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gt.geometric topology – Thurston measure under finite covers

Let $S= S_{g,n}$ be a finite type orientable surface of genus $g$ and $n$ punctures and let $mathcal{ML}(S)$ denote the corresponding space of measured laminations. The Thurston measure, $mu^{Th},$ is a mapping class group invariant and locally finite Borel measure on $mathcal{ML}(S)$ which is obtained as a weak-$star$ limit of (appropriately weighted and rescaled) sums of Dirac measures supported on the set of integral multi-curves.

The Thurston measure arises in Mirzakhani’s curve counting framework. Concretely, given a hyperbolic metric $rho$ on $S_{g,n}$, let $B_{rho} subset mathcal{ML}(S)$ denote the set of measured laminations with $rho$-length at most $1$. Then $mu^{Th}(B_{rho})$ controls the top coefficient of the polynomial that counts multi-curves up to a certain $rho$-length and living in a given mapping class group orbit.

Question: Fix a hyperbolic metric $rho$ on $S$ and a finite (not necessarily regular) cover $p: Y rightarrow S$. Let $rho_{p}$ denote the hyperbolic metric on $Y$ obtained by pulling $rho$ back to $Y$ via $p$. Is there a straightforward relationship between $mu^{Th}(B_{rho_{p}})$ and $mu^{Th}(B_{rho})$? For example, is the ratio
$$ frac{mu^{Th}(B_{rho})}{mu^{Th}(B_{rho_{p}})} $$
uniformly bounded away from $0$ and $infty$? Does it equal a fixed value, independent of $rho$? If so, can it be easily related to the degree of the cover $Y rightarrow S$?

It seems hard to approach the above by thinking about counting curves on $Y$ versus $S$, because “most” simple closed curves on $Y$ project to non-simple curves on $S$. But, maybe the generalizations of curve counting for non-simple curves due to Mirzakhani (https://arxiv.org/pdf/1601.03342.pdf) or Erlandsson-Souto (https://arxiv.org/pdf/1904.05091.pdf) could be useful. Of course, both apply to counting curves in a fixed mapping class group orbit, so it’s not clear (to me) how to apply these results either since multi-curves on $Y$ can project to curves on $S$ with arbitrarily large self intersection.

Thanks for reading! I appreciate any ideas or reading suggestions.

usability – on-screen keyboard covers input field

Are there best practices for forms with HTML input elements when they must be used on a mobile device?

On my in-development web application, when running on an Android phone, the input field is often covered by the software keyboard when the field gets focus. The result of this is that the user must enter the text blindly, only seeing what was typed after the keyboard is dismissed.

My online searches have only turned up one treatment of the problem, and it seems rather complicated (I am wary of complicated solutions: they seem fragile and prone to breaking unpredictably).

I have a couple of ideas, one being to always shift the input field to the top of the view window when it receives focus, the second to open a subordinate dialog with a single text element under which the keyboard should always fit. However, these also trigger my complicated solution anxiety.

With internet-connected mobile phones as ubiquitous as they are, I suspect that this problem of the keyboard covering input field has a solution. I just can’t find it. Please help me. 🙁

Proving the following shrinking lemma for binary covers

Prove a topological space is normal if and only if for each pair of open sets $U,V$ such that $U cup V=X$, there exists open sets $U’$ and $V’$ such that $overline U’ subset U$ and $overline V’ subset V$ and $U’ cup V’=X$

Attempt:

$(Rightarrow)$ Since $U cup V=X implies (X-U) cap (X-V)=varnothing$. $X-U$ and $X-V$ are disjoint closed sets. Then by normality of $X$, there are disjoint open sets $W,Z$ with $X-U subset W$ and $X-V subset Z$, and an equivalent definition of normality states there are open sets $W’,Z’$ containing $X-U$ and $X-V$ with $overline W’ subset W$ and $overline Z’ subset Z$. Now I am unsure how to show that $W’ cup Z’=X$, which is what I need to show in the problem.

$(Leftarrow)$ Suppose the second part of the statement holds. Let $A$ be closed in $X$ and $U$ be any open set containing $X$. Then the existence of an open set $U’$ containing $A$ such that $overline U’ subset U$ implies $X$ is normal.

Comment:I am having difficulty understanding (why/if) $W’ cup Z’=X$ holds in the first direction. Also I am skeptical whether I started the second direction correctly by starting with a closed set, and an arbitrary open set containing this closed set. I also realize it may not be true that $U’$ contains the closed set $A$, which adds to my skepticism. Any ideas?

ag.algebraic geometry – admissible covers vs. stable maps to P^1

The paper Stable maps and tautological classes of Faber-Pandharipande shows that the Gromov-Witten classes on spaces of relative stable maps of the projective line push forward to tautological classes on moduli spaces of stable curves.

