The document describes a presentation on an algorithm for exact synthesis of single qubit unitaries generated by Clifford and T gates. The algorithm reduces the problem of implementing a unitary to the problem of state preparation. It then uses a series of HT gates to iteratively decrease the smallest denominator exponent of the state entries until it reaches a base case that can be looked up. The algorithm runs in time linear in the initial smallest denominator exponent and provides an optimal sequence of H and T gates for implementing the input unitary exactly.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...Ali Ajouz
Jacobi forms of lattice index, whose theory can be viewed as extension of the theory of classical Jacobi forms, play an important role in various theories, like the theory of orthogonal modular forms or the theory of vertex operator
algebras. Every Jacobi form of lattice index has a theta expansion which implies, for index of odd rank, a connection to half integral weight modular forms and then via Shimura lifting to modular forms of integral weight, and implies a direct connection to modular forms of integral weight if the rank is
even. The aim of this thesis is to develop a Hecke theory for Jacobi forms of lattice index extending the Hecke theory for the classical Jacobi forms, and to study how the indicated relations to elliptic modular forms behave under Hecke operators. After defining Hecke operators as double coset operators,
we determine their action on the Fourier coefficients of Jacobi forms, and we determine the multiplicative relations satisfied by the Hecke operators, i.e. we study the structural constants of the algebra generated by the Hecke operators. As a consequence we show that the vector space of Jacobi forms
of lattice index has a basis consisting of simultaneous eigenforms for our Hecke operators, and we discover the precise relation between our Hecke algebras and the Hecke algebras for modular forms of integral weight. The
latter supports the expectation that there exist equivariant isomorphisms between spaces of Jacobi forms of lattice index and spaces of integral weight modular forms. We make this precise and prove the existence of such liftings
in certain cases. Moreover, we give further evidence for the existence of such liftings in general by studying numerical examples.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...Ali Ajouz
Jacobi forms of lattice index, whose theory can be viewed as extension of the theory of classical Jacobi forms, play an important role in various theories, like the theory of orthogonal modular forms or the theory of vertex operator
algebras. Every Jacobi form of lattice index has a theta expansion which implies, for index of odd rank, a connection to half integral weight modular forms and then via Shimura lifting to modular forms of integral weight, and implies a direct connection to modular forms of integral weight if the rank is
even. The aim of this thesis is to develop a Hecke theory for Jacobi forms of lattice index extending the Hecke theory for the classical Jacobi forms, and to study how the indicated relations to elliptic modular forms behave under Hecke operators. After defining Hecke operators as double coset operators,
we determine their action on the Fourier coefficients of Jacobi forms, and we determine the multiplicative relations satisfied by the Hecke operators, i.e. we study the structural constants of the algebra generated by the Hecke operators. As a consequence we show that the vector space of Jacobi forms
of lattice index has a basis consisting of simultaneous eigenforms for our Hecke operators, and we discover the precise relation between our Hecke algebras and the Hecke algebras for modular forms of integral weight. The
latter supports the expectation that there exist equivariant isomorphisms between spaces of Jacobi forms of lattice index and spaces of integral weight modular forms. We make this precise and prove the existence of such liftings
in certain cases. Moreover, we give further evidence for the existence of such liftings in general by studying numerical examples.
It is a new theory based on an algorithmic approach. Its only element
is called nokton. These rules are precise. The innities are completely
absent whatever the system studied. It is a theory with discrete space
and time. The theory is only at these beginnings.
Some Other Properties of Fuzzy Filters on Lattice Implication Algebrasijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
What is greenhouse gasses and how many gasses are there to affect the Earth.
Fast and efficient exact synthesis of single qubit unitaries generated by clifford and t gates
1. Fast and efficient
exact synthesis for single qubit unitaries
generated by Clifford and T gates
Yi-Hsin Ma
March 12, 2019
This is a presentation refers to the paper:
”Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates”
Vadym Kliuchnikov, Dmitri Maslov, Michele Mosca.
