $\scalebox{0.8}[1.3]{$\sqsubset$}$ be a tropical curve.

\begin{definition}
A \textbf{tropical map} from $\scalebox{0.8}[1.3]{$\sqsubset$}\to \Sigma$ is a continuous piecewise linear map such that every face of $\scalebox{0.8}[1.3]{$\sqsubset$}$ maps to a face of $\Sigma$.
\end{definition}

Families of tropical maps are defined in the analogous fashion, and one obtains a moduli space $\mathsf T_\Gamma(\Sigma)$ of tropical stable maps.

\begin{remark}
Observe that continuity of the map $\scalebox{0.8}[1.3]{$\sqsubset$}\to \Sigma$ imposes nontrivial restrictions on the edge lengths of $\scalebox{0.8}[1.3]{$\sqsubset$}$. For instance, if a collection of edges $e_1,\ldots, e_k$ form a cycle in $\scalebox{0.8}[1.3]{$\sqsubset$}$, continuity forces that the total displacement around the cycle is zero, and therefore a linear relationship between the edge lengths of the $e_i$.
\end{remark}

Given a logarithmic scheme $X$ with tropicalization $\Sigma$, there is a fibered category $\mathsf K^{\mathrm{log}}_\Gamma(X)$ over logarithmic schemes, whose fiber over a logarithmic scheme is the groupoid of families of logarithmic curves over $S$ together with a map to $X$. There is again a morphism
\[
\mathsf K^{\mathrm{log}}_\Gamma(X)\to \mathsf T_\Gamma(\Sigma).
\]

Abramovich--Chen and Gross--Siebert have developed a theory of minimal (resp. basic) logarithmic maps~\cite{AC11,Che10,GS13}. There is a subcategory $\mathsf K^{\mathrm{}}_\Gamma(X)$ of minimal logarithmic maps from within the category all logarithmic maps $\mathsf K^{\mathrm{log}}_\Gamma(X)$. A family is said to be minimal if, at each geometric point, the associated tropical family is a cone of this tropical moduli space.

\begin{remark}
The dual viewpoint to the statement above is that {a logarithmic family of maps is minimal precisely when the associated tropical family of maps is maximal.} It is immediate from this that if a family of logarithmic stable maps to $X$ is minimal, the underlying family of curves is typically not minimal when $X$ has nontrivial logarithmic structure. Equivalently, in the induced map from $\mathsf T_\Gamma(\Sigma)\to \mathfrak{M}_{g,n}^\mathrm{trop}$, there are cones of the source that do not surject onto their target cones.
\end{remark}

\subsection{Variation of minimality and subcategories} Subdivisions of tropical moduli spaces produce subcategories in the following manner. Given a refinement of cone stacks
\[
{\cal M}^\mathrm{trop}_\Gamma\to{\cal M}_{g,n}^\mathrm{trop},
\]
base change along the tropicalization map ${\cal M}_{g,n}^{\mathrm{log}}\to {\cal M}_{g,n}^\mathrm{trop}$ gives rise to a subcategory
\[
{\cal M}^{\mathrm{log}}_\Gamma\to {\cal M}_{g,n}^{\mathrm{log}}.
\]
Inside ${\cal M}^{\mathrm{log}}_\Gamma$, we may pick out a further subcategory of $\Gamma$-minimal objects. Specifically, an object in this category is still a family of logarithmic curves $\mathscr C\to S$ over a logarithmic base. The tropicalization is then a family $\scalebox{0.8}[1.3]{$\sqsubset$}\to S^\mathrm{trop}$, and we obtain a moduli map
\[
S^\mathrm{trop}\to {\cal M}^\mathrm{trop}_\Gamma.
\]
A family of curves is $\Gamma$-minimal if, at every point of $S$, the tropicalization of the base is identified with a cone of the moduli space ${\cal M}^\mathrm{trop}_\Gamma$. This subcategory of $\Gamma$-minimal objects is representable by a stack, and in fact by a logarithmic modification of $\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$.

Once again, there are parallel statements for tropical maps, which we employ in the next section.

\section{Curves in product}

To begin, we examine the product formula in an entirely combinatorial setting. Fix fans $\Sigma_X$ and $\Sigma_Y$ associated to the target varieties $X$ and $Y$, and let $\Sigma_Z$ be their product. As in the introduction, fix discrete data $\Gamma$, which, in a mild abuse of notation, will be used to describe the compatible discrete data on $X$, $Y$, and the product.

\subsection{Combinatorial products}\label{sec: comb-prod} We require a basic finiteness statement.

\begin{lemma}
Fix discrete data $\Gamma$. The cone complex $\mathsf T_\Gamma(\Sigma_Z)$ associated to the space $\mathsf K_\Gamma(Z)$ of logarithmic stable maps to $Z$ is a finite-type cone complex.
\end{lemma}

\begin{proof}
The targets have simple normal crossings logarithmic structures, so under these hypotheses, the statement is equivalent to the combinatorial finiteness proved by Gross and Siebert~\cite[Section~3.1]{GS13}.
\end{proof}

The forgetful morphism $\mathsf K_\Gamma(Z)\to \overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$ induces a morphism of cone complexes
\[
\mathsf T_\Gamma(\Sigma_Z)\to {\cal M}_{g,n}^\mathrm{trop}.
\]

\begin{lemma}
The image of the forgetful morphism
\[
\mathsf T_\Gamma(\Sigma_Z)\to {\cal M}_{g,n}^\mathrm{trop}.
\]
is supported on a finite type conical subset of the cone complex ${\cal M}_{g,n}^\mathrm{trop}$.
\end{lemma}

\begin{proof}
This follows immediately from the combinatorial finiteness in the previous lemma.
\end{proof}

Although the image of the morphism is a conical subset, it is not a \textit{subcomplex} of the tropical moduli space of curves.

