Breaking international economic news from this past week has caused Las Vegas-based Nine Elms to increase efforts in the jewelry and mining sectors.

The established UK landscaping supplier has achieved the world's most recognised quality management system standard, ISO 9001, as issued by the British Standards Institution (BSI).

Recognition Earned for Exceptional Growth and Performance as an IT Channel Provider

The Wait is Over - Dubba-AA drops 'Flex Tape 3', the longtime anticipated solo project Drops July 24th - Sounds by DJ Scream

Drain 247 Group have achieved UKAS approved certification to ISO 9001:2015 and ISO 14001:2015. There are many drainage companies with ISO certification but very few drainage companies in the UK have been able to achieve UKAS approved certification.

How can you keep your focus and drive during a global crisis? Take on a 90-day learning challenge for data science and check out this list of books and courses to follow.

While tax-exempt not-for-profit organizations don’t typically have federal tax liability, most are still responsible for filing an annual information return with the IRS. The IRS Forms 990-EZ, and 990 are public documents that include information about the organization’s mission, programs… Read More

The post Using Form 990 to Market Your Not-for-Profit appeared first on Anders CPAs.

If you have ever caught even one minute of the history channel, you have seen fraktur. You’ve seen the font on Nazi posters, on Nazi office buildings, on Nazi roadwork signs. Today in Germany, blackletter typefaces are frequently used by Neo-Nazi groups and for many Germans, they bring to mind the dark times of the country’s fascist past. This is ironic because fraktur has a long and strange history that includes the font actually being banned by the Nazis.

Plus, we get an opinion from Kate Wagner (McMansion Hell) about “Making Federal Buildings Beautiful Again.”

Fraktur

First Ride: Triumph Tiger 900 Rally Pro & Tiger 900 GT Pro

First Ride review: 2020 KTM 200 Duke & 390 Duke BS6

The borrower has SBI Max Gain home loan at 8.55 per cent under the MCLR option.

Making products meet international standards requires ISO Certifications, but getting it is an expensive affair. To get around this, the government provides reimbursement to SMEs seeking ISO 9000/ISO 14001Certification.

The issue price for Series I (April 20 to 24, 2020) was Rs 4,639 per gram of gold.

The issue price for Series I (April 20 to 24, 2020) was Rs 4,639 per gram of gold.

WBAL NewsRadio 1090 hosted a town hall with a number of WBAL listeners to discuss President Donald Trump’s first 100 days in office.

I am testing this new format for when I travel. So here are, as a test, my ramblings from the Viewfinder Villa kitchen table. Links: Download the MP3 for this episode Receive TFTTF updates via email

The post 890 New Format (Slice) appeared first on PHOTOGRAPHY TIPS FROM THE TOP FLOOR.

The Miaimi Dolphins announced Monday that coach Don Shula has died. He was 90 years old. 

A new National Park Service (NPS) report shows that 5,969,811 visitors to Grand Canyon National Park in 2016 spent $648,170,900 in communities near the park. https://www.nps.gov/grca/learn/news/econ-benefit-2016.htm

Not only did the patients experience high rates of remission, their cognitive function also showed marked improvement as well.

The post Magnetic Brain Treatment Found to Relieve Depression in 90% of Participants With No Negative Side Effects appeared first on Good News Network.

Seniors in the 80s, 90s, and 100s are surviving COVID-19 coronavirus and share advice on how to handle stress of cancelled weddings, and the future.

The post Elders Around the World in Their 80s, 90s, and 100s Are Bouncing Back From Virus – and Sharing Advice appeared first on Good News Network.