3. Introduction
Unitaries Synthesis Problems.
Single-Qubit Unitary Decomposition
Exact Decomposition
Approximate Decomposition
Solovay-Kitaev Algorithm
Multiple-Qubit Unitary Synthesize
What Solovay-Kitaev Algorithm Do?
Compile an quantum algorithm into an efficient fault-tolerance
form using Hadamard, T(π
8 ), and C-NOT gates.
Produces sequence of gate with length Ologc(1
), and requires
time Ologd (1
), c, d ∈ Z.
It does not guarantee
Finding an exact decomposition if there is one.
Whether the exact decomposition exists.
4. Introduction
What this Paper Want to Solve?
Show equivalence between the set of unitaries which are
computable by the circuit by Clifford and T gates and the set
of unitaries over Z[ 1√
2
, i].
Report an exact single-qubit circuit synthesis algorithm with H
and T optimality.
Note: T =
1 0
0 ω
, ω = e
2πi
8 .
5. Formulation
Theorem 2.1
The set of 2 × 2 unitaries over the ring Z[ 1√
2
, i] is equivalent to
the set of those unitaries implementable exactly as single-qubit
circuits constructed using H and T gates only.
Note that ”⊇” is straightforward, since both H and T is over the
ring Z[ 1√
2
, i]. The otherside is more difficult to proof.
Conjecture 2.2
For n > 1, the set of 2n × 2n unitaries over Z[ 1√
2
, i] is equivalent
to the set of unitaries implementable exactly as circuits with
Clifford and T gates built using (n+1) qubits, where the last qubit
is an ancillary qubit which is set to |0 at the beginning and is
required to be |0 at the end.
6. Reducing unitary implementation to state preparation
Any unitary U can be represented as
U =
z −w∗eiφ
w z∗eiφ ,
where det(U) = eiφ should belong to Z[ 1√
2
, i], which gives that
eiφ = ωk.
Hence we have the general form of the matrix we interested as
U =
z −w∗ωk
w z∗ωk
7. Reducing unitary implementation to state preparation
Lemma 3.1
Any single-qubit state with entries in the ring Z[ 1√
2
, i] can be
prepared using only H and T gates given the initial state |0 .
Using Lemma 3.1 we can reduce the problem of finding the
implementation of an unitary into state preparation. Suppose we
have a circuit that prepares state z
w ,(of course z, w ∈ Z[ 1√
2
, i]).
There exist k ∈ Z such that the unitaries is equal to
z −w∗ωk
w z∗ωk
8. Sequence for state preparation
Consider sequence of (HT)n|0 . It is an infinite sequence, since
in the Bloch sphere picture unitary HT corresponds to rotation
over an angle that is an irrational fraction of π.
Observation in Table 1.
Power of
√
2 in the denominator of the entries is the same.
Power of
√
2 in the denominator of |zn|2 increases by 1 after
multiplication by HT.
In general, multiplication by HTk cannot change the power of√
2 in the denominator by more than one.(Lemma 4.3)
10. Definitions of Sde and Gde
Notation: Z[ω] := {a + bω + cω2 + dω3, a, b, c, d ∈ Z}.
Divisibility to element of Z[ω]: x divides y if ∃x ∈ Z[ω] such that
xx = y.
Definition 4.1 (smallest denominator exponent)
The smallest denominator exponent, sde(z, x), of base x ∈ Z[ω]
with respect to z ∈ Z[ 1√
2
, i] is the smallest integer value k such
that zxk ∈ Z[ω]. If there is no such k, the smallest denominator
exponent is infinite.
11. Definitions of Sde and Gde
Notation: Z[ω] := {a + bω + cω2 + dω3, a, b, c, d ∈ Z}.
Divisibility to element of Z[ω]: x divides y if ∃x ∈ Z[ω] such that
xx = y.
Definition 4.1 (smallest denominator exponent)
The smallest denominator exponent, sde(z, x), of base x ∈ Z[ω]
with respect to z ∈ Z[ 1√
2
, i] is the smallest integer value k such
that zxk ∈ Z[ω]. If there is no such k, the smallest denominator
exponent is infinite.