\begin{lemma}
There exists a smooth (unimodular) subdivision of conical subcomplexes
\[
{\cal M}_\Gamma^{\mathrm{trop}}\to {\cal M}_{g,n}^\mathrm{trop}
\]
such that the images of the morphisms
\[
\mathsf T_\Gamma(\Sigma_Z)\to {\cal M}_{g,n}^\mathrm{trop}, \ \ \ \mathsf T_\Gamma(\Sigma_X)\to {\cal M}_{g,n}^\mathrm{trop}, \ \ \textrm{ and} \ \  \ \mathsf T_\Gamma(\Sigma_Y)\to {\cal M}_{g,n}^\mathrm{trop}
\]
are each unions of cones in the subdivision ${\cal M}_\Gamma^{\mathrm{trop}}$.
\end{lemma}

\begin{proof}
The lemma is a consequence of~\cite[Lemma~4.3]{AK00}, which asserts that for a morphism of polyhedral complexes, there is always a projective subdivision of the target that ensures that the image of each cone of the source is a union of cones. Toric resolution of singularities guarantees the existence of a smooth subdivision.
\end{proof}\bigskip

\begin{center}
{\it
From this point forward, we fix a choice of subdivision ${\cal M}_\Gamma^{\mathrm{trop}}$ of the tropical moduli space of curves realizing the lemma above.
}
\end{center}
\bigskip

As constructed, the morphism
\[
\mathsf{T}_\Gamma(\Sigma_X)\to {\cal M}_\Gamma^\mathrm{trop}
\]
may not be a morphism of cone complexes, as the image of a cone may be a union of cones, and therefore not contained in a single cone. However, we choose subdivisions $\mathsf{T}_\Gamma(X)^\dagger$, $\mathsf{T}_\Gamma(Y)^\dagger$, and $\mathsf{T}_\Gamma(Z)^\dagger$ that make these morphisms maps of cone complexes. Indeed, this may be obtained as the fiber product in the category of cone complexes. By construction, the cone complexes satisfy the following condition:

\begin{lemma}
Each of the morphisms
\[
\mathsf T_\Gamma(\Sigma_Z)^\dagger\to {\cal M}_{\Gamma}^\mathrm{trop}, \ \ \ \mathsf T_\Gamma(\Sigma_X)^\dagger\to {\cal M}_{\Gamma}^\mathrm{trop}, \ \ \textrm{ and} \ \  \ \mathsf T_\Gamma(\Sigma_Y)^\dagger\to {\cal M}_{\Gamma}^\mathrm{trop}
\]
have the property that they map cones surjectively onto cones.
\end{lemma}

\subsection{Virtual strict transforms} The formalism in Section~\ref{sec: subdivisions}, together with the subdivision ${\cal M}_\Gamma^{\mathrm{trop}}$ produced in the previous section, give rise to a birational models of the moduli spaces of curves and maps. These will be the spaces claimed to exist in the introductory statements.

In order to lift our combinatorial statements to virtual classes, we use a formalism developed by Abramovich and Wise~\cite{AW}. If $W$ is a toric variety with dense torus $T$, the \textit{Artin fan} of $W$ is the toric Artin stack $\mathscr A_W = [W/T]$. Both $W$ and $\mathscr A_W$ are equipped with divisorial logarithmic structure, and the morphism
\[
W\to \mathscr A_W
\]
is strict. The Artin fan is logarithmically \'etale over a point. For an arbitrary logarithmic scheme $W$, there is a replacement of the global quotient construction above, producing an Artin fan, which continues to be an Artin stack,  logarithmically \'etale over a point; $W$ has a strict morphism
\[
W\to \mathscr A_W.
\]
The stack $\mathscr A_W$ is essentially combinatorial, constructed from the local toric models that give charts for the logarithmic structure for $W$. Details may be found in~\cite{ACMUW}.

Manolache has a defined a functorial virtual pullback in Chow homology for morphisms equipped with relative perfect obstruction theories~\cite{Mano12}.

\begin{proposition}
Let $W$ be logarithmically smooth. The moduli stack of logarithmic maps $\mathsf K_\Gamma(\mathscr A_W)$  from curves to $\mathscr A_W$ is a  logarithmically smooth stack that is locally of finite type. The natural morphism
\[
\mathsf K_\Gamma(W)\to \mathsf K_\Gamma(\mathscr A_W)
\]
has a natural relative perfect obstruction theory. The virtual pullback of the fundamental class is equal to the virtual fundamental class of $\mathsf K_\Gamma(W)$.
\end{proposition}

The virtual class defined above has natural lifts to any logarithmic modification. Any subdivision
\[
\mathsf T_\Gamma(\Sigma_W)^\dagger\to \mathsf T_\Gamma(\Sigma_W)
\]
induces a logarithmic modifications
\[
\mathsf K_\Gamma(\mathscr A_W)^\dagger\to \mathsf K_\Gamma(\mathscr A_W)
\ \ \textrm{and} \ \ \ \mathsf K_\Gamma(W)^\dagger\to \mathsf K_\Gamma(W)
\]
by pulling back~\cite{AW,Kato94}. The moduli space $\mathsf K_\Gamma(\mathscr A_W)^\dagger$ is certainly logarithmically smooth, since it is a logarithmically \'etale modification of a logarithmically smooth stack.  We summarize the observations concerning the virtual structure of this stack from~\cite[Section~3.5]{R19}.

\begin{proposition}
The morphism $\pi: \mathsf K_\Gamma(W)^\dagger\to \mathsf K_\Gamma(\mathscr A_W)^\dagger$ has a relative perfect obstruction theory. The virtual pullback of the fundamental class defines a virtual fundamental class
\[
[\mathsf K_\Gamma(W)^\dagger]^{\mathrm{vir}}:=\pi^{!}[\mathsf K_\Gamma(\mathscr A_W)^\dagger]
\]
for $\mathsf K_\Gamma(W)^\dagger$. Moreover, pushforward along the modification
\[
\mathsf K_\Gamma(W)^\dagger\to \mathsf K_\Gamma(W)
\]
identifies virtual classes.
\end{proposition}

The virtual class $[\mathsf K_\Gamma(W)^\dagger]^{\mathrm{vir}}$ functions as a \textit{virtual strict transform} of the virtual class on $\mathsf K_\Gamma(W)$.