The New Zealand Cabinet will meet on Monday to decide whether restrictions can be eased allowing domestic travel to restart and most businesses to open. There have been four new recorded infections in the past five days, and 90 percent of approximately 1500 confirmed or probable cases, have recovered from the virus. As the New Zealand and Australian economies reopen, a Trans-Tasman travel bubble could emerge as a serious possibility, if both nations continue to effectively flatten their coronavirus curves. Image: Associated Press

Epson this week announced the SureColor P700 and P900 printers, updates to their SureColor 13- and 17-inch, pro-level photo printer line. The new desktop printers offer a smaller footprint; a new pigment-based inkset with 10 inks; improved color gamut; and enhanced connectivity and paper-handling options. In addition — and possibly the most important enhancement for […]

The post Epson debuts SureColor P700 and P900 photo printers appeared first on Complete Digital Photography.

The quick bread recipe tastes delicious and requires just five ingredients

Estimates of forest carbon stores and flux for California circa 1990 were modeled from forest inventory data in support of California's legislatively mandated greenhouse gas inventory. Reliable estimates of live-tree carbon stores and flux on timberlands outside of national forest could be calculated from periodic inventory data collected in the 1980s and 1990s; however, estimation of circa 1990 flux on national forests and forests other than timberland was problematic owing to a combination of changing inventory protocols and definitions and the lack of remeasurement data on those land categories. We estimate annual carbon flux on the 7.97 million acres of timberlands outside of national forests (which account for 24 percent of California's forest area and 28 percent of its live tree aboveground biomass) at 2.9 terragrams per year.

Longtime NEW YORK and syndicated talk show host BARRY FARBER died TUESDAY (5/5) on his 90th birthday, his daughter CELIA posted on FACEBOOK YESTERDAY. FARBER, a 2014 inductee of the NATIONAL … more

Manager Darren Mullen also delivered a club gift to the lifelong supporter

At the turn of the 20th century, a vaccine had been developed for smallpox, a virus that killed millions in the 1800s. Those who survived the disease were often left badly scarred or blind. The...

IOWA CITY — Investigators on Wednesday arrested a long-haul trucker from Iowa who they say is linked by DNA evidence to the killings of three women whose bodies were dumped in Wyoming and...

At the turn of the 20th century, a vaccine had been developed for smallpox, a virus that killed millions in the 1800s. Those who survived the disease were often left badly scarred or blind.

The vaccine’s protection, though, lasted only five years and had to be renewed. And people forgot to do that, leading to occasional epidemics, including a serious outbreak on the Meskwaki Settlement near Tama.

In February 1901, three dozen Iowa communities reported smallpox outbreaks. When Des Moines reported 200 smallpox cases in late February, the mayor closed the schools and prohibited public gatherings. Still, no one died. That would not be the case at the Meskwaki Settlement.

On Oct. 22, 1901, an area resident told Dr. Benjamin Thompson of Tama, “I believe the Indians have the smallpox.”

Thompson went to the nearby Meskwaki Settlement, home to 309 people, to investigate.

He learned that an Indian from the Winnebago tribe had visited the settlement Sept. 23. He had become ill, but no doctor was called and he continued meeting with tribe members.

Two weeks later, the Meskwaki were becoming ill.

Thompson went back to the settlement with Dr. George Carpenter of Toledo. The first wickiup they visited had four active smallpox cases and three people recovering from the virus. Two Meskwaki who’d lived there had died.

The doctors found another five cases and were told of two other deaths.

The doctors that night reported to a joint meeting of the boards of health in Tama, Toledo and Montour.

William Malin, the Interior Department’s Indian agent for the settlement, insisted there was no problem. But after another medical visit to the settlement, the three cities quarantined themselves.

By Nov. 2, 70 Meskwaki had smallpox, and nine members of the tribe had died. Two weeks later, the totals had risen to 90 Meskwaki with smallpox and 35 deaths.

To complicate matters, the Meskwaki declined offers to go to the hospital, they refused smallpox vaccinations and they refused to stay on the settlement. If they became sick, they would hide from the doctors.

The Tama Herald reported, “It looks as though the disease must run its course through the tribe, carrying off the aged and the infirm and weakening the constitution of those who may survive.”

The Iowa governor appealed to Interior Secretary Ethan Hitchcock for authority to enforce the quarantine among the Meskwaki. Hitchcock complied, giving the state the authority to “take any necessary action.”