Definition 4.2 (greatest dividing exponent)
The greatest dividing exponent, gde(z, x), of base x ∈ Z[ω] with
respect to z ∈ Z[ω] is the integer value k such that xk divides z
and x does not divide quotient z
xk . If no such k exists, the
greatest dividing exponent is said to be infinite.
12. Important Lemmas
Lemma 4.3
Let z
w be a state with entries in Z[ 1√
2
, i] and sde(|z|2) ≥ 4.
Then, for any integer k:
−1 ≤ sde
z + wωk
√
2
2
− sde(|z|2
) ≤ 1
13. Important Lemmas
Lemma 4.3
Let z
w be a state with entries in Z[ 1√
2
, i] and sde(|z|2) ≥ 4.
Then, for any integer k:
−1 ≤ sde
z + wωk
√
2
2
− sde(|z|2
) ≤ 1
Lemma 4.4
Let z
w be a state with entries in Z[ 1√
2
, i] and sde(|z|2) ≥ 4.
Then, for each number s ∈ {−1, 0, 1} there exits an integer
k ∈ {0, 1, 2, 3} such that:
sde
z + wωk
√
2
2
− sde(|z|2
) = s
14. Important Lemmas
Lemma 4.5
Let z
w be a state over Z[ 1√
2
, i] such that sde(|z|2) ≥ 1 or
sde(|w|2) ≥ 1,
then sde(|z|2) = sde(|w|2), sde(z) = sde(w).
Also, for x, y ∈ Z[ω], x = z(
√
2)sde(z) and y = w(
√
2)sde(w),
it holds that gde(|x|2) = gde(|y|2) ≤ 1.
Goal: Find the implementation of unitary matrix U using only H
and T gates.
First, we reduce the problem into state preparation.
U =
z −w∗ωk
w z∗ωk ⇒ U|0 =
z
w
15. Idea of Algorithm 1
Observe that HTk z
w = H z
wωk = 1√
2
z+wωk
z−wωk .
By Lemma 4.3, when we multiply the state with HTk, then
|sde(|
z + wωk
√
2
|2
) − sde(|z|2
)| ≤ 1
Moreover, Lemma 4.4 guarantees ∃k ∈ {0, 1, 2, 3} such that
sde(|
z + wωk
√
2
|2
) = sde(|z|2
) − 1
Hence, each time the state is multiplied by HTk , sde(|z|2) would
decrease by 1.
16. Idea of Algorithm 1
Lemma 4.5 implies that sde(|z−wωk
√
2
|2) = sde(|w|2) − 1 too.
Both entries of the state have same sde(| · |2) and they decrease by
1 at the same time if conditions of Lemma 4.5 are satisfied.
Iterating such procedure, at the end we can find
k1, ..., kn−3 ∈ {0, 1, 2, 3},
where n = sde(|z|2) such that
HTkn−3
...HTk1
z
w
=
zn−3
wn−3
, sde(|zn−3|2
) = 3
17. Idea of Algorithm 1
Define sde|·|2
( ˆU) = sde(|u|2) where u is an entry of U.
Since the set S3 := { ˆU ∈ U(2)|sde|·|2
( ˆU) = 3} is finite and small.
It takes little efforts to find ˆU ∈ S3 where its first column equals
zn−3
wn−3
.
Hence,
HTkn−3
...HTk1
z
w
= ˆU|0 ⇒ T−k1
H...T−kn−3
H ˆU|0 = U|0
U = T−k1
H...T−kn−3
H ˆU
18. Algorithm 1 Decomposition of a unitary matrix over Z[ 1√
2
, i]
Input: Unitary U =
z00 z01
z10 z11
over Z[ 1√
2
, i].
Output: Sequence Sout of H and T gates that implements U.