\subsection{Proof of Theorem~\ref{thm: product-formula}} Let $\overline{{\cal M}}\vphantom{{\cal M}}_\Gamma$ be the logarithmic modification of $\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$ defined by pulling back the tropical subdivision ${\cal M}_\Gamma^{\mathrm{trop}}\to {\cal M}_{g,n}^\mathrm{trop}$. Consider the following commutative diagram.
\[
\begin{tikzcd}
\mathsf K_\Gamma(Z)^\dagger\arrow{d}{\varphi}\arrow{r}{\vartheta} & \mathsf P(X\times Y) \arrow{r} \arrow{d} & \mathsf K_\Gamma(X)^\dagger\times \mathsf K_\Gamma(Y)^\dagger\arrow{d}{\phi} \\
\mathsf K_\Gamma(\mathscr A_Z)^\dagger \arrow{r}[swap]{\nu} & \mathsf P(\mathscr A_X\times \mathscr A_Y) \arrow{r}[swap]{g} \arrow{d} & \mathsf K_\Gamma(\mathscr A_X)^\dagger\times \mathsf K_\Gamma(\mathscr A_Y)^\dagger\arrow{d}{\psi} \\
& \overline{{\cal M}}\vphantom{{\cal M}}_\Gamma \arrow[swap]{r}{\Delta} & \overline{{\cal M}}\vphantom{{\cal M}}_\Gamma\times\overline{{\cal M}}\vphantom{{\cal M}}_\Gamma.
\end{tikzcd}
\]

We are now led to the main result, which asserts that after these logarithmic modifications, the virtual pullbacks in the above diagram are compatible.

\begin{theorem}
There is equality of Chow homology classes on the space $\mathsf P(X\times Y)$:
\[
\vartheta_\star [\mathsf K_\Gamma(Z)^\dagger]^{\mathrm{vir}} = \Delta^! [ \mathsf K_\Gamma(X)^\dagger\times \mathsf K_\Gamma(Y)^\dagger]^{\mathrm{vir}}.
\]
\end{theorem}

\begin{proof}
We first inspect the vertical map $\psi$, which, on each factor forgets the data of the map and stabilizes if necessary. We claim that this map is flat with reduced fibers. In order to see this, we first note that the morphism is a toroidal morphism of toroidal embeddings. This is established in~\cite[Proposition~3.2]{AW}. We can therefore use the polyhedral criteria for equidimensionality~\cite[Lemma~4.1]{AK00} and reducedness~\cite[Lemma~5.2]{AK00}. These are satisfied by construction.

The flatness of $\psi$ leads to the main conclusions necessary for the proof. The flatness and reducedness implies that the two squares on the right are Cartesian in the categories of fine and saturated logarithmic stacks, as well as in ordinary stacks. The morphisms $\varphi$ and $\phi$ each give relative obstruction theories and these are compatible, and the proof of compatibility is identical to~\cite[Proposition~6]{Beh97}. As noted above, the morphism $\psi$ is a flat morphism by construction, and therefore $g$ is the flat base change of $\Delta$. In turn, $\Delta$ is the diagonal morphism on a smooth Deligne--Mumford stack, and is therefore a local complete intersection morphism. Since the property of being local complete intersection is stable under flat base change, we see $g$ is also a local complete intersection morphism. The obstruction theories given by $\Delta$ and $g$ are therefore compatible~\cite[\href{https://stacks.math.columbia.edu/tag/069I}{Tag 069I}]{stacks-project}.

Finally, we consider the morphism $\nu$. Since the stack $\mathsf P(\mathscr A_X\times \mathscr A_Y)$ is a fiber product in ordinary schemes,  it parameterizes data
\[
(C,\widetilde C_1,\widetilde C_2, \widetilde C_1\to\mathscr A_X,\widetilde C_2\to \mathscr A_Y).
\]
That is families of curves, together with two destabilizations of the curve, respectively equipped with logarithmic morphisms to $\mathscr A_X$ and to $\mathscr A_Y$. We identify the tropicalization of this moduli problem analogously, as parameterizing data of piecewise linear families
\[
(\scalebox{0.8}[1.3]{$\sqsubset$},\widetilde \scalebox{0.8}[1.3]{$\sqsubset$}_1,\widetilde \scalebox{0.8}[1.3]{$\sqsubset$}_2, \widetilde \scalebox{0.8}[1.3]{$\sqsubset$}_1\to\Sigma_X,\widetilde \scalebox{0.8}[1.3]{$\sqsubset$}_2\to \Sigma_Y).
\]
Let $\sigma$ be the base of such a family. Note that fiberwise over $\sigma$, $\widetilde \scalebox{0.8}[1.3]{$\sqsubset$}_1$ and $\widetilde \scalebox{0.8}[1.3]{$\sqsubset$}_2$ are each simply subdivisions of $\scalebox{0.8}[1.3]{$\sqsubset$}$.  In particular, after a further subdivision of $\sigma$, there exists a common refinement $\widetilde \scalebox{0.8}[1.3]{$\sqsubset$}$ of these two curves, admitting maps to $\Sigma_X$ and $\Sigma_Y$, and therefore to $\Sigma_Z$. Observe that the morphism
\[
\mathsf K_\Gamma(\mathscr A_Z)\to \mathsf P(\mathscr A_X\times \mathscr A_Y)
\]
is pulled back from the morphism of their tropicalizations. The argument above shows that the analogous morphism of tropicalizations is a subdivision. The pullback is therefore a logarithmic modification, and we conclude that $\nu$ has pure degree $1$.

The product formula now follows from a standard diagram chase, using the established compatibilities. The fact that $\nu$ is pure degree $1$ shows that
\[
\vartheta_\star[\mathsf K_\Gamma(Z)^\dagger]^{\mathrm{vir}} = \phi^![\mathsf P(\mathscr A_X\times \mathscr A_Y)]
\]
by an application of Costello's pushforward theorem~\cite[Theorem~5.0.1]{Cos06}. We also have the equality
\[
\phi^![\mathsf P(\mathscr A_X\times \mathscr A_Y)] = g^![\mathsf K_\Gamma(X)^\dagger\times\mathsf K_\Gamma(Y)^\dagger],
\]
by functoriality of virtual pullbacks~\cite[Theorem~4.1]{Mano12}. The result follows.

\end{proof}

\subsection{Toric contact cycles} To deduce Theorem~\ref{thm: HPS-theorem}, we apply the argument with a minor change. Specifically, for the rubber geometry $\mathsf K_{\Gamma}(\PP^1)_{\mathbb G_m}$, we need a replacement for $\mathsf K_\Gamma(\mathscr A_X)$. The latter can be identified as fibered category over logarithmic schemes whose fibers are $(C,\scalebox{0.8}[1.3]{$\sqsubset$}\to \RR)$, where $C$ is a logarithmic curve, $\scalebox{0.8}[1.3]{$\sqsubset$}$ is its tropicalization, and the map $\scalebox{0.8}[1.3]{$\sqsubset$}\to \RR$ is piecewise linear, balanced, with asymptotic slopes given by $A$.

