There’s a certain joy in watching someone you hate
suffer. It was likely a German mind
that first cobbled together and joined
the words schaden and freude. I see

how it might have happened. Pointed helmet
on head, my great grandfather, Peter Kessel,
seated atop his rust stead in WWI
came crushing over a small patch

of openness through the forest
he guarded. Little yellow and pink Spring flowers
had commandeered the fields with their beauty.
His horse snorted and slowed. Ahead,

what had been hidden
in the small heights of grass
lay a man sprawled on his back.
Peter dismounted, pushed a polished boot

into the man, who jerked and moaned,
left hand, zigzagging like a baguette
conducting the breeze. He began to sing
“C’est la vie.” Grandfather kneeled,

ungloved his left hand, touching the man’s forehead.
Schade — he said,
too bad – it was schade. His own supplies
meager, he’d been told not to bring

enemy combatants back to camp.
The Frenchmen was rotten with fever.
Peter trotted on leaving him. Without much thought
he’d have shot a crippled horse. Behind him,

the man’s cries rose like larks into the meadow.

copyright © 2015 Kurt Lovelace All Rights Reserved

I’ve never really felt in control of my life.
So unlike Will Robinson, who walks alone
confident in the dark, navigating rocks

jutting up from soil like jagged giants.
Will moves without tripping, welcomes
vast rivers blocking his way, ahead

where his cleverness already owns solutions
waiting to unfold from his brain, his small frame
skedaddling before enormous painted draperies

of a B&W planet. Chalk light
shines on slate grass. Black insects
ooze over the black leaves. Undaunted, Will

squeezes his walkie-talkie, hailing
Alpha Control. As if we could call out
waiting light years for an answer

to find its vast way back
between the vacuum and the stars
for what is control but the hand’s reaching

out, manipulating, from the rotation
and twisting wheels of the shoulder
the things we love that sit right before us

when there’s no one there to tell us how?

Originally posted on What's new:

Many problems and results in analytic prime number theory can be formulated in the following general form: given a collection of (affine-)linear forms $latex {L_1(n),dots,L_k(n)}&fg=000000$, none of which is a multiple of any other, find a number $latex {n}&fg=000000$ such that a certain property $latex {P( L_1(n),dots,L_k(n) )}&fg=000000$ of the linear forms $latex {L_1(n),dots,L_k(n)}&fg=000000$ are true. For instance:

• For the twin prime conjecture, one can use the linear forms $latex {L_1(n) := n}&fg=000000$, $latex {L_2(n) := n+2}&fg=000000$, and the property $latex {P( L_1(n), L_2(n) )}&fg=000000$ in question is the assertion that $latex {L_1(n)}&fg=000000$ and $latex {L_2(n)}&fg=000000$ are both prime.
• For the even Goldbach conjecture, the claim is similar but one uses the linear forms $latex {L_1(n) := n}&fg=000000$, $latex {L_2(n) := N-n}&fg=000000$ for some even integer $latex {N}&fg=000000$.
• For Chen’s theorem, we use the same linear forms $latex {L_1(n),L_2(n)}&fg=000000$ as in the previous two cases, but now…

View original 2,531 more words

i

I can almost hear the waves sweep in, their soft susurrations,
tips of their lips breaking between my curled toes.

Noise is now everywhere I want to be
without it. Cars swoosh past Galveston beach roaring their inept monstrous lungs. I can barely

breathe. Or think. Why do trees and blades of every green thing shudder?
Because we are a hyper-intelligent insidious poison? Cats and dogs cling to us in shock and awe.

Ninety-five percent of a car’s energy goes towards moving simply itself not the passengers.
Or rather that’s 2,500 pounds of wastefulness before the crux of tissue steering the steel.

In Hermann Memorial Park a yellow-blue finch tries to sing and fails
in the roar cars shed in their wake on the I-10 adjoining the beige greenery.

I nod off under a canker tree. A whale whistles out of its water spout, breathing. I roll under
such plushness, floating with barnacles and sticky ambergris. So glued are our dream’s illogical logic.