1: //S3 : set of all unitaries over Z[ 1√
2
, i] such that for all ˆU ∈ S3, sde|·|(ˆ(U))
2: queue Sout ← ∅
3: s ← sde(|z00|2)
4: while s > 3 do
5: state ← unfound
6: for all k ∈ {0, 1, 2, 3} do
7: while state = unfound do
8: z00 = [HT−k U]00
9: if sde(|z00|2) = s − 1 then
10: state ← found
11: add Tk H to Sout
12: s ← sde(|z00|2)
13: U ← HT−k U
14: end if
15: end while
16: end for
17: end while
18: lookup sequence Srem for U in S3
19: add Srem into Sout
20: return Sout
19. Some Useful Properties
Here, we introduce some properties which will be used to proof the
following theorems.
(I) sde( z
xk , x) = k − gde(z, x)
20. Some Useful Properties
Here, we introduce some properties which will be used to proof the
following theorems.
(I) sde( z
xk , x) = k − gde(z, x)
(II) gde(y + y , x) ≥ min{gde(y, x), gde(y , x)}
21. Some Useful Properties
Here, we introduce some properties which will be used to proof the
following theorems.
(I) sde( z
xk , x) = k − gde(z, x)
(II) gde(y + y , x) ≥ min{gde(y, x), gde(y , x)}
(III) gde(y, x) < gde(y , x) =⇒ gde(y + y , x) = gde(y, x)
22. Some Useful Properties
Here, we introduce some properties which will be used to proof the
following theorems.
(I) sde( z
xk , x) = k − gde(z, x)
(II) gde(y + y , x) ≥ min{gde(y, x), gde(y , x)}
(III) gde(y, x) < gde(y , x) =⇒ gde(y + y , x) = gde(y, x)
(IV) gde(yxk, x) = k + gde(y, x)
23. Some Useful Properties
Here, we introduce some properties which will be used to proof the
following theorems.
(I) sde( z
xk , x) = k − gde(z, x)
(II) gde(y + y , x) ≥ min{gde(y, x), gde(y , x)}
(III) gde(y, x) < gde(y , x) =⇒ gde(y + y , x) = gde(y, x)
(IV) gde(yxk, x) = k + gde(y, x)
(V) gde(x) = gde(|x|2, 2)
24. Some Useful Properties
Here, we introduce some properties which will be used to proof the
following theorems.
(I) sde( z
xk , x) = k − gde(z, x)
(II) gde(y + y , x) ≥ min{gde(y, x), gde(y , x)}
(III) gde(y, x) < gde(y , x) =⇒ gde(y + y , x) = gde(y, x)
(IV) gde(yxk, x) = k + gde(y, x)
(V) gde(x) = gde(|x|2, 2)
(VI) 0 ≤ gde(|x|2) − 2gde(x) ≤ 1
25. Some Useful Properties
Here, we introduce some properties which will be used to proof the
following theorems.
(I) sde( z
xk , x) = k − gde(z, x)
(II) gde(y + y , x) ≥ min{gde(y, x), gde(y , x)}
(III) gde(y, x) < gde(y , x) =⇒ gde(y + y , x) = gde(y, x)
(IV) gde(yxk, x) = k + gde(y, x)
(V) gde(x) = gde(|x|2, 2)
(VI) 0 ≤ gde(|x|2) − 2gde(x) ≤ 1
(VII) gde(Re(
√
2xy∗)) ≥ 1
2 gde |x|2 + gde |y|2
26. Some Useful Properties
Here, we introduce some properties which will be used to proof the
following theorems.
(I) sde( z
xk , x) = k − gde(z, x)
(II) gde(y + y , x) ≥ min{gde(y, x), gde(y , x)}
(III) gde(y, x) < gde(y , x) =⇒ gde(y + y , x) = gde(y, x)
(IV) gde(yxk, x) = k + gde(y, x)
(V) gde(x) = gde(|x|2, 2)
(VI) 0 ≤ gde(|x|2) − 2gde(x) ≤ 1
(VII) gde(Re(
√
2xy∗)) ≥ 1
2 gde |x|2 + gde |y|2
(VIII) gde(y, x) is invariant with respect to multiplication by ω and
complex conjugate of both x, y.