We replace $\mathsf K_\Gamma(\mathscr A_X)$ with the category $\mathsf K_\Gamma(\mathscr A_X)_{\mathbb G_{\mathrm{trop}}}$ where (1) the data $\scalebox{0.8}[1.3]{$\sqsubset$}\to \RR$ is only required to be determined up to additive translation on the target, and (2) $C$ is required to be stable. The corresponding category of minimal objects of schemes is an open subset of a blowup of $\overline{{\cal M}}\vphantom{{\cal M}}_{g,n}$. The moduli space is constructed in detail in~\cite[Section~5]{MW17}. With this replacement, an identical diagram chase as above yields the claimed result.

\qed

\begin{remark}
The cycle of curves in any toric variety, not necessarily one of product type, can be decomposed into products of strict transforms of double ramification cycles. Indeed, for $X$ toric of dimension $r$, equipped with its toric boundary logarithmic structure, the moduli space $\mathsf K_\Gamma(X)$ of logarithmic maps is virtually birational to $\mathsf K_\Gamma((\PP^1)^r)$ by the invariance statements of~\cite{AW}. The result above may then be applied to each factor of the product.
\end{remark}

\subsection{A sample subdivision} The subdivisions constructed in Section~\ref{sec: comb-prod} can be described explicitly for the double ramification and toric contact cycles. We include a figure below, which describes a piece of the subdivision for genus $2$ curves mapping to rubber $\PP^1$ with degree $3$, totally ramified over $0$ and $\infty$.

\begin{figure}[h!]
\begin{tikzpicture}
\tikzset{me/.style={to path={
\pgfextra
 \pgfmathsetmacro{\startf}{-(#1-1)/2}
 \pgfmathsetmacro{\endf}{-\startf}
 \pgfmathsetmacro{\stepf}{\startf+1}}
 \ifnum 1=#1 -- (\tikztotarget)  \else
     let \p{mid}=($(\tikztostart)!0.5!(\tikztotarget)$)
         in
\foreach \i in {\startf,\stepf,...,\endf}
    {%
     (\tikztostart) .. controls ($ (\p{mid})!\i*6pt!90:(\tikztotarget) $) .. (\tikztotarget)
      }
      \fi
     \tikztonodes
}}}
\node[circle,fill=black,inner sep=1pt,draw] (a) at (0,0) {};
\node[circle,fill=black,inner sep=1pt,draw] (b) at (3,0) {};

\draw (0,0)--(-0.5,0);
\draw [densely dotted] (-0.5,0)--(-0.7,0);
\draw (3,0)--(3.5,0);
\draw [densely dotted] (3.5,0)--(3.7,0);
\draw (a) edge[me=3] (b);

\draw [densely dotted] (3.5,-1.5)--(3.7,-1.5);
\draw (3.5,-1.5)--(-0.5,-1.5);
\draw [densely dotted] (-0.5,-1.5)--(-0.7,-1.5);

\end{tikzpicture}
\caption{A dual graph of a relative stable map to rubber $\PP^1$ of degree $3$ with total ramification. The slopes along the outer edges are each $3$ while the slopes along the internal edges are all $1$.}\label{fig: degree-3-cover}
\end{figure}

The cone associated to the cover in Figure~\ref{fig: degree-3-cover} is a ray, since by continuity of the tropical map, the three bounded edges must have the same edge length. However, this ray maps into a $3$-dimensional cone in ${\cal M}_{2,2}^\mathrm{trop}$. This $3$-dimensional cone is dual to a codimension $3$ stratum, consisting of curves with two genus $0$ components and three mutual nodes. Perform a stellar subdivision of this cone. After this subdivision, the ray of the tropical rubber mapping space now maps onto a subcomplex of this cone. Similar subdivisions can be made in other cones of the moduli space.

After subdividing ${\cal M}_{2,2}^\mathrm{trop}$, we obtain a birational model $\overline{{\cal M}}\vphantom{{\cal M}}_\Gamma$ of the space of stable curves. The refinement may be pulled back to the tropical rubber space. The resulting virtual birational model of the rubber mapping space has ``virtually proper intersections'' with strata. That is, the preimage of a stratum under the map
\[
\mathsf K_{\Gamma}(\PP^1)_{\mathbb G_m}\to \overline{{\cal M}}\vphantom{{\cal M}}_\Gamma,
\]
if nonempty, has the expected virtual dimension. The pushforward of the virtual class is a \textit{transverse double ramification cycle}, which we view as a strict transform of the usual double ramification cycle under the blowup. Our main theorem asserts that products of pullbacks (under point-forgetting) of such transverse cycles are toric contact cycles. We note that the combinatorics that arises in identifying the locus in the tropical moduli space of curves that admit a piecewise linear map to $\RR$ with the given slopes is essentially a manifestation of the \textit{admissible weightings} that appear in Pixton's formula for the double ramification cycle~\cite{JPPZ}.

