Exomala (Neoblitopertha) campestris (Latreille, 1804) and Exomala (Neoblitopertha) succincta (Castelnau, 1840): two distinct European species (Coleoptera, Scarabaeidae, Rutelinae, Anomalini).

Es wird gezeigt, dass entsprechend dem Code Melolontha campestris Latreille, 1804, obwohl ein jüngeres primäres Homonym von Melolontha campestris Herbst, 1783, als gültiger Name für die Art Exomala (Neoblitopertha) campestris verwendet werden muss. Exomala (Neoblitopertha) succincta (Castelnau, 1840), welche bislang als Synonym von Exomala (Neoblitopertha) campestris Latreille, 1804 betrachtet wurde, ist eine valide Art. Diagnostische Merkmale, die die Separation von succincta und campestris ermöglichen, sowie ein Bestimmungschlüssel für die Arten der Untergattung Neoblitopertha Baraud, 1991, werden präsentiert.StichwörterTaxonomy, nomenclature, new synonymy, key to species, Coleoptera, Scarabaeidae, Anomalini, Exomala (Neoblitopertha), Palaearctic region.Nomenklatorische Handlungencampestris (Latreille, 1804) (Exomala (Neoblitopertha)), nom. protectum over Melontha campestris Herbst, 1783: nom. oblitumsuccincta (Castelnau, 1840) (Exomala (Neoblitopertha)), stat. rev. hitherto a synonym of Exomala (Neoblitopertha) campestris (Latreille, 1804)abbreviata Mulsant, 1842 (Phyllopertha campestris var.), syn. n. of Exomala (Neoblitopertha) succincta (Castelnau, 1840)cruciata Mulsant, 1842 (Phyllopertha campestris var.), syn. n. of Exomala (Neoblitopertha) succincta (Castelnau, 1840)maculata Mulsant, 1842 (Phyllopertha campestris var.), syn. n. of Exomala (Neoblitopertha) succincta (Castelnau, 1840)pauperata Mulsant, 1842 (Phyllopertha campestris var.), syn. n. of Exomala (Neoblitopertha) succincta (Castelnau, 1840)Evidence is presented showing that, according to the Code, Melolontha campestris Latreille, 1804, albeit a junior primary homonym of Melolontha campestris Herbst, 1783 must to be used as the valid name for the species currently known as Exomala (Neoblitopertha) campestris. Exomala (Neoblitopertha) succincta (Castelnau, 1840), currently considered synonym of Exomala (Neoblitopertha) campestris (Latreille, 1804), is rehabilitated as a good species. Diagnostic features enabling the separation of succincta from campestris are provided, as well as a key to the species of Neoblitopertha Baraud, 1991.KeywordsTaxonomy, nomenclature, new synonymy, key to species, Coleoptera, Scarabaeidae, Anomalini, Exomala (Neoblitopertha), Palaearctic region.Nomenclatural Actscampestris (Latreille, 1804) (Exomala (Neoblitopertha)), nom. protectum over Melontha campestris Herbst, 1783: nom. oblitumsuccincta (Castelnau, 1840) (Exomala (Neoblitopertha)), stat. rev. hitherto a synonym of Exomala (Neoblitopertha) campestris (Latreille, 1804)abbreviata Mulsant, 1842 (Phyllopertha campestris var.), syn. n. of Exomala (Neoblitopertha) succincta (Castelnau, 1840)cruciata Mulsant, 1842 (Phyllopertha campestris var.), syn. n. of Exomala (Neoblitopertha) succincta (Castelnau, 1840)maculata Mulsant, 1842 (Phyllopertha campestris var.), syn. n. of Exomala (Neoblitopertha) succincta (Castelnau, 1840)pauperata Mulsant, 1842 (Phyllopertha campestris var.), syn. n. of Exomala (Neoblitopertha) succincta (Castelnau, 1840

