arXiv subjects

Gagik Amirkhanyan

Publications and source records attributed to Gagik Amirkhanyan.

5 recordsLinked to original sources

arXiv2026

Rank-one convexity, polyconvexity, and extremality for three-dimensional elasticity tensors in various symmetry classes

It has been known that, for quadratic functions, quasiconvexity equals rank-one convexity but need not equal polyconvexity. While there is an explicit characterization of polyconvexity for quadratic energies, a similar characterization for quasiconvexity is not known in dimensions $n,N \geq 3.$ This paper is concerned with the search of extremal quasiconvex quadratic forms regarded as a linear elastic energy in dimensions $N=n=3$ in various symmetry classes of the elasticity tensor. In particular, we prove that for tensors with orthotropic symmetry, quasiconvexity implies polyconvexity, and thus the orthotropic class contains no nontrivial extremals. We show that this equivalence fails first at trigonal symmetry, where among some statemets, we provide a one-parameter family of non-polyconvex trigonal extreme rays. We also provide some general criteria for proving rank-one convexity, polyconvexity, and extremality, improving a result in [\ref{bib:Har.Hov.}].

math.AG↗

arXiv2025

Gemma 3 Technical Report

We introduce Gemma 3, a multimodal addition to the Gemma family of lightweight open models, ranging in scale from 1 to 27 billion parameters. This version introduces vision understanding abilities, a wider coverage of languages and longer context - at least 128K tokens. We also change the architecture of the model to reduce the KV-cache memory that tends to explode with long context. This is achieved by increasing the ratio of local to global attention layers, and keeping the span on local attention short. The Gemma 3 models are trained with distillation and achieve superior performance to Gemma 2 for both pre-trained and instruction finetuned versions. In particular, our novel post-training recipe significantly improves the math, chat, instruction-following and multilingual abilities, making Gemma3-4B-IT competitive with Gemma2-27B-IT and Gemma3-27B-IT comparable to Gemini-1.5-Pro across benchmarks. We release all our models to the community.

cs.CL↗

arXiv2016

Order-preserving Freiman isomorphisms

An order-preserving Freiman 2-isomorphism is a map $ϕ:X \rightarrow \mathbb{R}$ such that $ϕ(a) < ϕ(b)$ if and only if $a < b$ and $ϕ(a)+ϕ(b) = ϕ(c)+ϕ(d)$ if and only if $a+b=c+d$ for any $a,b,c,d \in X$. We show that for any $A \subseteq \mathbb{Z}$, if $|A+A| \le K|A|$, then there exists a subset $A' \subseteq A$ such that the following holds: $|A'| \gg_K |A|$ and there exists an order-preserving Freiman 2-isomorphism $ϕ: A' \rightarrow [-c|A|,c|A|] \cap \mathbb{Z}$ where $c$ depends only on $K$. Several applications are also presented.

math.CO↗

arXiv2013

Dichotomy Results for the L1 Norm of the Discrepancy Function

It is a well-known conjecture in the theory of irregularities of distribution that the L1 norm of the discrepancy function of an N-point set satisfies the same asymptotic lower bounds as its L^2 norm. In dimension d=2 this fact has been established by Halasz, while in higher dimensions the problem is wide open. In this note, we establish a series of dichotomy-type results which state that if the L^1 norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be large in some other function space.

math.NT↗

arXiv2013

Estimates of the Discrepancy Function in Exponential Orlicz Spaces

We prove that in all dimensions n at least 3, for every integer N there exists a distribution of points of cardinality $ N$, for which the associated discrepancy function D_N satisfies the estimate an estimate, of sharp growth rate in N, in the exponential Orlicz class exp)L^{2/(n+1)}. This has recently been proved by M.~Skriganov, using random digit shifts of binary digital nets, building upon the remarkable examples of W.L.~Chen and M.~Skriganov. Our approach, developed independently, complements that of Skriganov.

math.NT↗