\bibliographystyle{siam}

redpajama_set_name: RedPajamaArXiv


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The iStar Charity Foundation - Hoops For Kids
By Daniel Dodaro and John Michael Dodaro
Think of a time when you knew that someone was in need and you didn't know how to help. Did you think about helping them by playing basketball! You could do both by playing basketball and looking up the iStar Charity Foundation.
This special foundation helps many people in need involving children and basketball. Some of the Knicks and Harlem Wizards players volunteer to come to this event every year at Madison Square Garden. These players play basketball with different sponsors and kids' organizations to raise money. Everyone comes to have fun, meet some of the players and visiting stars and at the same contribute to iStar.
My brother John and I were excited to be part of such a worthy cause. This year the charities they were raising money for included Baby Buggy, Human Rights first, Robin Hood, Garden of Dreams and Teach for America. The Knicks and the Harlem Wizards have been supporting the iStar Charity Foundation for seven years. Companies such as Morgan Stanley and the Sugarman Foundation donated funds to the organizations and participated in the event.
The energy at the Madison Square Garden iStar charity event was charged. There were many different kid teams of all ages roaring to get to play with the Knicks and Harlem Wizards to raise money for this event. Besides the excitement of playing basketball, the kids were entertained by the dancing group, the Fab 5 and the rap group, Kid Rocke.
The Knicks players that we interviewed were very kind and had a soft spot for raising money for kids. My brother and I asked the Knicks Legend, Charles Smith, "Why did your team pick these four charities out of many others?"
Smith answered, "I still feel part of the Garden family and I play here. The Garden of Dreams is one of my charities that I like to work with. Any time they do something in collaboration with any other charitiy, I like to get involved and support them."
When asked how these organizations affected his life, he immediately answered, "I know they had an impact on a lot of organizations and a lot of kids. When you look at what these children's charities are doing; just the name alone, the Garden of Dreams, explains it. It helps kids know they have a frame to meet a certain player or travel to a certain place. They make dreams come true."
John then asked Charles Smith, "What is the greatest part of being a professional basketball player?"
Smith answered, "The greatest part of being a professional basketball player is you are considered the best at what you do in the world. If someone was an investment banker they would want to be one of the top three or four hundred investment bankers. You become an elite class where you can say you are the best."
We also interviewed another Knicks legend, Anthony Mason. We asked him at what age did you begin to play basketball.
"I was a little different. Baseball and football were my first sports. I didn't start playing basketball until my junior year in high school. I pretty much was one of the lucky ones."
Our next question was, "How did you transition from football and baseball to basketball?"
Mason answered, "Part of it was where I was living. I moved back to Jersey and when I was young, football and baseball were the big sports. Then I moved to Queens and I said let's play some baseball and football. They said that was weird. So basketball became the thing to try."
I then asked Mason, "Do you get to choose what charities you sponsor or do the managers?"
Mason replied, "Actually, it's pretty much both. I love to look out for kids as far as education, and helping them accomplish their dreams. At times of crisis they know that we are there."
We then wanted to know which charity he stands for the most.
Mason responded, "I would say the Garden of Dreams because it facilitates everything within the Garden. I feel that we (the players) have accomplished our dreams, so it's our job to try to help kids accomplish their dreams."
I wondered if any of these types of organizations helped Mason growing up.
Mason then said, "The Boys Club and stuff like that kept me out of trouble. That's why I'm have a real fascination about helping kids. When my mother was at work I would have gotten into a lot of trouble doing other things. So the Boys Cub was real good to me."
I then wanted to know how it feels to represent these charities.
"Mason said, "Any time you can help kids you can see the bright smile and dreams come true. We have been so blessed as professional athletes. It is always good to give back! The National Boys Club helped keep me off the streets. In essence, because of them I became a professional basket ball player."
My brother and I also interviewed a newer Knicks Player, Danilo Gallinari. We asked, "What is the greatest part of being a professional basketball player?"
He answered, "I think it was the best thing that ever happened to me. "I started playing basketball when I was five years old."
The last Knicks player we interviewed was Wilson Chandler. We asked him some questions too! We wanted to know how these charity organizations affected his life."
"It helped me communicate with the younger people. Not many people get to become a professional basketball player," he said.
We wanted to know what the greatest part of being a professional basketball player was.
He responded, "I really like playing all the time; that's the greatest part of basketball! I was around twelve years old when I began to play."
He also told me how he felt about representing these charities.
He explained, "It feels good to be in this position that we are in, because when I was young I didn't have anyone come around and help me."
One of the kids I spoke with was a 10 year old boy named Joshua who went to school at PS 142 in New York. This was the first time he did this.
He said "I like this event and I am happy to be here. I started playing basketball when I was nine."
He said he was excited, and happy to be raising money for charity and to play with his icon, Gallinari. He then went out on the court and played with his team and the Knicks players. Each group raised money every time they scored a basket.
We also spoke to 13 year old Jack Stallsmen from Trinity School in N.Y.C. He was very excited to be there and stated his top player was LeBron James.
He stated, "This is my third time at this event. I am anxious to shoot some hoops with the players and raise some money."
We also interviewed Jack, 11, from New York City, who has never done this sort of thing before. He said he is inspired by Carmello Anthony. When Jack's team was called to play, he rushed on to the court with his teammates and gave it his all.
We asked Tojo from the Harlem Wizards how do these organizations affect his life.
He replied, "It gave him a chance to travel all over the world. To do what I loved to do the most and that is basketball, and I have a lot of fun."
We asked him if he performs at different charities throughout the year, and he answered, "We go to many charities throughout the year and work a lot with Special Olympics and Children's Miracle Network and many other charities."
We then inquired what was the greatest part of being a professional basketball player.
Tojo answered,"Well, for me especially playing for the Wizards and bringing smiles to the faces of the kids. It's great to make a living doing what you love to do so much. Most charities have special meanings, especially the ones we do with Special Olympics, diabetes, and cancer. We also do a lot of work with United Way. My family has been affected by cancer and diabetes. It's a great feeling having a chance to give back. "
There were stars there lending their support to the event. Kidsnewsinc.com got to interview Hugh Jackman, Vivica Fox, and Emme!
Hugh Jackman was one of the stars that came out to lend his support for iStar. This was his first iStar Charity Foundation game. If you have seen the X-Men movies, he plays Wolverine and recently was seen in the movie, Australia, and many other films. He was a real gentleman and he was very friendly and easy to speak to. He came to the event with his family and participated in rating the players. As a Laker's fan, Mr. Jackman told us in his opinion that the guard was the most important position in basketball but he feels that Stephon Marbury is the best player on the Knicks.
He stated, "As an ambassador for the World Mission, I want to help disadvantaged kids all over the world. I also want all kids to have the same chances in life."
Vivica Fox and Hugh Jackman both rated the players and the kids at the event. Vivica Fox also lent her vivacious personality to this event. She was lots of fun to talk to with lots of energy and opinions. She played basketball in 1982 in the City Girl Championships in Indianapolis, Indiana. Vivica is a big basketball fan. Her top player on the Knicks is Jared Jeffries. Ms. Fox refers to him as .J.J.
Vivaca stated that in her opinion, "The center is the most important position on a basketball team. The center sets up the entire game for rebounds, sets up for defense and gets the ball back to offense."
Model, Emme, was also there lending her support with her daughter Tobey. She played very vigorously before she came to speak to us. Tobey told us that she loves the Harlem Wizards.
"I think the coach is the most important part of any team and I think the guard is the most important player on the team," Emme said.
In all, the charity raised over $350,000. This was accumulated with half-court shots, free throws and games between the players, the kids' teams and the celebrities. It was a great experience to be involved in. We even got to shoot basketballs! This event teaches us all to give our best for our fellow man and to give it your best in all you do.