27. Lemma 4.5
We state Lemma 4.5 again: if z or w satisfies sde(| · |2) ≥ 1 ⇒
sde(|z|2) = sde(|w|2), sde(|z|) = sde(|w|),
gde(|x|2) = gde(|y|2) ≤ 1, x = z(
√
2)sde(z), y = w(
√
2)sde(w)
Proof.
WLOG. Let sde(z) ≥ sde(w), sde(|z|2) ≥ 1.
x = z(
√
2)sde(z), y = w(
√
2)sde(w).
sde(z) = sde( x
√
2
sde(z) ) = sde(z) − gde(x) ⇒ gde(x) = 0.
By property(VI), 0 ≤ gde(|x|2) ≤ 1.
1 = |z|2 + |w|2 =
√
2
−2sde(z)
|x|2 +
√
2
−2sde(w)
|y|2
⇒
√
2
2sde(z)
− |x|2 =
√
2
2(sde(z)−sde(w))
|y|2
29. Lemma 4.6
Lemma 4.6
If x, y ∈ Z[ω] such that |x|2 + |y|2 = (
√
2)m,then
gde(|x + y|2
) ≥ min m, 1 +
1
2
gde |x|2
+ gde |y|2
30. Lemma 4.6
Lemma 4.6
If x, y ∈ Z[ω] such that |x|2 + |y|2 = (
√
2)m,then
gde(|x + y|2
) ≥ min m, 1 +
1
2
gde |x|2
+ gde |y|2
Proof.
gde(|x + y|2) = gde(|x|2 + |y|2 +
√
2Re(
√
2xy∗))
= gde(
√
2
m
+
√
2Re(
√
2xy∗)) = min{m, gde(
√
2Re(
√
2xy∗))}
≥ min m, 1 + 1
2 gde |x|2 + gde |y|2
, where we use property(VII) at the end.
31. Lemma 4.3
We state Lemma 4.3 again: for sde(|z|2) ≥ 4, then k ∈ Z ⇒
−1 ≤ sde
z + wωk
√
2
2
− sde(|z|2
) ≤ 1
Proof.
Since sde(|z|2) ≥ 4, by Lemma 4.5 ⇒ sde(|z|2) = sde(|w|2),
sde(z) = sde(w), and gde(|x|2) = gde(|y|2) ≤ 1.
where x = ω−kz
√
2
m
, y = w
√
2
m
, m = sde(z) = sde(w).
sde z+wωk
√
2
2
− sde(|z|2) = sde |xωk +yωk |2
√
2
2m+2 − sde |xωk |2
√
2
2m
= (2m + 2 − gde(|xωk + yωk|2)) − (2m − gde(|xωk|2))
= 2 − gde(|x + y|2) + gde(|x|2).
32. Lemma 4.3
Proof (Cont.)
We change the inequality which we want to proof into:
1 ≤ gde(|x + y|2
) − gde(|x|2
) ≤ 3
Since |x|2 + |y|2 =
√
2
2m
, x, y ∈ Z[ω]. By Lemma 4.6 ⇒
gde(|x + y|2
) ≥ min 2m, 1 +
1
2
gde(|x|2
) + gde(|y|2
)
Note that gde(|x|2) = gde(|y|2) ≤ 1, and 2m ≥ 4 + gde(|x|2) ⇒
gde(|x + y|2
) ≥ 1 + gde(|x|2
)
33. Lemma 4.3
Proof (Cont.)
Now we have proven one side of the inequality. Next, we want to
show gde(|x + y|2) − gde(|x|2) ≤ 3.
Consider x + y, x − y ∈ Z[ω],
|x + y|2 + |x − y|2
= |x|2 + |y|2 +
√
2Re(
√
2xy∗) + |x|2 + |y|2 −
√
2Re(
√
2xy∗)
=
√
2
2m+2
.