Kettenbruchentwicklung in beliebiger Dimension, Stabilität und Approximation

Wir behandeln Kettenbruchentwicklungen in beliebiger Dimension. Wir geben einen Kettenbruchalgorithmus an, der für beliebige Dimension n simultane diophantische Approximationen berechnet, die bis auf den Faktor 2 exp (n+2)/4 optimal sind. Für einen reellen Eingabevektor x := (x1,...,X n-1, 1) berechnet der Algorithmus eine Folge ganzzahliger Vektoren ....., so daß für i =1, ...., n-1 : | q exp (k) xi -pi exp (k)| 0 findet der Algorithmus entweder eine Relation zu x, das heißt einen ganzzahligen zu x orthogonalen Vektor (ungleich Null), mit euklidischer Länge kleiner oder gleich E exp -1, oder er schließt Relationen zu x mit euklidischer Länge kleiner als E exp -1 aus. Der Algorithmus führt in der Dimension n und |log E| polynomial viele arithmetische Operationen auf rellen Zahlen in exakter Arithmetik aus. Für rationale Eingaben x := (p1, ....., pn)/pn, E>0 mit p1,.....,pn Teil von Z besitzt der Algorithmus polynomiale Bitkomplexität in O........ Eine Variante dieses Algorithmus konstruiert für Eingabevektoren x einen (von x nicht notwendigerweise verschiedenen) Nahebeipunkt x' zu x und eine kurze Relation zu x'. Im Falle xx können wir die Existenz von Relationen kleiner als (2E)exp -1 für Punkte in einer kleinen offenen Umgebung um x' ausschließen. Wir erhalten in diesem Sinne eine stetige untere Schranke für die Länge der kürzesten Relation zu Punkten in dieser Umgebung. Die für x' berechnete Relation ist bis auf einen in der Dimension n exponentiellen Faktor kürzeste Relation für x'. Zur Implementierung des Kettenbruchalgorithmus stellen wir ein numerisch stabiles Verfahren vor und berichten über experimentelle Ergebnisse. Wir geben untere Schranken für die Approximierbarkeit kürzester Relationen in der Maximum-Norm und minimaler diophantischer Approximationen an: Unter der Annahme, daß die Klasse NP nicht in der deterministischen Zeitklasse O(n exp poly log n) enthalten ist, zeigen wir: Es existiert kein Algorithmus, der für rationale Eingabevektoren x polynomial in der Bitlänge bin(x) von x ist und die in der Maximum-Norm kürzeste Relation bis auf einen Faktor 2 exp (log 0.5 - zeta bin(x)) approximiert. Dabei ist zeta eine beliebig kleine positive Konstante. Wir übertragen dieses Resultat auf das Problem, zu gegebenen rationalen Zahlen x1,....,xn-1 und einem rationalen E > 0 gute simultane diophantische Approximationen zu finden, das heißt rationale Zahlen p1/q,...; (p n-1/)q mit möglichst kleinem Hauptnenner q zu konstruieren, so daß max 1 <=i <= n-1 |q xi - pi| <= E. Wir zeigen unter obiger Annahme, daß kein Algorithmus existiert, der für gegebene rationale Zahlen x1,........,x n-1 und natürlicher Zahl N polynomial-Zeit in der Bitlänge bin(x) von x ist und simultane diophantische Approximationen berechnet, so daß max 1 <=i <= n-1 |q xi - pi| für q gehört zu [1, N] bis auf den Faktor 2 exp (log 0.5 - zeta bin(x)) minimal ist. Hierbei ist zeta wieder eine beliebig kleine positive Konstante

Factoring via strong lattice reduction algorithm : technical report

We address to the problem to factor a large composite number by lattice reduction algorithms. Schnorr has shown that under a reasonable number theoretic assumptions this problem can be reduced to a simultaneous diophantine approximation problem. The latter in turn can be solved by finding sufficiently many l_1--short vectors in a suitably defined lattice. Using lattice basis reduction algorithms Schnorr and Euchner applied Schnorrs reduction technique to 40--bit long integers. Their implementation needed several hours to compute a 5% fraction of the solution, i.e., 6 out of 125 congruences which are necessary to factorize the composite. In this report we describe a more efficient implementation using stronger lattice basis reduction techniques incorporating ideas of Schnorr, Hoerner and Ritter. For 60--bit long integers our algorithm yields a complete factorization in less than 3 hours

Factoring via Strong Lattice Reduction Algorithms

We address to the problem to factor a large composite number by lattice reduction algorithms. Schnorr [Sc93] has shown that under a reasonable number theoretic assumptions this problem can be reduced to a simultaneous diophantine approximation problem. The latter in turn can be solved by finding sufficiently many ` 1 --short vectors in a suitably defined lattice. Using lattice basis reduction algorithms Schnorr and Euchner applied the reduction technique of [Sc93] to 40--bit long integers. Their implementation needed several hours to compute a 5% fraction of the solution, i.e., 6 out of 125 congruences which are necessary to factorize the composite. In this report we describe a more efficient implementation using stronger lattice basis reduction techniques incorporating ideas of [SH95] and [R97]. For 60--bit long integers our algorithm yields a complete factorization in less than 3 hours. 1 Introduction The security of many public key cryptosystems relies on the hardness of factoring ..