I am a sticky carbuncle tearing through the earth’s thin breathability. It’s afternoon in Houston.
I shower again. I scrunch into a starched shirt. I rope my throat with a dead worm’s shiny excrement.

ii

My right ear is dead. When I was three
German measles like dappled freckles

grew in me, killing the nerve. My left ear
still good, at thirteen, I hear pretty well

the unprettiness in my parent’s voices as they divorce
and I listen in, in the mosquito bitten dark

roof above the living room window, then roll
on my back to swallow insignificance

in the drifting milky way above. Now
the frogs have started up. A few ducks quack. A splash

might be catfish come to nibble at the stars
tangled in cheap tabloid, suspended on the pond’s scum. My chest

makes a soft squelching sound like tossed gravel granite
settling into the decay layer at the ponds pitch black bottom.

iii

Some sounds have no feet, like running in a dream
with something chasing behind. Once, as a boy

in the Bahamas, in Freeport, in a wooded area
two older boys forced me to be

naked, and dance for them, my penis
slapping around like a snake in the beak

or eye of some predatory bird, I forget
which one it was that kept me, held

squirming, until I ran screaming and
pounding my way past the low palm trees. Power

is holding the thing that does not want you
to rape it into a display for you to play

with, you’d think. If you could think.
Those are pearls that were his eyes

nothing of him that doth fade but suffers.

– after the 2014 mid-term elections

Tuesday was wet. We couldn’t have bothered to vote.
Panhandlers huddle, unlike football players, anonymous

under a corporate freeway. Chubby constables
watch them from their warm cars. Is politics

what happens on FOX or CNN
in some distant country? We wait

in lines for a hot latte to warm our hands
at Starbucks, muttering before starting second jobs

assembling complaints of failures on a phone line.

Copyright © 2014 Kurt Lovelace, All Rights Reserved

It used to be beautiful but people got there
with ideas. I don’t know why a parking lot

should cover the green velvet moss that wrapped
the long slippery slate-stone path to the water

under thick green sun-spackled trees that was like walking
through golden pollen hovering inside the vest of a vast leprechaun

before opening out onto a beige pebbled beach
of bodies bobbing naked in the sunned shallows

or reclining like purposeful porpoises that Manet
or Seurat would gladly have painted, hips

and breasts, with their delicate French brushstrokes.
I decline the five dollar asking price

and drive on, back to Austin, talking to myself
feeling like Matthew McConaughey in a Mercedes commercial,

famous in my own mind, alone, and bewildered.

In Soonest Mended, through a humbling out of self by means of a sequence of self-referential questions and answers, through a process of bringing forth by rejecting and flipping answers and questions around, through a discourse that levels itself out by qualification as it proceeds, John Ashbery achieves a beautiful and stunning sublimation of self through the mere act of talking, thereby discovering, almost as if by accident, the nature of the poetic truth he had, apparently, been aiming for all along.

I posit that Ashbery hones a reductionist, almost mathematical, technique for approaching the truth. His speakers engage in a series of approximations to the “truth” using argumentative qualifications. It is precisely these qualifications that allow the speakers to wend their way to truth by discovering, recovering, and discarding the many “truths” — these weaker version of themselves — that argumentatively lie along the path of such self-referential discourse. This is a key technique used within almost all of Ashbery’s poetry: talk that uncovers truth by qualifying itself at every turn.

Poetry cannot be translated. So too, other than for giving a reader the raw idea of what is being said, any literal translation is especially negligible. The translator must be someone capable of writing decent if not inspired English poetry, whatever that might mean. Therefore, if one does attempt the translation of a poem, it must be ruthless, a culling from the bloody heart of both tongues. In other words, one may only create a poem in the target language that is its own poem with echoes, distortions, and intentions pointing to those places that the original also points out. So, that being said, here are three of my favorite poems from the Spanish, German, and French by Neruda, Rilke and Prevert, respectively.

These translations are very much, in an effort of love and intellect, an attempt to convey the beauty, wordplay and sound-play in the originals — and this explains some of my  perhaps more daring  “choices” — first person tense, ellipses, metaphor shifts, occasional sound emphasis over word sense — choices I prefer to call “intelligently risky”  as they are collectively my attempts to “transmogrify” these lovely poems into some semblance of a worthy English simulacrum.  Caveat emptor!

Árbol
Anoche al apagar la luz
se me durmieron las raíces
y se me quedaron los ojos
enredados entre las hojas
hasta que, tarde, con la sombra
se me cayó una rama al sueño
y por el tronco me subió
la fría noche de cristal
como una iguana transparente.
Entonces me quedé dormido.
Cerré los ojos y las hojas.
Pablo Neruda

Tree

Last night, putting the light
out, my deep roots slept
but my eyes strayed, open
tangled in between leaves
until, later, the shadow
of a branch fell over my dream
and I rose up into the trunk
of the cold night, a crystal
transparent iguana.