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We pay tribute this month to David Bintley, who stepped down as director of Birmingham Royal Ballet on June 29 after 24 years of service. He was, of course, associated with the company long before that, both as a dancer and choreographer when it was named Sadler's Wells Royal Ballet, but – as can be seen in his interview with Paul Arrowsmith – he plans to continue with his career as a choreographer for the foreseeable future.
We also report back on the 2019 Blackpool Dance Festival, learn about the commercial dance scene, and find out about Iain MacKay's plans for Yorkshire Ballet Seminars, which return to Askham Bryan College this month. In addition, there is an extended Dance Scene section, highlighting the vast amount of dance that has recently been on offer on stage in the UK and abroad.
Finally, to mark this month's Pride celebrations in the UK, we include an article – requested by our readers – on voguing, a dance style that has recently made a comeback in the popular imagination. Enjoy the summer!
JONATHAN GRAY
Dancing Times, July 2019 Volume 109 Issue 1307
15 – Missing the hurly-burly. Paul Arrowsmith asks David Bintley to look back at his time as director of Birmingham Royal Ballet
19 – Turning point. Jonathan Gray talks to Iain Mackay about his ambitions for the Yorkshire Ballet Seminars
23 – Strike a pose. Nicola Rayner explores the re-emergence of voguing
76 – Blackpool 2019! Martin Cutler reports from Blackpool Dance Festival
26 – Be your own brand. Laura Dodge discovers the growing commercial dance industry
29 – Kalamata – what's not to like? Paul Arrowsmith meets Linda Kapetanea, director of the Kalamata Dance Festival
33 – Collective interest. Fátima Nollén looks at Ecuador's contemporary dance scene
36 – Method of the Maestro. Diane van Schoor talks to Jonathan Gray about the release on DVD of Complete Cecchetti Diploma
Dance Today
80 – All change, please. Alison Gallagher-Hughes attends the British Open
82 – Tips on technique. By James Whitehead
83 –Technique clinic. By Phil Meacham
84 – Simon's Guide to Swing. By Simon Selmon
85 – Stepping Out. By Marianka Swain
86 – Notes from the Dance Floor. By Vikki Jane Vile
88 – Somewhere in time. By Jack Reavely
6 – News. Northern Ballet's 50th anniversary plans
10 – Letters
12 – Talking Point. By Northern Ballet's Gavin McCaig
39 – Dance Scene. Reviews include Seeta Patel, Rambert, Northern Ballet, Rosie Kay Dance Company, Masters of Choreography, The Royal Ballet, San Francisco Ballet and Norwegian National Ballet
67 – Variety Lights. By Barbara Newman
69 – FRANCE/dance. By Laura Cappelle
71 – Letter from St Petersburg. By Igor Stupnikov
73 – Notes from New York. By Jack Anderson
91 – Dancer of the Month. Margaret Willis meets The Royal Ballet's Joseph Sissens
95 – People. Marcelino Sambé, the Queen's Birthday Honours
96 – Obituaries. John O'Brien
99 – Media. Nicola Rayner reviews Force of Nature Natalia
101 – Products
103 – Education. British Ballet Grand Prix
108 – Health. Debbie Malina considers the development of the Dance for Parkinson's Network
111 – Classified
112 – Calendar. Our guide to what's on this month in the UK and abroad
118 – Listings
122 – Last Dance. We look back to July 1979
Simon Oliver has been production editor of Dancing Times since 2010 and is highly experienced in design across print and online magazine production. Throughout his career, Simon has worked on a diverse range of subjects including music, family history, book collecting and poker.
Nourishment for the imagination