National Guard hospital tents and cots were shipped to the settlement. The Meskwaki were vaccinated, and they acquiesced to the quarantine.

The local towns began raising money for medicines and provisions not covered by the federal government.

Meanwhile, Congress appropriated $50,000 — more than $1 million in today’s dollars — to eradicate smallpox on the settlement. The bill gave the Interior Department the authority to quarantine the village and, if necessary, burn Meskwaki wickiups and clothing to halt the contagion.

The state of Iowa chipped in another $7,000 to burn the clothing, blankets, wickiups and other property belonging to the Tama tribe and to provide replacements for the Meskwaki.

The money paid for disinfectants — formaldehyde gas and corrosive sublimate fluid — to bathe tribal members and their dogs. Members of the tribe were forced to move to a camping ground while their village was decontaminated.

Malin, the Indian agent, reported to the Interior Department it took seven days to clean and renovate “the Indian camp.”

“During this process, a large number of the wickiups, where the disease had been prevalent, also large quantities of clothing, bedding and other infected property, were committed to the flames and new goods of similar character supplied,” he reported.

“Twenty-four new board houses, built of good lumber, and some 2,700 square yards of very heavy duck for tents, to those who preferred tents to houses, were given in lien of the wickiups destroyed. ... The Indians emerged from the trying ordeal through which they had passed and came out into the world again, after having been confined to the limited area of their camping grounds ... with a higher and better conception of the white man’s civilization.”

That assessment aside, the Meskwaki Settlement survived and grew. It now covers 8,000 acres and is home to 800 of the tribe’s 1,300 members.

It wasn’t until 1980 that smallpox was declared eradicated worldwide.

l Comments: d.fannonlangton@gmail.com

IOWA CITY — Investigators on Wednesday arrested a long-haul trucker from Iowa who they say is linked by DNA evidence to the killings of three women whose bodies were dumped in Wyoming and Tennessee in the early 1990s.

Police arrested Clark Perry Baldwin, 58, at his home in Waterloo, Iowa, on murder charges filed in Wyoming and Tennessee in the deaths of the women, including two who were pregnant. Investigators said they were looking into whether Baldwin could be responsible for other unsolved slayings.

Baldwin was arrested after investigators used semen and other material recovered from the victims to develop DNA profiles of their perpetrators, according to court documents in Wyoming. Last year, they learned that the same profile matched all three cases.

Investigators zeroed in on Baldwin after finding DNA in commercial genealogy databases of someone related to the suspect’s profile, court documents say. Last month in Waterloo, the FBI secretly collected DNA from Baldwin’s trash and a shopping cart he used at Walmart and it matched the profile.

In Wyoming, Baldwin is charged in the deaths of two unidentified women whose bodies were found in 1992 roughly 400 miles apart.

A female trucker discovered the nude body of the first victim in March 1992 near the Bitter Creek Truck turnout on Interstate 80 in southwestern Wyoming. An autopsy determined the woman suffered head trauma consistent with strangulation and her body had likely been in the snow for weeks.

A month later, Wyoming Department of Transportation workers found the partially mummified body of a pregnant woman in a ditch off of Interstate 90, near Sheridan in northern Wyoming.

An autopsy didn’t determine the cause of death but found the victim had an injury potentially consistent with suffering a blow to the head.

Investigators never identified the women and referred to them as “Bitter Creek Betty” and “I-90 Jane Doe.” Both were believed to be in their late teens or early 20s, Wyoming Division of Criminal Investigation Cmdr. Matt Waldock said.

In Tennessee, Baldwin is charged with two counts of murder in the 1991 killing of a 32-year-old pregnant woman from Virginia, Pamela McCall, and her fetus.

McCall was found in woods off Interstate 65 in Spring Hill, Tenn., in March 1991. An autopsy determined McCall had neck injuries and died of strangulation. Sperm was recovered from pantyhose worn by McCall, who was last seen at a Tennessee truck stop days earlier.