I slept soundly then.

I closed my eyes and my leaves.

translated by Kurt Lovelace,
Copyright 2014 All Rights Reserved

Herbsttag

Herr: Es ist Zeit. Der Sommer war sehr groß.
Leg deinen Schatten auf die Sonnenuhren,
und auf den Fluren laß die Winde los.

Befiehl den letzten Früchten voll zu sein;
gieb ihnen noch zwei südlichere Tage,
dränge sie zur Vollendung hin und jage
die letzte Süße in den schweren Wein.

Wer jetzt kein Haus hat, baut sich keines mehr.
Wer jetzt allein ist, wird es lange bleiben,
wird wachen, lesen, lange Briefe schreiben
und wird in den Aleen hin und her
unruhig wandern, wenn die Blätter treiben

Rainer Maria Rilke

Fall Day

God! Is it time? Summer was so thick:
dragging its slow shadows over sundials,
and in the meadows its winds still rip loose!

Hear us! Let our last fruits fatten into fullness;
give us two more sun-drenched days
plumping all into ripeness till hither and thither
the last sweet drops drain into swarthy wine.

Whoever has no house, you’ll not build it now.
If you’re alone, it‘ll stay that way, a long time
you will stay awake, reading, writing long letters
as you walk alone shuffling, here and there
disturbed, wandering where leaves tremble.
translated by Kurt Lovelace,
Copyright 2014 All Rights Reserved

Déjeuner du matin

Il a mis le café
Dans la tasse
Il a mis le lait
Dans la tasse de café
Il a mis le sucre
Dans le café au lait
Avec la petite cuiller
Il a tourné
Il a bu le café au lait
Et il a reposé la tasse
Sans me parler
Il a allumé
Une cigarette
Il a fait des ronds
Avec la fumée
Il a mis les cendres
Dans le cendrier
Sans me parler
Sans me regarder
Il s’est levé
Il a mis
Son chapeau sur sa tête
Il a mis
Son manteau de pluie
Parce qu’il pleuvait
Et il est parti
Sous la pluie
Sans une parole
Sans me regarder
Et moi j’ai pris
Ma tête dans ma main
Et j’ai pleuré.

Jacques Prevert

Breakfast at the Dinner

He pours coffee
into the cup.
He pours milk
into the cup of coffee.
He sprinkles sugar
atop the café au lait,
and with a little spoon
stirs it round.
He finishes his café au lait.
He reposes cup in saucer.
Not one word spoken,
he lights up
a cigarette.
He blows round
smoke rings.
He taps ash
into the ashtray.
Never speaking to me,
never regarding me,
he readies to leave, places
hat on head,
throws on his coat
because rain is splashing down
pouring into puddles
as he leaves me,
turning into the splashing
rain, never speaking
nor regarding me once.
And I, eyes
splashing into the ground
of my hands,
cry.

translation by Kurt Lovelace
copyright 2014 All Rights Reserved

Note: this article is intended for undergraduate math majors, preferably seniors, but freshman will benefit from the look ahead, at what is expected to be known by them when they are seniors, as well as entering graduate students.

I am readying myself for the GRE math subject exam in the late Spring, and therefore reviewing all four years of undergraduate mathematics at this time.

In what follows, I will be summarizing every major area of undergraduate mathematics, as follows:

1. Geometry: Plane, Elliptic, Hyperbolic, and Affine
2. Linear Algebra
3. Vector Analysis
4. Real Analysis
5. Complex Analysis
6. Topology
7. Ordinary Differential Equations
8. Fourier Analysis, Lebesgue Integration and General Transform Theory
9. Probability Theory
10. Abstract Algebra
11. Graph Theory
12. Combinatorics and Algorithmic Complexity
13. Set Theory and Transfinite Arithmetic
14. Basic & Analytic Number Theory
15. Partial Differential Equations
16. Differential Geometry

If you are a student of any subject, then I hope you have already asked yourself at times the question “What does all of this mean? What is it for?” if you are as I am, a mathematics major, then I believe that this question is particularly important. For one can too often get lost in the forest, standing amid the numberless trees, each one a bit different from its neighbors yet all oddly familiar, somehow similar, like wondering through a waking dream.

This is neither an idle nor a superfluous question. Call it “the big picture.” Call it what you will. It is important to know the gist and the connections between different areas of ones subject, and to know each area for what it really is about.