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How the Nazis Failed to Destroy the Life and Legacy of Artist David Friedmann
Daughter of famed Holocaust-era artist on a mission to find his Nazi-looted and lost artwork
By Miriam Friedman Morris
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David Friedmann in 1936 in his apartment in Berlin-Wilmersdorf. In the background, his painting of the Berlin Cathedral appears. After World War II, it was found in his sister-in-law's apartment. Friedmann's painting of the Schlossbrücke und Zeughaus (castle bridge and arsenal), today the German Historical Museum, also appears. His paintings are among hundreds of Nazi-looted and lost artworks | Photo: Courtesy National Library of Israel
Since childhood I watched my father paint with an intensity and passion that struck a chord within me. I was intrigued by his successful prewar career and the fate of his Nazi-looted art. He had little to show from a collection of hundreds of paintings, drawings, lithographs and etchings. This fueled my passion to find these works and to rescue him from obscurity.
David Friedmann was born on December 20, 1893, in Mährisch Ostrau, Austria-Hungary, now Ostrava, Czech Republic. He studied etching with Hermann Struck and painting with Lovis Corinth in Berlin. He painted some of the most important events in modern history, surviving World War I and World War II as an artist. Friedmann produced late impressionist landscapes, still lifes, interiors, nudes and achieved acclaim as a painter known for his portraits drawn from life. He exhibited at the Akademie der Kunst, Berliner Secession and numerous galleries throughout Germany and Czechoslovakia.
His use of light and dark, his ability to convey expressions on faces, the composition, are all hallmarks of his work. With pencil and paper, he captured the great chess champions of the 1920s. In 1924, his quick-sketching skills launched a secondary career as a freelance press artist. He sketched hundreds of famous contemporary personalities from the arts, music, theater, sports, politics, and industry, published mainly in the Berlin newspapers and the radio-program magazine, Der Deutsche Rundfunk. Among the portrayed luminaries were Albert Einstein, Arnold Schoenberg, Thomas Mann, Max Liebermann and Emanuel Lasker.
Friedmann's flourishing career in Berlin was terminated in 1933 by the Nazi regime. As each of his options narrowed, he continued to produce art illustrating the events and his personal experiences of the time. In 1938, Friedmann fled with his family to Prague, escaping from the Nazis with only his artistic talent as a means to survive. He depicted human fate as a refugee in Prague, as a prisoner in the Lodz Ghetto, in the Auschwitz subcamp, Gleiwitz I, and as a survivor. His wife Mathilde and little daughter Mirjam Helene were murdered in Auschwitz.
In 1941, the Gestapo looted his left-behind oeuvre in Berlin. He lost his studio furniture and materials, hundreds of oil paintings, watercolours, drawings, etching prints and lithographs. After Friedmann's deportation to the Lodz Ghetto, Nazi authorities looted his Prague art production.
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In 1946, when mail service from Berlin to Prague was finally restored, Friedmann received portrait prints and photos of his work in an album. The Prague portraits dated 1940 to 1941 gave face to numerous known and unknown victims — historically significant evidence of a dynamic Jewish community destroyed by the Nazi regime. Additional portrait prints were found at the National Museum in Prague, Beit Theresienstadt in Givat Haim (Ihud), Israel, and in two family-owned collections. Numerous works, including portraits and landscapes, surfaced at the Jewish Museum in Prague.
Artwork was systematically confiscated and sold at auction by the Nazi regime. The whereabouts of the remainder of Friedmann's looted art is unknown.
"Richard Réti at the Chessboard", Lithograph, 1923. This appeared in a portfolio entitled "Das Schachmeister Turnier in Mährisch Ostrau" and alternatively "Köpfe berühmter Schachmeister". Five portfolios have been found | Photo: © Miriam Friedman Morris; image courtesy of the National Library of the Netherlands
From his incarceration period, a portrait drawing of a Polish prisoner in Gleiwitz I was discovered at the State Museum Auschwitz-Birkenau in Poland. Evidence also surfaced of Friedmann's work in the ghetto. His 1942 etching of the Lodz Ghetto bridge appeared as a header on pages of The Chronicle of the Lodz Ghetto, 1941-1944. A handmade album with thirty-three drawings documenting the activities of a hat-manufacturing workshop ("ressort") in the Lodz Ghetto in 1943 is also held in the collection of the Jewish Historical Institute in Warsaw.
Liberated at age 51, an age significantly older than most survivors, Friedmann believed there was a reason he lived. The responsibility to bear witness weighed heavily on his conscience even before deportation. His burning desire was to show to the world the ruthless persecution and inhumanity as practiced by the Nazis, in the hope such barbarism would never happen again. Friedmann captured the scenes he could not erase from his memory — forced labor, torture, killings and the death march. He called the series, Because They Were Jews!
Descriptive title: "At the Water's Edge", Oil on wood panel. Signed Dav. Friedmann lower left and dated 1932
Friedmann continued to paint throughout his postwar journey. In 1948, in Prague, he wed Hildegard Taussig, a survivor of several concentration camps. Their marriage began at a refugee's pace. One year later, the couple fled communist Czechoslovakia to Israel, where their daughter, also named Miriam, was born. He worked in a sign shop and contributed to the founding of Israel's commercial art industry.
Every spare moment he painted for himself. Friedmann's colour palette changed to brighter, sun-filled hues as he left behind his old dark world to explore his newly adopted country. After two years, he established his own advertising business and freelanced for the newspapers, permitting more time for artistic pursuits. Besides portraits, he painted landscapes of Lake Kinneret, Jaffa, Haifa, Tel Aviv, Netanya, Naharia and Tiberias. Some works are signed "Dfri" in Hebrew letters Daled, Peh, Resh, Yod.
This colourized drawing is from a 1943 album by David Friedmann documenting the activities of a hat-manufacturing workshop ("ressort") in the Lodz Ghetto | Photo: E. Ringelblum Jewish Historical Institute, Warsaw; Inventory No. MŻIH B-419/24)
He also enjoyed painting the Yarkon River views and Hadar Yosef, where we lived. Sympathetic to the impoverished Jews who had emigrated from Yemen, he portrayed beggars on the streets to express their plight. David Friedmann had captured the landscape of the beginnings of the Jewish state. Decades later, I had immense pleasure tracking down the dramatically changed scenery he painted, now difficult to find or nonexistent.
Israel was a new state in poor economic circumstances. Undeterred by his being 61-years-old, Friedmann set his ambitions on America, arriving in New York in 1954. He had to forget what was hidden in his heart, the paintings from the concentration camps and make a living. Straight from the boat, he auditioned for the billboard company, General Outdoor Advertising (GOA). He painted as fast as possible because only this would save our family from poverty. GOA did not care about his age or that he barely spoke English.
"Havel River Landscape, Berlin", Oil, 1923. This painting hung for decades in the home of Andrea Kress, who became curious about David Friedmann. She learned about the artist's daughter's pursuit of lost art and sent this photo.
They were impressed with an accomplished artist who painted with astonishing speed — the same skill that saved his life in 1944 in Gleiwitz I where Friedmann had improvised with primitive materials, making his own paints and brushes out of camp supplies to paint a mural across a barrack's wall in order to show the SS officers his artistic ability and spare him from death. What could he produce to impress them? He thought of the Havel River, painted in Berlin with "white clouds in the blue sky, trees, and in-between a few small houses with red roofs, water, white sailboats and their reflections on the surface of the water."
GOA moved the family first to Chicago and then to St. Louis. After only fifteen months in America, Friedmann had been appointed to the top artist position at this branch. Instead of pictures from the concentration camps, he painted the iconic Clydesdales and happy folks selling beer on two-story-tall billboards. The new career brought recognition and satisfaction with life in America. In 1960, the Friedmann family became proud United States citizens and symbolically dropped the double "n" spelling of the surname.
After retirement in 1962, his art would not be silent. He produced a second series of Holocaust art to fight antisemitism and race hatred of all people. The David Friedman Exhibition opened in Baltimore, Maryland in 1965, marking 20 years after liberation, and was even reported in the Israeli press.
David Friedmann adds final touches to his charcoal drawing, "Liberation?" The artist depicts himself as the prisoner with eyeglasses | Photo: Peter Rosvik, St. Louis, Missouri, 1964