Court documents say that Baldwin allegedly raped a female hitchhiker in Wheeler County, Texas, at gunpoint in his truck in 1991. The 21-year-old woman told police that Baldwin struck her on the head, bound her hands and mouth and tried to choke her to death. He allegedly admitted to the assault but was released pending grand jury proceedings. Court documents do not indicate whether he was charged or prosecuted.

Baldwin, who has previously lived in Nashua, Iowa, and Springfield, Mo., was a cross-country truck driver for Marten Transport at the time.

Baldwin’s name also surfaced during a 1992 homicide investigation in Iowa. His ex-wife told police then that Baldwin once bragged about “killing a girl out west by strangulation and throwing her out of his truck,” court documents say.

Waldock said investigators were “hopeful” to solve other cases with Baldwin’s arrest.

One case of interest is the 1992 death of Tammy Jo Zywicki, 21, an Iowa college student who was last seen after her car broke down on an Illinois highway. A white man who was driving a semi-trailer was seen near her vehicle. Zywicki’s body was found in rural Missouri, stabbed to death.

Another is the 1992 killing of Rhonda Knutson, 22, a truck stop convenience store clerk in northern Iowa who was bludgeoned to death during an overnight shift. Investigators have released sketches of two men who were in the store, including one trucker. Baldwin lived in nearby Nashua then.

In 1997, Secret Service agents raided Baldwin’s apartment in Springfield, Mo., after learning he was making counterfeit U.S. currency on a personal computer. He and two female associates were indicted on counterfeiting charges. Baldwin was sentenced to 18 months in prison and released in 1999.

In 2008, a fire destroyed a Nashua building where Baldwin operated a candle business and damaged two adjacent buildings, including one that housed the town’s newspaper. The cause of the fire was never determined.

Baldwin is being held at the Black Hawk County jail pending extradition to Tennessee.

The charges stunned Jazz Baldwin, 32, of New Hampton, Iowa, who said she learned two years ago that Baldwin was her father after he purchased a DNA test kit. The two had been in contact over Facebook since then, she said.

“I heard rumors about his ‘possible crimes’ but always thought they were bogus,” she wrote in a Facebook message. “Murder was NOT on the list of things we thought he had done and gotten away with.”

Let $mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $mathrm{rex}(n, F)$, that are best possible up to a constant factor, when $F$ is one of $C_4$, $K_{2,t}$, $K_{3,3}$ or $K_{s,t}$ when $t>s!$.

It is well-known that a celebrated J"{o}rgens-Calabi-Pogorelov theorem for Monge-Amp`ere equations states that any classical (viscosity) convex solution of $det(D^2u)=1$ in $mathbb{R}^n$ must be a quadratic polynomial. Therefore, it is an interesting topic to study the existence and uniqueness theorem of such fully nonlinear partial differential equations' Dirichlet problems on exterior domains with suitable asymptotic conditions at infinity. As a continuation of the works of Caffarelli-Li for Monge-Amp`ere equation and of Bao-Li-Li for $k$-Hessian equations, this paper is devoted to the solvability of the exterior Dirichlet problem of Hessian quotient equations $sigma_k(lambda(D^2u))/sigma_l(lambda(D^2u))=1$ for any $1leq l<kleq n$ in all dimensions $ngeq 2$. By introducing the concept of generalized symmetric subsolutions and then using the Perron's method, we establish the existence theorem for viscosity solutions, with prescribed asymptotic behavior which is close to some quadratic polynomial at infinity.

We prove the following result: Let $(M,g_0)$ be a compact manifold of dimension $ngeq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a compact quotient manifold of $mathbb{S}^{n-1} imes mathbb{R}$ by diffeomorphisms, or a connected sum of a finite number of such manifolds. This extends a recent work of Brendle, and implies a conjecture of Schoen in dimensions $ngeq 12$. The proof uses Ricci flow with surgery on compact orbifolds with isolated singularities.