———————————————————————————————————————————-
Linear Algebra

Let’s look at linear algebra. The matrix is the lingua franca tool of linear algebra, and so linear algebra is the study of vector spaces and their transformations using matrices. After a first course, one should know the axioms that define a vector space, the algebra of matrices, how to express the structure of vector spaces using matrices, and especially how, given the basis of a vector space, to represent any transformation of that vector space using a matrix, and how to use and calculate eigenvalues and eigenvectors. Lastly, the key theorem of linear algebra is to know the 8 distinct conditions that each alone can guarantee a matrix is inevitable. And that is it, the heart of an entire semester of undergraduate linear algebra summarized in one paragraph. Again, in a nutshell: basic linear algebra is the study of the transformation of vector spaces into one another and understanding equivalent conditions under which a matrix is invertible.

What is linear algebra used for? Everything, and almost everywhere, is the short answer. Anytime you seek a first approximation to some problem, it is likely that you will be using linear algebra. In graph theory matrices are used to represent a graph’s structure in a precise and concise manner. In abstract algebra, matrices are used in the representation of groups and other algebraic structures. And linear algebra comes up in both ordinary and partial differential equations, differential geometry, knot theory, number theory, and almost everywhere else. Learn it well, know it well if you are going to be doing research in pure or applied mathematics.

If one takes a fourth year, second semester of linear algebra, often considered “part 2″ of linear algebra studies, then one will likely encounter — inner product spaces, direct sums of subspaces, primary decomposition, reduction to triangular and Jordan forms, both rational and classical forms, dual spaces, orthogonal direct sums, bilinear and quadratic forms, and real normality — among the main topics.

Here are some great free full textbooks to download for reference or study:

Linear Algebra by Jim Hefferon. 499 pages. [direct link to whole book.]

Linear Algebra. David Cherney, Tom Denton & Andrew Waldron. 410 pages. [direct link to whole book]

-=KuRt=- © 2013 Kurt Lovelace – All Rights Reserved

## Euler’s Characteristic Formula V-E+F = 2

How is it that for thousands of years the best minds in mathematics did not see the fundamental relationship that, in any regular polyhedron, the sum of the vertices and faces minus the edges equals two? Although this question is interesting, no attempt will be made to answer it. Yet the question, merely by being asked, serves to highlight the tremendous stature of Leonard Euler.

Euler worked from first principles, digging into a topic by performing calculations to get a feel for the shape and edges of a problem, then developed conjectures and proofs based on such research. And so, Euler must have tabulated the edges, faces and vertices of shapes such as the platonic solids to thereby notice this relationship. Perhaps it took an almost childlike playfulness to discover this relationship. Yet it is all speculation. The only fact is that Euler did discover that $V-E+F=2$ where V, E, and F are, respectively, the vertices, edges and faces of a regular polyhedron

I first learned of Euler’s formula in a senior course on graph theory taught by the Polish graph theorist Dr. Siemion Fajtlowicz. Therefore, let me provide a few definitions before offering a compact proof that $V-E+F=2$ using basic graph theoretical methods.

Definition. A graph consists of a non-empty set of vertices and a set of edges, possibly empty. If an edge E exists, then it will be related to an unordered pair (a,b) of vertices in V. We may write $G=(\{V\},\{E\})$ to denote the graph.

A graph is finite if both the vertex set and edge set are finite. A graph that is not finite is infinite. The size of a graph is the number of vertices. The order of a graph is the number of edges. All graphs discussed in this article are finite.

Immediately note that if we are given one edge, then there exist two vertices — because an edge must connect to something and that something will always be a vertex. Two vertices are incident if they share a common edge and said to be adjacent. An edge with identical ends is called a loop while an edge with distinct ends is called a link. A graph is simple if it has no loops.

The degree $d(v)$ of a vertex in a graph G is equal to the number of edges in G incident with v where each loop counts as two edges. Thus, a vertex with two edges has degree 2 because the degree of the other two vertices would be one each, and so the sum of the degrees of all the vertices would be 4. From this we have immediately the following two theorems.

Theorem 1. The sum of the degree of the vertices of a graph equals twice the number of edges. That is

$\displaystyle \sum\limits_{i/G}{d({{v}_{i}})}=2e$

proof. If a graph has no edges, then e=0 and the theorem is true because the sum of the degree of the vertices of a graph with no edges must, by definition, be zero. If a given vertex has an edge, then there must be a second vertex connected to the first vertex by the given edge and so the sum of the degree of the vertexes will be 2, even if the edge were a loop beginning and returning to a single vertex the degree of that vertex due to the loop would still be 2.