Friedmann died at the age of eighty-six on February 27, 1980. He is recognized internationally with works on permanent display at the Holocaust History Museum, Yad Vashem, Jerusalem; the St. Louis Holocaust Museum & Learning Center; and the Sokolov Museum in the Czech Republic. His works are in the collections of the State Museum Auschwitz-Birkenau, Poland; and the United States Holocaust Memorial Museum in Washington DC, among other institutions and museums. Exhibition venues include the Berlin Philharmonic Hall in Germany, the Terezín Memorial in the Czech Republic, the United Nations Headquarters and the German Consulate General in New York.
In 1954, Friedmann was among the first to win restitution from Germany for Nazi-looted art. The sum incorporated claims for all his looted property. He continued to fight for justice. In 1961, the International Supreme Restitution Court in Berlin adjudicated an upward adjustment.
David Friedmann was a successful artist with both Jewish and non-Jewish clientele. Art was sold privately, at galleries, exhibitions and auctions. Fleeing the German Reich, most emigrants found it necessary to sell their art to finance an escape. Others managed to flee with their art.
Artwork often continues to find new owners — sold at auction or through private sales — purchased by people who are not known as collectors. Pieces are displayed on walls of family homes for generations, art they enjoyed all these years, not knowing the paintings have a history and the artist's daughter is searching to find them. David Friedmann artwork has surfaced all over the world — the Netherlands, Denmark, Austria, Belgium, England, France, Germany, Italy, Czech Republic, Poland, Israel, Australia, China, Canada and the United States. I have started to find his prewar art just over the last two decades.
Every painting to emerge is a victory against the German Reich. David Friedmann made important contributions both in the realms of 20th-century art and in the creation of materials that play a powerful humanitarian role in educating people about the reality of the Holocaust.
My goal is to publish a catalogue of his works, evidence of the brilliant career the Nazis could not destroy.
For more about David Friedmann and to provide information you may have about existing works, please visit: www.davidfriedmann.org or the "David Friedmann—Artist As Witness" Facebook page.
This article has been published as part of Gesher L'Europa, the National Library of Israel's initiative to share stories and connect with people, institutions and communities in Europe and beyond. It originally appeared in The National Library of Israel website.
Miriam Friedman Morris is the daughter of Auschwitz survivors David and Hildegard Friedman(n). Art shaped her life. Fascination with her father's Nazi-looted art launched a four-decade quest to find lost works and ensure his rightful place in history. Her pursuit launched a worldwide revival of an artist obscured by the Nazi regime. It even led to an episode "Holocaust Diary" on the Fox Business News show, Strange Inheritance. Morris, who lives in New York, facilitates exhibitions, lectures and writes — and is dedicated to the preservation of her father's art legacy. She has co-written a book, traveled the world, is a recognized lecturer and author of several articles. Morris is currently writing a book about her father's life through art: http://z.umn.edu/davidfriedmann.
Tags: David FriedmannHolocaust ArtMiriam Friedman MorrisNazi Confiscated Art
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'Essential' Status
Employees & Visitors
COVID-19 response & readiness:
As shown in the service map below, Douglas has 7 service regions in the United States as well as coverage for Canada and Mexico. All regions are supported by an experienced and talented team of remote and factory-based service technicians. Despite the potential for further travel restrictions we are confident that these resources can continue to support any service need you may have within the U.S.
The following key service capabilities and hours (CST) remain in effect:
Remote troubleshooting from 6:00 a.m. to 4:30 p.m.
Live phone support 24 hours per day, 7 days per week.
Service parts shipments from 7:00 a.m. to 5:30 p.m. 5 days per week; airline or courier flights can be arranged on an as-needed basis outside of standard operating hours.
Online troubleshooting based on machine capability and connectivity availability; as detailed in the 'Risk Mitigation' section below, this is an area we continue to enhance.
Need to contact Douglas service personnel? Main Service Line: 320.763.5507
At present, we have not seen and are not anticipating disruptions to service levels. In reality we are currently experiencing excess capacity due to cancelled trips and travel/visitor restrictions.
To support the social distancing guidance our current travel preference is via rental car. However, absent future government-mandated flight restrictions we will travel via air in cases of emergency.
We are following the US State Department's level 4 travel advisory which dictates that our Service Technicians remain in the United States. However, we are working on an as-needed basis with members of our Sales team to support customers outside the United States.
Potential risks
Looking forward, there is a possibility that domestic air travel becomes further restricted. If this happens, our technicians will be forced to drive, which could delay normal arrival times.
As our customers' production facilities grapple with the threat of direct and/or indirect exposure to COVID-19, we are taking steps to ensure the continued safety of our traveling service technicians. We will soon roll out a process that requests information transparency from plants that our technicians plan to visit. While we don't anticipate problems with this approach we recognize that is an additional layer of communication that needs to be navigated prior to initiating service trips.
For customers outside the United States, third party partners are being utilized to manage all service requests. While international requests continue to be fulfilled, in some cases there have been delays due to this arrangement.
Long-term, we have also considered the potential for service shortages in the event that restrictions lift and normal business conditions resume. In this scenario we anticipate reacting to high levels of pent up service demand.
Service offerings to mitigate risk
Many of our customers are working hard to keep up with surging demand. At the same time, needed COVID-19 safety policies place limitations on suppliers that are allowed to enter their facilities.
With these constraints in mind, our Technical Services team has been hard at work leveraging existing remote service support capabilities and developing new ones - all with an eye toward maintaining excellent levels of service. Below, we list the remote support capabilities and offerings we provide to our customers. If your company has a preference for other tools, let us know - we're flexible.
We've expanded our investment and utilization of Zoom, one of the world's leading web conferencing solutions. With access to fully equipped ZoomRooms and to individual service support personnel, customers can engage Douglas experts in a virtual support environment via voice, video, chat, screen shares, and live annotations - all with the simple click of a Douglas-provided link.
To ensure hassle-free connection and engagement, Zoom work instructions are also available.
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Ewon is a full end-to-end remote access router that allows our technical experts to access your machine's diagnostic information via a simple and secure connection. Specifically, Douglas utilizes the Ewon Cosy 131 Router. No IT configuration tools or command-line skills are required to implement and/or utilize this highly secured, ISO27001 and STAR certified product.
For customers who already have machines with Ewon routers, we will provide work instructions to guide the interaction and ensure a smooth connection with a member of our Technical Services team.
For customers that do not have machines equipped with an Ewon router, Douglas will provide a kit that enables Ewon connection for temporary use during this pandemic. This kit includes everything you'll need to get up and running with a remote connection to a Douglas technician. (Note: a $1,500 refundable security deposit will be required to receive this kit.)
For more information on Ewon, please visit their website, contact your Service Advisor, or reach out to one of the Technical Services leaders listed on our Key Contacts page.
For secure remote PC access and support, Douglas can also deploy TeamViewer. Like Ewon, TeamViewer allows Douglas technicians to do more than just show and tell, it allows qualified Douglas personnel to guide and direct a customer's PC when connected to a machine's controller and its diagnostic information.
Complete work instructions are available for TeamViewer download, install, and use.
For those customers that prefer to maintain control of network access, Douglas can also connect and engage through Virtual Private Networks, or VPNs. This option allows all of the benefits of Ewon and/or TeamViewer, plus the added sense of security that comes with controlling the means of access to your network.
To limit the financial burden of remote connectivity and engagement during these unprecedented times, we're taking some liberties with our fee structures. Until travel conditions improve or Douglas determines there is a need for change, service support via Zoom, Ewon, TeamViewer and/or VPN will be conducted free of charge. To learn more, please contact your Service Advisor or one of the Technical Services leaders listed on our Key Contacts page.
As always, given the dynamic and shifting environment we all find ourselves in, we encourage customers to communicate proactively with our Technical Services team whenever possible.