It is claimed on p. 6 that $overline{M}_{g,0}(mu_1,ldots,mu_m)=overline{H}_g(mu_1,ldots,mu_m)$, that is, the space of relative stable maps with ramification profiles $mu_i$ over marked points of $mathbb{P}^1$ is equal to the space of admissible covers of a genus 0 curve. However, I am confused as to how this can be correct: if the degree is 1, then I believe the space on the right is empty, but the space on the left has many components in general, coming from starting with the identity map on $mathbb{P}^1$ and attaching ghost components of positive genus to the source. More generally, I don’t see how the presence of ghost components allow one to identify a stable map with an admissible cover.

Later Proposition 1 it is stated as a corollary the fundamental class of $overline{H}_g$ pushes forward to a tautological class on $overline{M}_{g,n}$, but because of the discrepancy above I don’t see why this is immediate.

Am I missing something?

graph theory – Complexity of heaviest 2-optimal vertex-disjoint cycle covers

Calculating lightest vertex-disjoint cycle covers of finite complete symmetric graphs with weighted edges can be done efficiently and also renders the edge set of the calculated cycles free of pairs of crossing edges.

Calculating the unconstrained heaviest cycle cover will however generate edge sets with many pairs of crossing edges, leading to the

Question:
what is the complexity of calculating heaviest 2-optimal vertex-disjoint cycle covers, i.e. if constraints are added that make crossing edges mutually exclusive?

The motivation for the question is the observation that the minimal cycle cover may contain cycles that are far apart, thus making it hard to merge these cycle in an optimally to get an approximation of the shortest Hamilton cycle.
For heaviest 2-optimal cycle covers we may expect many pairs of long edges for which the minimum-weight matching of the $K_4$ that is induced by their adjacent vertices is small, thus easing the task of optimally merging the cycles to a short tour.

Galois covers of curves of arbitrary degree.

For smooth projective algebraic curves over finite fields. Do they admit a finite Galois cover of any degree, that is not induced by just base extension?

travel insurance covers between flight trip?

I have a travel insurance that would cover flight ie the departure (NYC) and the return (LA).

ie My departure date is DEC 15 and Return date is Jan 5

Trip details on the insurance document
Departure date  12/15/2020
Return date 01/05/2021
Trip deposit date   12/02/2020

If I make a round trip from LA Departure 12/23/2020 to Seattle Return LA 12/27/2020

Does a travel insurance cover between flight too ?

Many thanks!

These are an additional details –

Insured on Policy Trip

Coverages & Benefit Limits
Standard Packages
BAGGAGE COVERAGE
$750.00 Per Insured
BAGGAGE DELAY
$200.00 Per Insured
CONCIERGE SERVICES
Included    Per Insured
EMERGENCY EVACUATION AND REPATRIATION OF REMAINS
$150000.00  Per Insured
EMERGENCY TRAVEL ASSISTANCE
Included    Per Insured
NON-FLIGHT ACCIDENTAL DEATH & DISMEMBERMENT
$30000.00   Per Insured
PERSONAL SECURITY ASSISTANCE
Included    Per Insured
SINGLE OCCUPANCY
100% Trip Cost  Per Insured
TRAVEL MEDICAL ASSISTANCE
Included    Per Insured
TRAVEL MEDICAL EXPENSE
$15000.00   Per Insured
TRIP CANCELLATION
100% Trip Cost  Per Insured
TRIP DELAY
$500.00 Per Insured
$100.00 Payout Limit Per Day
TRIP INTERRUPTION
100% Trip Cost  Per Insured
TRIP INTERRUPTION-RETURN TRANSPORTATION ONLY
$500.00 Per Insured
WORLDWIDE TRAVEL ASSISTANCE
Included    Per Insured
15 Day Benefits
MISSED CONNECTION
$150.00 Per Insured
PRE-EXISTING CONDITION WAIVER
Included    Per Insured
TRIP CANCELLATION FOR DEFAULT
100% Trip Cost  Per Insured
TRIP INTERRUPTION FOR DEFAULT
100% Trip Cost  Per Insured

travel insurance covers between flight?

I have a travel insurance that would cover flight ie the departure (NYC) and the return (LA).

ie My departure date is DEC 15 and Return date is Jan 5

Trip details on the insurance document
Departure date  12/15/2020
Return date 01/05/2021
Trip deposit date   12/02/2020

If I make a round trip from LA Departure 12/23/2020 to Seattle Return LA 12/27/2020

Does the above insurance cover between flight too ?

Many thanks!

What are the scenes & monsters on the covers?

What are the scenes being portrayed on the covers of the Player’s Handbook (PHB), Dungeon Master’s Guide (DMG) and Monster Manual (MM) and what monsters do the scenes contain?

(Click for larger images.)
Player's Handbook Cover
Dungeon Master's Guide Cover
Monster Manual Cover