Computation of highly regular nearby points

We call a vector x/spl isin/R/sup n/ highly regular if it satisfies =0 for some short, non-zero integer vector m where is the inner product. We present an algorithm which given x/spl isin/R/sup n/ and /spl alpha//spl isin/N finds a highly regular nearby point x' and a short integer relation m for x'. The nearby point x' is 'good' in the sense that no short relation m~ of length less than /spl alpha//2 exists for points x~ within half the x'-distance from x. The integer relation m for x' is for random x up to an average factor 2/sup /spl alpha//2/ a shortest integer relation for x'. Our algorithm uses, for arbitrary real input x, at most O(n/sup 4/(n+log A)) many arithmetical operations on real numbers. If a is rational the algorithm operates on integers having at most O(n/sup 5/+n/sup 3/(log /spl alpha/)/sup 2/+log(/spl par/qx/spl par//sup 2/)) many bits where q is the common denominator for x

An Optimal, Stable Continued Fraction Algorithm for Arbitrary Dimension

. We analyse a continued fraction algorithm (abbreviated CFA) for arbitrary dimension n showing that it produces simultaneous diophantine approximations which are up to the factor 2 (n+2)=4 best possible. Given a real vector x =(x1 ; : : : ; xn\Gamma1 ; 1) 2R n this CFA generates a sequence of vectors (p (k) 1 ; : : : ; p (k) n\Gamma1 ; q (k) ) 2Z n ; k = 1; 2; : : : with increasing integers jq (k) j satisfying for i = 1; : : : ; n \Gamma 1 jx i \Gamma p i (k) =q (k) j 2 (n+2)=4 p 1 + x 2 i = jq (k) j 1+ 1 n\Gamma1 : By a theorem of Dirichlet this bound is best possible in that the exponent 1 + 1 n\Gamma1 can in general not be increased. 1 Introduction We analyse a CFA which computes for real vectors x 2 R n diophantine approximations to x that are up to the factor 2 (n+2)=4 best possible. Given x 2 R n this CFA constructs a sequence of lattice bases of the lattice Z n consisting of vectors that approximate the line xR: For given ffl ? 0 ; this C..

Approximating Good Simultaneous Diophantine Approximations is almost NP-hard

. Given a real vector ff=(ff1 ; : : : ; ff d ) and a real number &quot; ? 0 a good Diophantine approximation to ff is a number Q such that kQff mod Zk1 &quot;, where k \Delta k1 denotes the `1-norm kxk1 := max 1id jx i j for x = (x1 ; : : : ; xd ). Lagarias [12] proved the NP-completeness of the corresponding decision problem, i.e., given a vector ff 2 Q d , a rational number &quot; ? 0 and a number N 2 N+ , decide whether there exists a number Q with 1 Q N and kQff mod Zk1 &quot;. We prove that, unless NP ` DTIME(n poly(log n) ), there exists no polynomial -time algorithm which computes on inputs ff 2 Q d and N 2 N+ a number Q with 1 Q 2 log 0:5\Gammafl d N and kQ ff mod Zk1 2 log 0:5\Gammafl d min 1qN jjqff mod Zk1 ; where fl is an arbitrary small positive constant. To put it in other words, it is almost NP--hard to approximate a minimum good Diophantine approximation to ff in polynomial-time within a factor 2 log 0:5\Gammafl d for an arbitrary small positive const..

On the hardness of approximating shortest integer relations among rational numbers

Given x small epsilon, Greek Rn an integer relation for x is a non-trivial vector m small epsilon, Greek Zn with inner product = 0. In this paper we prove the following: Unless every NP language is recognizable in deterministic quasi-polynomial time, i.e., in time O(npoly(log n)), the &#8467;infinity-shortest integer relation for a given vector x small epsilon, Greek Qn cannot be approximated in polynomial time within a factor of 2log0.5 &#8722; small gamma, Greekn, where small gamma, Greek is an arbitrarily small positive constant. This result is quasi-complementary to positive results derived from lattice basis reduction. A variant of the well-known L3-algorithm approximates for a vector x small epsilon, Greek Qn the &#8467;2-shortest integer relation within a factor of 2n/2 in polynomial time. Our proof relies on recent advances in the theory of probabilistically checkable proofs, in particular on a reduction from 2-prover 1-round interactive proof-systems. The same inapproximability result is valid for finding the &#8467;infinity-shortest integer solution for a homogeneous linear system of equations over Q

A stable integer relation algorithm

We study the following problem: given x element Rn either find a short integer relation m element Zn, so that =0 holds for the inner product , or prove that no short integer relation exists for x. Hastad, Just Lagarias and Schnorr (1989) give a polynomial time algorithm for the problem. We present a stable variation of the HJLS--algorithm that preserves lower bounds on lambda(x) for infinitesimal changes of x. Given x \in {\RR}^n and \alpha \in \NN this algorithm finds a nearby point x' and a short integer relation m for x'. The nearby point x' is 'good' in the sense that no very short relation exists for points \bar{x} within half the x'--distance from x. On the other hand if x'=x then m is, up to a factor 2^{n/2}, a shortest integer relation for \mbox{x.} Our algorithm uses, for arbitrary real input x, at most \mbox{O(n^4(n+\log \alpha))} many arithmetical operations on real numbers. If x is rational the algorithm operates on integers having at most \mbox{O(n^5+n^3 (\log \alpha)^2 + \log (\|q x\|^2))} many bits where q is the common denominator for x