This paper addresses the problem of identifying the graph structure of a dynamical network using measured input/output data. This problem is known as topology identification and has received considerable attention in recent literature. Most existing literature focuses on topology identification for networks with node dynamics modeled by single integrators or single-input single-output (SISO) systems. The goal of the current paper is to identify the topology of a more general class of heterogeneous networks, in which the dynamics of the nodes are modeled by general (possibly distinct) linear systems. Our two main contributions are the following. First, we establish conditions for topological identifiability, i.e., conditions under which the network topology can be uniquely reconstructed from measured data. We also specialize our results to homogeneous networks of SISO systems and we will see that such networks have quite particular identifiability properties. Secondly, we develop a topology identification method that reconstructs the network topology from input/output data. The solution of a generalized Sylvester equation will play an important role in our identification scheme.

Let $chi_{Delta}(mathbb{R}^{n})$ denote the minimum number of colors needed to color $mathbb{R}^{n}$ so that there will not be a monochromatic equilateral triangle with side length $1$. Using the slice rank method, we reprove a result of Frankl and Rodl, and show that $chi_{Delta}left(mathbb{R}^{n} ight)$ grows exponentially with $n$. This technique substantially improves upon the best known quantitative lower bounds for $chi_{Delta}left(mathbb{R}^{n} ight)$, and we obtain [ chi_{Delta}left(mathbb{R}^{n} ight)>(1.01446+o(1))^{n}. ]

In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We successfully detect the product of two species masses as a feature to determine boundedness, gradient estimates, blow-up and $W^{j,infty}(1leq jleq 3)$-exponential convergence of classical solutions for the corresponding IBVP. More specifically, we first show generally a smallness on the product of both species masses, thus allowing one species mass to be suitably large, is sufficient to guarantee global boundedness, higher order gradient estimates and $W^{j,infty}$-convergence with rates of convergence to constant equilibria; and then, in a special case, we detect a straight line of masses on which blow-up occurs for large product of masses. Our findings provide new understandings about the underlying model, and thus, improve and extend greatly the existing knowledge relevant to this model.

In a multitype branching process, it is assumed that immigrants arrive according to a nonhomogeneous Poisson or a generalized Polya process (both processes are formulated as a nonhomogeneous birth process with an appropriate choice of transition intensities). We show that the renormalized numbers of objects of the various types alive at time $t$ for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, some transient moment analysis when there are only two types of particles is provided. AMS 2000 subject classifications: Primary 60J80, 60J85; secondary 60K10, 60K25, 90B15.

"Quasi-elliptic" functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated and given by Eisenstein series. A related structure has appeared recently in the computation of Feynman integrals. The two approaches are related by a sequence of polynomials with interlacing zeroes.

We study the regularity of exceptional actions of groups by $C^{1,alpha}$ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $alpha$. Let $G$ be a finitely generated group admitting a $C^{1,alpha}$ action $ ho$ with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if $G$ has spherical growth bounded by $c n^{d-1}$ and if the function $1/alpha^d$ is integrable near zero, then under some mild technical assumptions on $alpha$, there is a sequence of exceptional $C^{1,alpha}$ actions of $G$ which converge to $ ho$ in the $C^1$ topology. As a consequence for a single diffeomorphism, we obtain that if the function $1/alpha$ is integrable near zero, then there exists a $C^{1,alpha}$ exceptional diffeomorphism of the circle. This corollary accounts for all previously known moduli of continuity for derivatives of exceptional diffeomorphisms. We also obtain a partial converse to our main result. For finitely generated free abelian groups, the existence of an exceptional action, together with some natural hypotheses on the derivatives of group elements, puts integrability restrictions on the modulus $alpha$. These results are related to a long-standing question of D. McDuff concerning the length spectrum of exceptional $C^1$ diffeomorphisms of the circle.