Now, each additional edge added to the graph will increase either the degree of the vertex it is connect to and add one additional vertex or else it will add one degree each to two vertices that already exist. Thus, each time an edge is added, the degree count of the vertices increases by 2. Therefore, the sum of the degree of all the vertices will always be simply twice the number of edges present.

Theorem 1.1 The number of vertices in a graph of odd degree is even.

proof. Let ${{v}_{1}},{{v}_{2}}$ be sets of vertices where the ${{v}_{1}}$ is of odd degree and ${{v}_{2}}$ is of even degree. Then, from Theorem 1, we must have that

$\sum\limits_{i/G}{d({{v}_{1}})}+\sum\limits_{i/G}{d({{v}_{2}})}=\sum\limits_{i/G}{d(v)}$

but we have already established that

$\sum\limits_{i/G}{d({{v}_{i}})}=2e$

and therefore we must have

$\sum\limits_{i/G}{d({{v}_{1}})}+\sum\limits_{i/G}{d({{v}_{2}})}=2e$

$\sum\limits_{i/G}{d({{v}_{1}})}=2e-\sum\limits_{i/G}{d({{v}_{2}})}$

but

$\sum\limits_{i/G}{d({{v}_{2}})}=2{{e}_{2}}$

so

$\sum\limits_{i/G}{d({{v}_{1}})}=2e-2{{e}_{2}}=2(e-{{e}_{2}})=2{{e}_{1}}$

is even. Thus, we have that the sum of the vertices in a graph of odd degree is even. Now, we will need the following definitions to proceed.

Definition. A cycle is a closed chain of edges. A connected graph that contains no cycles is a tree.

Definition. A spanning tree of a graph G is one that uses every vertex of G but not all of the edges of G.

Every connected graph G contains a spanning tree T as a subgraph of G.

Definition. A planer graph is one that can be drawn in the plane without crossing any edges.

Definition. A face in the plane consists of both unbounded and bounded regions.

The last definition allows us to say that the number of “faces” of a finite line segment in the plane is 1, but that the number of faces of an infinite line is 2, one for each side that the line divides the plane into, and that the number of faces of a circle in the plane is 2, one corresponding to the inside and the other to the outside of the circle one being the unbounded and the other the bounded region.

Now we are ready to prove Euler’s formula as it may be stated in graph theory.

Theorem 2. If G is a connected planer graph with vertices v, edges e, and faces f, then

$\displaystyle v-e+f=2$

proof. Let T be a spanning tree of a graph G. Then the number of faces f=1 and the number of edges e= v – 1 is true for the spanning tree T of G and so we have $v-(v+1)+(1)=2$ and our formula is true. Now, G may be constructed from T by adding edges to T, but each time we do so we are adding a new face to T. Therefore, with the addition of each new edge, e will increase by 1 and f will increase by 1. Therefore, all of these additions cancel out and we have our formula.

Now, a homomorphism exists between polyhedron and graphs in that a connected plane graph can be uniquely associated with a polyhedron by making any face the flat unbounded part of the plane. Therefore, Euler’s formula is true for polyhedron. We can therefore immediately prove:

Theorem 3. Exactly five platonic solids exist.

proof. Let the number of edges and faces be n and the number of edges each vertex is incident to be d. If we multiply nf, we get twice as many edges because each edge belongs to two faces, so nf is the number of faces multiplied by the number of edges on each face. Likewise, with dv each edge is incident to two faces so that we have $dv=2e$ so that we have the equality

$nf=2e=dv$

or

$e=\frac{dv}{2}$ and $f=\frac{dv}{n}$

Now, placing these into Euler’s formula, we get

$v+\frac{dv}{n}-\frac{dv}{2}=2$

or

$(2n+2d-dn)v=4n$

But both v and n are positive integers and for $(2n+2d-dn)v=4n$ to be true, we must have that

$2n+2d-dn>0$

or

$(n-2)(d-2)<4$

and so there are just five possibilities for the values of n and d and each of these corresponds to one of the five platonic solids, so that we have

$n=3,d=3$ is a tetrahedron
$n=3,d=4$ is a hexahedron also known as a cube
$n=4,d=3$ is a octahedron
$n=3,d=5$ is a dodecahedron
$n=5,d=3$ is a icosahedron

QED.

© 2012 Kurt Lovelace – All Rights Reserved