In this paper, we provide some topological criteria for the Poisson Dixmier-Moeglin equivalence for $A$ in terms of the poset $({ m P. spec A}, subseteq)$ and the symplectic leaf or core stratification on its maximal spectrum. In particular, we prove that the Zariski topology of the Poisson prime spectrum and of each symplectic leaf or core can detect the Poisson Dixmier-Moeglin equivalence for any complex affine Poisson algebra. Moreover, we generalize the weaker version of the Poisson Dixmier-Moeglin equivalence for a complex affine Poisson algebra proved in [J. Bell, S. Launois, O.L. S'anchez, and B. Moosa, Poisson algebras via model theory and differential algebraic geometry, J. Eur. Math. Soc. (JEMS), 19(2017), no. 7, 2019-2049] to the general context of a commutative differential algebra.

The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine processes as solutions to infinite dimensional stochastic differential equations with values in Hilbert spaces. This requires a suitable version of the Yamada-Watanabe theorem, which we will provide in this paper. Several examples of infinite dimensional affine processes accompany our results.

We study an equivariant extension of the Batalin-Vilkovisky formalism for quantizing gauge theories. Namely, we introduce a general framework to encompass failures of the quantum master equation, and we apply it to the natural equivariant extension of AKSZ solutions of the classical master equation (CME). As examples of the construction, we recover the equivariant extension of supersymmetric Yang-Mills in 2d and of Donaldson-Witten theory.

In this paper we consider the "quasidensity" of a subset of the product of a Banach space and its dual, and give a connection between quasidense sets and sets of "type (NI)". We discuss "coincidence sets" of certain convex functions and prove two sum theorems for coincidence sets. We obtain new results on the Fitzpatrick extension of a closed quasidense monotone multifunction. The analysis in this paper is self-contained, and independent of previous work on "Banach SN spaces". This version differs from the previous version because it is shown that the (well known) equivalence of quasidensity and "type (NI)" for maximally monotone sets is not true without the monotonicity assumption and that the appendix has been moved to the end of Section 10, where it rightfully belongs.

This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime $mathbb{R}^{1+3}$. We find that there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime $mathbb{R}^{1+3}$, the geometry of them are two spheres. The linear mode unstable of those lightlike self-similar solutions for the radially symmetric membranes equation is given. After that, we show those self-similar solutions of the radially symmetric membranes equation are nonlinearly stable inside a strictly proper subset of the backward lightcone. This means that the dynamical behavior of those two spheres is as attractors. Meanwhile, we overcome the double roots case (the theorem of Poincar'{e} can't be used) in solving the difference equation by construction of a Newton's polygon when we carry out the analysis of spectrum for the linear operator.

We introduce an infinite-dimensional $p$-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.

We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node in the class of methods with optimal number of communication steps takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique we show that all proposed methods with stochastic oracle can be additionally parallelized at each node.

The central aim of this paper is to study (regional) fractional Poincar'e type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results are established depending on various conditions on domains and on the range of $s in (0,1)$. The best constant in both regional fractional and fractional Poincar'e inequality is characterized for strip like domains $(omega imes mathbb{R}^{n-1})$, and the results obtained in this direction are analogous to those of the local case. This settles one of the natural questions raised by K. Yeressian in [ extit{Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89, (2014), no 1-2}].

We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $psiin E^*$ satisfying $$|T(x)| leq K , |T| , |x|_{psi},$$ for all $xin E$. Applying this result we show that, given $G > 8 (1+2sqrt{3})$ and a bounded bilinear form $V$ on the Cartesian product of two JB$^*$-triples $E$ and $B$, there exist norm-one functionals $varphiin E^{*}$ and $psiin B^{*}$ satisfying $$|V(x,y)| leq G |V| , |x|_{varphi} , |y|_{psi}$$ for all $(x,y)in E imes B$. These results prove a conjecture pursued during almost twenty years.

The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel-Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel-Roiter measure $mu$ in an abelian length category $mathcal{A}$, there exists an object $X'$ which depends on $X$ and $mu$, such that $Gamma = operatorname{End}_{mathcal{A}}(X oplus X')$ has finite global dimension. Analogously to Iyama's original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.